Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

### The abstract of the Mizar article:

### The Theorem of Weierstrass

**by****Jozef Bialas, and****Yatsuka Nakamura**- Received July 10, 1995
- MML identifier: WEIERSTR

- [ Mizar article, MML identifier index ]

environ vocabulary METRIC_1, PRE_TOPC, BOOLE, ABSVALUE, ARYTM_1, SETFAM_1, FINSEQ_1, FINSET_1, TOPMETR, RELAT_1, TARSKI, FUNCT_1, SUBSET_1, ORDINAL2, COMPTS_1, EUCLID, LATTICES, PCOMPS_1, SEQ_2, SEQ_1, SEQM_3, SEQ_4, WEIERSTR, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, TOPMETR, METRIC_1, COMPTS_1, SEQ_1, SEQ_2, SEQM_3, SEQ_4, ABSVALUE, FINSEQ_1, FINSET_1, STRUCT_0, PRE_TOPC, TOPS_2, RCOMP_1, EUCLID, PCOMPS_1; constructors REAL_1, NAT_1, TOPMETR, COMPTS_1, SEQ_2, SEQM_3, SEQ_4, ABSVALUE, TOPS_2, RCOMP_1, EUCLID; clusters STRUCT_0, TOPMETR, PRE_TOPC, PCOMPS_1, EUCLID, XREAL_0, METRIC_1, RELSET_1, ARYTM_3, FINSET_1, MEMBERED, NUMBERS, ORDINAL2; requirements REAL, SUBSET, BOOLE, NUMERALS, ARITHM; begin theorem :: WEIERSTR:1 for M being MetrSpace, x1,x2 being Point of M, r1,r2 being Real ex x being Point of M, r being Real st Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r); theorem :: WEIERSTR:2 for M being MetrSpace, n being Nat, F being Subset-Family of M, p being FinSequence st (F is finite & F is_ball-family & rng p = F & dom p = Seg(n+1)) holds ex G being Subset-Family of M st (G is finite & G is_ball-family & ex q being FinSequence st (rng q = G & dom q = Seg(n)) & ex x being Point of M st ex r being Real st (union F c= union G \/ Ball(x,r))); theorem :: WEIERSTR:3 for M being MetrSpace, F being Subset-Family of M st (F is finite & F is_ball-family) holds ex x being Point of M, r being Real st union F c= Ball(x,r); definition let T,S be non empty TopSpace; let f be map of T,S; let G be Subset-Family of S; func f"G -> Subset-Family of T means :: WEIERSTR:def 1 for A being Subset of T holds A in it iff ex B being Subset of S st (B in G & A = f"B); end; theorem :: WEIERSTR:4 for T,S being non empty TopSpace, f being map of T,S, A,B being Subset-Family of S st A c= B holds f"A c= f"B; theorem :: WEIERSTR:5 for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of S st f is continuous & B is open holds f"B is open; definition let T,S be non empty TopSpace; let f be map of T,S; let G be Subset-Family of T; func f.:G -> Subset-Family of S means :: WEIERSTR:def 2 for A being Subset of S holds A in it iff ex B being Subset of T st (B in G & A = f.:B); end; theorem :: WEIERSTR:6 for T,S being non empty TopSpace, f being map of T,S, A,B being Subset-Family of T holds A c= B implies f.:A c= f.:B; theorem :: WEIERSTR:7 for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of S, P being Subset of S st f.:(f"B) is_a_cover_of P holds B is_a_cover_of P; theorem :: WEIERSTR:8 for T,S being non empty TopSpace, f being map of T,S, B being Subset-Family of T, P being Subset of T st B is_a_cover_of P holds f"(f.:B) is_a_cover_of P; theorem :: WEIERSTR:9 for T,S being non empty TopSpace, f being map of T,S, Q being Subset-Family of S holds union(f.:(f"(Q))) c= union Q; theorem :: WEIERSTR:10 for T,S being non empty TopSpace, f being map of T,S, P being Subset-Family of T holds union P c= union(f"(f.:P)); theorem :: WEIERSTR:11 for T,S being non empty TopSpace holds for f being map of T,S holds for Q being Subset-Family of S holds Q is finite implies f"Q is finite; theorem :: WEIERSTR:12 for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset-Family of T holds P is finite implies f.:P is finite; theorem :: WEIERSTR:13 for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset of T holds for F being Subset-Family of S holds (ex B being Subset-Family of T st (B c= f"F & B is_a_cover_of P & B is finite)) implies (ex G being Subset-Family of S st (G c= F & G is_a_cover_of f.:P & G is finite)); begin :: :: The Weierstrass` theorem :: theorem :: WEIERSTR:14 for T,S being non empty TopSpace holds for f being map of T,S holds for P being Subset of T holds (P is compact & f is continuous) implies f.:P is compact; theorem :: WEIERSTR:15 for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds P is compact & f is continuous implies f.:P is compact; theorem :: WEIERSTR:16 for f being map of TOP-REAL 2,R^1 holds for P being Subset of TOP-REAL 2 holds P is compact & f is continuous implies f.:P is compact; definition let P be Subset of R^1; func [#](P) -> Subset of REAL equals :: WEIERSTR:def 3 P; end; theorem :: WEIERSTR:17 for P being Subset of R^1 holds P is compact implies [#](P) is bounded; theorem :: WEIERSTR:18 for P being Subset of R^1 holds P is compact implies [#](P) is closed; theorem :: WEIERSTR:19 for P being Subset of R^1 holds P is compact implies [#](P) is compact; definition let P be Subset of R^1; func upper_bound(P) -> Real equals :: WEIERSTR:def 4 upper_bound([#](P)); func lower_bound(P) -> Real equals :: WEIERSTR:def 5 lower_bound([#](P)); end; theorem :: WEIERSTR:20 for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds (P <> {} & P is compact & f is continuous) implies ex x1 being Point of T st (x1 in P & f.x1 = upper_bound(f.:P)); theorem :: WEIERSTR:21 for T being non empty TopSpace holds for f being map of T,R^1 holds for P being Subset of T holds (P <> {} & P is compact & f is continuous) implies ex x2 being Point of T st (x2 in P & f.x2 = lower_bound(f.:P)); begin :: :: The measure of the distance between compact sets :: definition let M be non empty MetrSpace; let x be Point of M; func dist(x) -> map of TopSpaceMetr(M),R^1 means :: WEIERSTR:def 6 for y being Point of M holds it.y = dist(y,x); end; theorem :: WEIERSTR:22 for M being non empty MetrSpace holds for x being Point of M holds dist(x) is continuous; theorem :: WEIERSTR:23 for M being non empty MetrSpace holds for x being Point of M holds for P being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in P & (dist(x)).x1 = upper_bound((dist(x)).:P)); theorem :: WEIERSTR:24 for M being non empty MetrSpace holds for x being Point of M holds for P being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in P & (dist(x)).x2 = lower_bound((dist(x)).:P)); definition let M be non empty MetrSpace; let P be Subset of TopSpaceMetr(M); func dist_max(P) -> map of TopSpaceMetr(M),R^1 means :: WEIERSTR:def 7 for x being Point of M holds it.x = upper_bound((dist(x)).:P); func dist_min(P) -> map of TopSpaceMetr(M),R^1 means :: WEIERSTR:def 8 for x being Point of M holds it.x = lower_bound((dist(x)).:P); end; theorem :: WEIERSTR:25 for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st P is compact for p1,p2 being Point of M holds p1 in P implies dist(p1,p2) <= upper_bound((dist(p2)).:P) & lower_bound((dist(p2)).:P) <= dist(p1,p2); theorem :: WEIERSTR:26 for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds abs(upper_bound((dist(p1)).:P) - upper_bound((dist(p2)).:P)) <= dist(p1,p2); theorem :: WEIERSTR:27 for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds for x1,x2 being Real holds x1 = (dist_max(P)).p1 & x2 = (dist_max(P)).p2 implies abs(x1 - x2) <= dist(p1,p2); theorem :: WEIERSTR:28 for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds abs(lower_bound((dist(p1)).:P) - lower_bound((dist(p2)).:P)) <= dist(p1,p2); theorem :: WEIERSTR:29 for M being non empty MetrSpace holds for P being Subset of TopSpaceMetr(M) st (P <> {} & P is compact) holds for p1,p2 being Point of M holds for x1,x2 being Real holds x1 = (dist_min(P)).p1 & x2 = (dist_min(P)).p2 implies abs(x1 - x2) <= dist(p1,p2); theorem :: WEIERSTR:30 for M being non empty MetrSpace holds for X being Subset of TopSpaceMetr(M) st (X <> {} & X is compact) holds dist_max(X) is continuous; theorem :: WEIERSTR:31 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in Q & (dist_max(P)).x1 = upper_bound((dist_max(P)).:Q)); theorem :: WEIERSTR:32 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in Q & (dist_max(P)).x2 = lower_bound((dist_max(P)).:Q)); theorem :: WEIERSTR:33 for M being non empty MetrSpace holds for X being Subset of TopSpaceMetr(M) st X <> {} & X is compact holds dist_min(X) is continuous; theorem :: WEIERSTR:34 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x1 being Point of TopSpaceMetr(M) st (x1 in Q & (dist_min(P)).x1 = upper_bound((dist_min(P)).:Q)); theorem :: WEIERSTR:35 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) holds (P <> {} & P is compact & Q <> {} & Q is compact) implies ex x2 being Point of TopSpaceMetr(M) st (x2 in Q & (dist_min(P)).x2 = lower_bound((dist_min(P)).:Q)); definition let M be non empty MetrSpace; let P,Q be Subset of TopSpaceMetr(M); func min_dist_min(P,Q) -> Real equals :: WEIERSTR:def 9 lower_bound((dist_min(P)).:Q); func max_dist_min(P,Q) -> Real equals :: WEIERSTR:def 10 upper_bound((dist_min(P)).:Q); func min_dist_max(P,Q) -> Real equals :: WEIERSTR:def 11 lower_bound((dist_max(P)).:Q); func max_dist_max(P,Q) -> Real equals :: WEIERSTR:def 12 upper_bound((dist_max(P)).:Q); end; theorem :: WEIERSTR:36 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = min_dist_min(P,Q); theorem :: WEIERSTR:37 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = min_dist_max(P,Q); theorem :: WEIERSTR:38 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = max_dist_min(P,Q); theorem :: WEIERSTR:39 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds ex x1,x2 being Point of M st x1 in P & x2 in Q & dist(x1,x2) = max_dist_max(P,Q); theorem :: WEIERSTR:40 for M being non empty MetrSpace holds for P,Q being Subset of TopSpaceMetr(M) st P is compact & Q is compact holds for x1,x2 being Point of M st x1 in P & x2 in Q holds min_dist_min(P,Q) <= dist(x1,x2) & dist(x1,x2) <= max_dist_max(P,Q); :: B i b l i o g r a p h y :: J.Dieudonne, "Foundations of Modern Analysis",Academic Press, 1960. :: J.L.Kelley, "General Topology", von Nostnend, 1955. ::

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