Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Groups

**by****Wojciech A. Trybulec**- Received July 3, 1990
- MML identifier: GROUP_1

- [ Mizar article, MML identifier index ]

environ vocabulary INT_1, VECTSP_1, ABSVALUE, BINOP_1, FUNCT_1, ARYTM_1, REALSET1, RELAT_1, SETWISEO, FINSEQOP, ARYTM_3, CARD_1, FINSET_1, RLVECT_1, GROUP_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, RELAT_1, FUNCT_1, REAL_1, FUNCT_2, STRUCT_0, RLVECT_1, VECTSP_1, INT_1, NAT_1, FINSEQOP, SETWISEO, CARD_1, FINSET_1, BINOP_1, ABSVALUE; constructors REAL_1, VECTSP_1, INT_1, NAT_1, FINSEQOP, SETWISEO, BINOP_1, ABSVALUE, MEMBERED, XBOOLE_0; clusters FINSET_1, VECTSP_1, INT_1, STRUCT_0, RELSET_1, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve k,m,n for Nat; reserve i,j for Integer; reserve S for non empty HGrStr; reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S; definition let i be Integer; redefine func abs i -> Nat; end; definition let A be non empty set, m be BinOp of A; cluster HGrStr(#A,m#) -> non empty; end; definition let IT be non empty HGrStr; attr IT is unital means :: GROUP_1:def 1 ex e being Element of IT st for h being Element of IT holds h * e = h & e * h = h; canceled; attr IT is Group-like means :: GROUP_1:def 3 ex e being Element of IT st for h being Element of IT holds h * e = h & e * h = h & ex g being Element of IT st h * g = e & g * h = e; end; definition cluster Group-like -> unital (non empty HGrStr); end; definition cluster strict Group-like associative (non empty HGrStr); end; definition mode Group is Group-like associative (non empty HGrStr); end; canceled 4; theorem :: GROUP_1:5 ((for r,s,t holds (r * s) * t = r * (s * t)) & ex t st for s1 holds s1 * t = s1 & t * s1 = s1 & ex s2 st s1 * s2 = t & s2 * s1 = t) implies S is Group; theorem :: GROUP_1:6 (for r,s,t holds r * s * t = r * (s * t)) & (for r,s holds (ex t st r * t = s) & (ex t st t * r = s)) implies S is associative Group-like; theorem :: GROUP_1:7 HGrStr (# REAL, addreal #) is associative Group-like; reserve G for Group-like (non empty HGrStr); reserve e,h for Element of G; definition let G be unital (non empty HGrStr); func 1.G -> Element of G means :: GROUP_1:def 4 for h being Element of G holds h * it = h & it * h = h; end; canceled 2; theorem :: GROUP_1:10 (for h holds h * e = h & e * h = h) implies e = 1.G; reserve G for Group; reserve e,f,g,h for Element of G; definition let G,h; func h" -> Element of G means :: GROUP_1:def 5 h * it = 1.G & it * h = 1.G; end; canceled; theorem :: GROUP_1:12 h * g = 1.G & g * h = 1.G implies g = h"; canceled; theorem :: GROUP_1:14 h * g = h * f or g * h = f * h implies g = f; theorem :: GROUP_1:15 h * g = h or g * h = h implies g = 1.G; theorem :: GROUP_1:16 (1.G)" = 1.G; theorem :: GROUP_1:17 h" = g" implies h = g; theorem :: GROUP_1:18 h" = 1.G implies h = 1.G; theorem :: GROUP_1:19 h"" = h; theorem :: GROUP_1:20 h * g = 1.G or g * h = 1.G implies h = g" & g = h"; theorem :: GROUP_1:21 h * f = g iff f = h" * g; theorem :: GROUP_1:22 f * h = g iff f = g * h"; theorem :: GROUP_1:23 ex f st g * f = h; theorem :: GROUP_1:24 ex f st f * g = h; theorem :: GROUP_1:25 (h * g)" = g" * h"; theorem :: GROUP_1:26 g * h = h * g iff (g * h)" = g" * h"; theorem :: GROUP_1:27 g * h = h * g iff g" * h" = h" * g"; theorem :: GROUP_1:28 g * h = h * g iff g * h" = h" * g; reserve u for UnOp of the carrier of G; definition let G; func inverse_op(G) -> UnOp of the carrier of G means :: GROUP_1:def 6 it.h = h"; end; canceled 2; theorem :: GROUP_1:31 for G being associative (non empty HGrStr) holds the mult of G is associative; theorem :: GROUP_1:32 for G being unital (non empty HGrStr) holds 1.G is_a_unity_wrt the mult of G; theorem :: GROUP_1:33 for G being unital (non empty HGrStr) holds the_unity_wrt the mult of G = 1.G; theorem :: GROUP_1:34 for G being unital (non empty HGrStr) holds the mult of G has_a_unity; theorem :: GROUP_1:35 inverse_op(G) is_an_inverseOp_wrt the mult of G; theorem :: GROUP_1:36 the mult of G has_an_inverseOp; theorem :: GROUP_1:37 the_inverseOp_wrt the mult of G = inverse_op(G); definition let G be unital (non empty HGrStr); func power G -> Function of [:the carrier of G,NAT:], the carrier of G means :: GROUP_1:def 7 for h being Element of G holds it.(h,0) = 1.G & for n holds it.(h,n + 1) = it.(h,n) * h; end; definition let G,i,h; func h |^ i -> Element of G equals :: GROUP_1:def 8 power(G).(h,abs(i)) if 0 <= i otherwise (power(G).(h,abs(i)))"; end; definition let G,n,h; redefine func h |^ n equals :: GROUP_1:def 9 power(G).(h,n); end; canceled 4; theorem :: GROUP_1:42 (1.G) |^ n = 1.G; theorem :: GROUP_1:43 h |^ 0 = 1.G; theorem :: GROUP_1:44 h |^ 1 = h; theorem :: GROUP_1:45 h |^ 2 = h * h; theorem :: GROUP_1:46 h |^ 3 = h * h * h; theorem :: GROUP_1:47 h |^ 2 = 1.G iff h" = h; theorem :: GROUP_1:48 h |^ (n + m) = h |^ n * (h |^ m); theorem :: GROUP_1:49 h |^ (n + 1) = h |^ n * h & h |^ (n + 1) = h * (h |^ n); theorem :: GROUP_1:50 h |^ (n * m) = h |^ n |^ m; theorem :: GROUP_1:51 h" |^ n = (h |^ n)"; theorem :: GROUP_1:52 g * h = h * g implies g * (h |^ n) = h |^ n * g; theorem :: GROUP_1:53 g * h = h * g implies g |^ n * (h |^ m) = h |^ m * (g |^ n); theorem :: GROUP_1:54 g * h = h * g implies (g * h) |^ n = g |^ n * (h |^ n); theorem :: GROUP_1:55 0 <= i implies h |^ i = h |^ abs(i); theorem :: GROUP_1:56 not 0 <= i implies h |^ i = (h |^ abs(i))"; canceled 2; theorem :: GROUP_1:59 i = 0 implies h |^ i = 1.G; theorem :: GROUP_1:60 i <= 0 implies h |^ i = (h |^ abs(i))"; theorem :: GROUP_1:61 (1.G) |^ i = 1.G; theorem :: GROUP_1:62 h |^ (- 1) = h"; theorem :: GROUP_1:63 h |^ (i + j) = h |^ i * (h |^ j); theorem :: GROUP_1:64 h |^ (n + j) = h |^ n * (h |^ j); theorem :: GROUP_1:65 h |^ (i + m) = h |^ i * (h |^ m); theorem :: GROUP_1:66 h |^ (j + 1) = h |^ j * h & h |^ (j + 1) = h * (h |^ j); theorem :: GROUP_1:67 h |^ (i * j) = h |^ i |^ j; theorem :: GROUP_1:68 h |^ (n * j) = h |^ n |^ j; theorem :: GROUP_1:69 h |^ (i * m) = h |^ i |^ m; theorem :: GROUP_1:70 h |^ (- i) = (h |^ i)"; theorem :: GROUP_1:71 h |^ (- n) = (h |^ n)"; theorem :: GROUP_1:72 h" |^ i = (h |^ i)"; theorem :: GROUP_1:73 g * h = h * g implies (g * h) |^ i = g |^ i * (h |^ i); theorem :: GROUP_1:74 g * h = h * g implies g |^ i * (h |^ j) = h |^ j * (g |^ i); theorem :: GROUP_1:75 g * h = h * g implies g |^ n * (h |^ j) = h |^ j * (g |^ n); canceled; theorem :: GROUP_1:77 g * h = h * g implies g * (h |^ i) = h |^ i * g; definition let G,h; attr h is being_of_order_0 means :: GROUP_1:def 10 h |^ n = 1.G implies n = 0; antonym h is_not_of_order_0; synonym h is_of_order_0; end; canceled; theorem :: GROUP_1:79 1.G is_not_of_order_0; definition let G,h; func ord h -> Nat means :: GROUP_1:def 11 it = 0 if h is_of_order_0 otherwise h |^ it = 1.G & it <> 0 & for m st h |^ m = 1.G & m <> 0 holds it <= m; end; canceled 2; theorem :: GROUP_1:82 h |^ ord h = 1.G; canceled; theorem :: GROUP_1:84 ord 1.G = 1; theorem :: GROUP_1:85 ord h = 1 implies h = 1.G; theorem :: GROUP_1:86 h |^ n = 1.G implies ord h divides n; definition let G; func Ord G -> Cardinal equals :: GROUP_1:def 12 Card(the carrier of G); end; definition let S be 1-sorted; attr S is finite means :: GROUP_1:def 13 the carrier of S is finite; antonym S is infinite; end; definition let G; assume G is finite; func ord G -> Nat means :: GROUP_1:def 14 ex B being finite set st B = the carrier of G & it = card B; end; canceled 3; theorem :: GROUP_1:90 G is finite implies ord G >= 1; definition cluster strict commutative Group; end; canceled; theorem :: GROUP_1:92 HGrStr (# REAL, addreal #) is commutative Group; reserve A for commutative Group; reserve a,b for Element of A; canceled; theorem :: GROUP_1:94 (a * b)" = a" * b"; theorem :: GROUP_1:95 (a * b) |^ n = a |^ n * (b |^ n); theorem :: GROUP_1:96 (a * b) |^ i = a |^ i * (b |^ i); definition let A be non empty set, m be BinOp of A, u be Element of A; cluster LoopStr(#A,m,u#) -> non empty; end; theorem :: GROUP_1:97 LoopStr (# the carrier of A, the mult of A, 1.A #) is Abelian add-associative right_zeroed right_complementable; reserve B for AbGroup; theorem :: GROUP_1:98 HGrStr (# the carrier of B, the add of B #) is commutative Group;

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