Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### On Powers of Cardinals

**by****Grzegorz Bancerek**- Received August 24, 1992
- MML identifier: CARD_5

- [ Mizar article, MML identifier index ]

environ vocabulary ORDINAL1, BOOLE, ORDINAL2, FINSEQ_1, FUNCT_1, CARD_1, PROB_1, RELAT_1, TARSKI, FINSET_1, WELLORD1, WELLORD2, ZFREFLE1, CARD_2, ORDINAL3, FUNCT_2, CARD_3, ZFMISC_1, FUNCOP_1, RLVECT_1, MCART_1, CARD_5; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, NAT_1, RELAT_1, FUNCT_1, FINSEQ_1, FINSET_1, ORDINAL1, FUNCT_2, WELLORD1, WELLORD2, MCART_1, FUNCOP_1, ORDINAL2, CARD_1, FUNCT_4, ORDINAL3, CARD_2, PROB_1, CARD_3, ORDINAL4, ZFREFLE1; constructors ZF_REFLE, NAT_1, WELLORD1, WELLORD2, MCART_1, FUNCOP_1, ORDINAL3, CARD_2, CARD_3, ZFREFLE1, XBOOLE_0; clusters SUBSET_1, FUNCT_1, ORDINAL1, ORDINAL2, CARD_1, CARD_3, ORDINAL3, ARYTM_3, ORDINAL4, FINSET_1, XREAL_0, ZFMISC_1, XBOOLE_0; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; begin :: Results of [(30),AXIOMS]. reserve k,n,m for Nat, A,B,C for Ordinal, X for set, x,y,z for set; theorem :: CARD_5:1 1 = {0} & 2 = {0,1}; canceled 6; theorem :: CARD_5:8 Seg n = (n+1) \ {0}; begin :: Infinity, alephs and cofinality reserve f,g,h,fx for Function, K,M,N for Cardinal, phi,psi for Ordinal-Sequence; theorem :: CARD_5:9 nextcard Card X = nextcard X; theorem :: CARD_5:10 y in Union f iff ex x st x in dom f & y in f.x; theorem :: CARD_5:11 alef A is infinite; theorem :: CARD_5:12 M is infinite implies ex A st M = alef A; theorem :: CARD_5:13 (ex n st M = Card n) or (ex A st M = alef A); definition let phi; cluster Union phi -> ordinal; end; theorem :: CARD_5:14 X c= A implies ex phi st phi = canonical_isomorphism_of(RelIncl order_type_of RelIncl X, RelIncl X) & phi is increasing & dom phi = order_type_of RelIncl X & rng phi = X; theorem :: CARD_5:15 X c= A implies sup X is_cofinal_with order_type_of RelIncl X; theorem :: CARD_5:16 X c= A implies Card X = Card order_type_of RelIncl X; theorem :: CARD_5:17 ex B st B c= Card A & A is_cofinal_with B; theorem :: CARD_5:18 ex M st M <=` Card A & A is_cofinal_with M & for B st A is_cofinal_with B holds M c= B; theorem :: CARD_5:19 rng phi = rng psi & phi is increasing & psi is increasing implies phi = psi; theorem :: CARD_5:20 phi is increasing implies phi is one-to-one; theorem :: CARD_5:21 (phi^psi)|(dom phi) = phi; theorem :: CARD_5:22 X <> {} implies Card { Y where Y is Element of bool X: Card Y <` M } <=` M*`exp(Card X,M); theorem :: CARD_5:23 M <` exp(2,M); definition cluster infinite set; cluster infinite Cardinal; end; definition cluster infinite -> non empty set; end; definition mode Aleph is infinite Cardinal; let M; canceled; func cf M -> Cardinal means :: CARD_5:def 2 M is_cofinal_with it & for N st M is_cofinal_with N holds it <=` N; let N; func N-powerfunc_of M -> Cardinal-Function means :: CARD_5:def 3 (for x holds x in dom it iff x in M & x is Cardinal) & for K st K in M holds it.K = exp(K,N); end; definition let A; cluster alef A -> infinite; end; begin :: Arithmetics of alephs reserve a,b for Aleph; theorem :: CARD_5:24 ex A st a = alef A; theorem :: CARD_5:25 a <> 0 & a <> 1 & a <> 2 & a <> Card n & Card n <` a & alef 0 <=` a; theorem :: CARD_5:26 a <=` M or a <` M implies M is Aleph; theorem :: CARD_5:27 a <=` M or a <` M implies a +` M = M & M +` a = M & a *` M = M & M *` a = M; theorem :: CARD_5:28 a +` a = a & a *` a = a; canceled 2; theorem :: CARD_5:31 M <=` exp(M,a); theorem :: CARD_5:32 union a = a; definition let a,M; cluster a +` M -> infinite; end; definition let M,a; cluster M +` a -> infinite; end; definition let a,b; cluster a *` b -> infinite; cluster exp(a,b) -> infinite; end; begin :: Regular alephs definition let IT be Aleph; attr IT is regular means :: CARD_5:def 4 cf IT = IT; antonym IT is irregular; end; definition let a; cluster nextcard a -> infinite; cluster -> ordinal Element of a; end; canceled; theorem :: CARD_5:34 cf alef 0 = alef 0; theorem :: CARD_5:35 cf nextcard a = nextcard a; theorem :: CARD_5:36 alef 0 <=` cf a; theorem :: CARD_5:37 cf 0 = 0 & cf Card (n+1) = 1; theorem :: CARD_5:38 X c= M & Card X <` cf M implies sup X in M & union X in M; theorem :: CARD_5:39 dom phi = M & rng phi c= N & M <` cf N implies sup phi in N & Union phi in N; definition let a; cluster cf a -> infinite; end; theorem :: CARD_5:40 cf a <` a implies a is_limit_cardinal; theorem :: CARD_5:41 cf a <` a implies ex xi being Ordinal-Sequence st dom xi = cf a & rng xi c= a & xi is increasing & a = sup xi & xi is Cardinal-Function & not 0 in rng xi; theorem :: CARD_5:42 alef 0 is regular & nextcard a is regular; begin :: Infinite powers reserve a,b for Aleph; theorem :: CARD_5:43 a <=` b implies exp(a,b) = exp(2,b); theorem :: CARD_5:44 exp(nextcard a,b) = exp(a,b) *` nextcard a; theorem :: CARD_5:45 Sum (b-powerfunc_of a) <=` exp(a,b); theorem :: CARD_5:46 a is_limit_cardinal & b <` cf a implies exp(a,b) = Sum (b-powerfunc_of a); theorem :: CARD_5:47 cf a <=` b & b <` a implies exp(a,b) = exp(Sum (b-powerfunc_of a), cf a);

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