Theorem pgpfac 18483

Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 18479. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
pgpfac.b	|- B = ( Base ` G )
pgpfac.c	|- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
pgpfac.g	|- ( ph -> G e. Abel )
pgpfac.p	|- ( ph -> P pGrp G )
pgpfac.f	|- ( ph -> B e. Fin )

Assertion

Ref	Expression
pgpfac	|- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )

Distinct variable groups:   C, s   s, r, G   B, s

Allowed substitution hints:   ph( s, r)   B( r)   C( r)   P( s, r)

Proof of Theorem pgpfac

Dummy variables t u x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac.g	. . 3 |- ( ph -> G e. Abel )
2		ablgrp 18198	. . 3 |- ( G e. Abel -> G e. Grp )
3		pgpfac.b	. . . 4 |- B = ( Base ` G )
4	3	subgid 17596	. . 3 |- ( G e. Grp -> B e. (SubGrp ` G ) )
5	1, 2, 4	3syl 18	. 2 |- ( ph -> B e. (SubGrp ` G ) )
6		pgpfac.f	. . 3 |- ( ph -> B e. Fin )
7		eleq1 2689	. . . . . 6 |- ( t = u -> ( t e. (SubGrp ` G ) <-> u e. (SubGrp ` G ) ) )
8		eqeq2 2633	. . . . . . . 8 |- ( t = u -> ( ( G DProd s ) = t <-> ( G DProd s ) = u ) )
9	8	anbi2d 740	. . . . . . 7 |- ( t = u -> ( ( G dom DProd s /\ ( G DProd s ) = t ) <-> ( G dom DProd s /\ ( G DProd s ) = u ) ) )
10	9	rexbidv 3052	. . . . . 6 |- ( t = u -> ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) <-> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) )
11	7, 10	imbi12d 334	. . . . 5 |- ( t = u -> ( ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) <-> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) )
12	11	imbi2d 330	. . . 4 |- ( t = u -> ( ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) <-> ( ph -> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) ) )
13		eleq1 2689	. . . . . 6 |- ( t = B -> ( t e. (SubGrp ` G ) <-> B e. (SubGrp ` G ) ) )
14		eqeq2 2633	. . . . . . . 8 |- ( t = B -> ( ( G DProd s ) = t <-> ( G DProd s ) = B ) )
15	14	anbi2d 740	. . . . . . 7 |- ( t = B -> ( ( G dom DProd s /\ ( G DProd s ) = t ) <-> ( G dom DProd s /\ ( G DProd s ) = B ) ) )
16	15	rexbidv 3052	. . . . . 6 |- ( t = B -> ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) <-> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) ) )
17	13, 16	imbi12d 334	. . . . 5 |- ( t = B -> ( ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) <-> ( B e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) ) ) )
18	17	imbi2d 330	. . . 4 |- ( t = B -> ( ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) <-> ( ph -> ( B e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) ) ) ) )
19		bi2.04 376	. . . . . . . . 9 |- ( ( t C. u -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) <-> ( t e. (SubGrp ` G ) -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) )
20	19	imbi2i 326	. . . . . . . 8 |- ( ( ph -> ( t C. u -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> ( ph -> ( t e. (SubGrp ` G ) -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
21		bi2.04 376	. . . . . . . 8 |- ( ( t C. u -> ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> ( ph -> ( t C. u -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
22		bi2.04 376	. . . . . . . 8 |- ( ( t e. (SubGrp ` G ) -> ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> ( ph -> ( t e. (SubGrp ` G ) -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
23	20, 21, 22	3bitr4i 292	. . . . . . 7 |- ( ( t C. u -> ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> ( t e. (SubGrp ` G ) -> ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
24	23	albii 1747	. . . . . 6 |- ( A. t ( t C. u -> ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> A. t ( t e. (SubGrp ` G ) -> ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
25		df-ral 2917	. . . . . 6 |- ( A. t e. (SubGrp ` G ) ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) <-> A. t ( t e. (SubGrp ` G ) -> ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) )
26		r19.21v 2960	. . . . . 6 |- ( A. t e. (SubGrp ` G ) ( ph -> ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) <-> ( ph -> A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) )
27	24, 25, 26	3bitr2i 288	. . . . 5 |- ( A. t ( t C. u -> ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) <-> ( ph -> A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) )
28		pgpfac.c	. . . . . . . . 9 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
29	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> G e. Abel )
30		pgpfac.p	. . . . . . . . . 10 |- ( ph -> P pGrp G )
31	30	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> P pGrp G )
32	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> B e. Fin )
33		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> u e. (SubGrp ` G ) )
34		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
35		psseq1 3694	. . . . . . . . . . . 12 |- ( t = x -> ( t C. u <-> x C. u ) )
36		eqeq2 2633	. . . . . . . . . . . . . 14 |- ( t = x -> ( ( G DProd s ) = t <-> ( G DProd s ) = x ) )
37	36	anbi2d 740	. . . . . . . . . . . . 13 |- ( t = x -> ( ( G dom DProd s /\ ( G DProd s ) = t ) <-> ( G dom DProd s /\ ( G DProd s ) = x ) ) )
38	37	rexbidv 3052	. . . . . . . . . . . 12 |- ( t = x -> ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) <-> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = x ) ) )
39	35, 38	imbi12d 334	. . . . . . . . . . 11 |- ( t = x -> ( ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) <-> ( x C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = x ) ) ) )
40	39	cbvralv 3171	. . . . . . . . . 10 |- ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) <-> A. x e. (SubGrp ` G ) ( x C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = x ) ) )
41	34, 40	sylib 208	. . . . . . . . 9 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> A. x e. (SubGrp ` G ) ( x C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = x ) ) )
42	3, 28, 29, 31, 32, 33, 41	pgpfaclem3 18482	. . . . . . . 8 |- ( ( ph /\ ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) /\ u e. (SubGrp ` G ) ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) )
43	42	exp32 631	. . . . . . 7 |- ( ph -> ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) -> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) )
44	43	a1i 11	. . . . . 6 |- ( u e. Fin -> ( ph -> ( A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) -> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) ) )
45	44	a2d 29	. . . . 5 |- ( u e. Fin -> ( ( ph -> A. t e. (SubGrp ` G ) ( t C. u -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) -> ( ph -> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) ) )
46	27, 45	syl5bi 232	. . . 4 |- ( u e. Fin -> ( A. t ( t C. u -> ( ph -> ( t e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) ) ) -> ( ph -> ( u e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = u ) ) ) ) )
47	12, 18, 46	findcard3 8203	. . 3 |- ( B e. Fin -> ( ph -> ( B e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) ) ) )
48	6, 47	mpcom 38	. 2 |- ( ph -> ( B e. (SubGrp ` G ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) ) )
49	5, 48	mpd 15	1 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C. wpss 3575   class class class wbr 4653   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   Fincfn 7955  Word cword 13291   Basecbs 15857   ↾_s cress 15858   Grpcgrp 17422  SubGrpcsubg 17588   pGrp cpgp 17946   Abelcabl 18194  CycGrpccyg 18279   DProd cdprd 18392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ghm 17658  df-gim 17701  df-ga 17723  df-cntz 17750  df-oppg 17776  df-od 17948  df-gex 17949  df-pgp 17950  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-cyg 18280  df-dprd 18394

This theorem is referenced by:  ablfaclem3  18486