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Theorem pgpfac 15562
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 15558. 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 )  |  ( Gs  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.
StepHypRef Expression
1 pgpfac.g . . 3  |-  ( ph  ->  G  e.  Abel )
2 ablgrp 15337 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
3 pgpfac.b . . . 4  |-  B  =  ( Base `  G
)
43subgid 14866 . . 3  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
51, 2, 43syl 19 . 2  |-  ( ph  ->  B  e.  (SubGrp `  G ) )
6 pgpfac.f . . 3  |-  ( ph  ->  B  e.  Fin )
7 eleq1 2440 . . . . . 6  |-  ( t  =  u  ->  (
t  e.  (SubGrp `  G )  <->  u  e.  (SubGrp `  G ) ) )
8 eqeq2 2389 . . . . . . . 8  |-  ( t  =  u  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  u ) )
98anbi2d 685 . . . . . . 7  |-  ( t  =  u  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  u ) ) )
109rexbidv 2663 . . . . . 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 ) ) )
117, 10imbi12d 312 . . . . 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 ) ) ) )
1211imbi2d 308 . . . 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 2440 . . . . . 6  |-  ( t  =  B  ->  (
t  e.  (SubGrp `  G )  <->  B  e.  (SubGrp `  G ) ) )
14 eqeq2 2389 . . . . . . . 8  |-  ( t  =  B  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  B ) )
1514anbi2d 685 . . . . . . 7  |-  ( t  =  B  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  B ) ) )
1615rexbidv 2663 . . . . . 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 ) ) )
1713, 16imbi12d 312 . . . . 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 ) ) ) )
1817imbi2d 308 . . . 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 bi2.04 351 . . . . . . . . 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 ) ) ) )
2019imbi2i 304 . . . . . . . 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 bi2.04 351 . . . . . . . 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 bi2.04 351 . . . . . . . 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 ) ) ) ) )
2320, 21, 223bitr4i 269 . . . . . . 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 ) ) ) ) )
2423albii 1572 . . . . . 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 2647 . . . . . 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 2729 . . . . . 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 ) ) ) )
2724, 25, 263bitr2i 265 . . . . 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 )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
291adantr 452 . . . . . . . . 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 )
3130adantr 452 . . . . . . . . 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 )
326adantr 452 . . . . . . . . 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 734 . . . . . . . . 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 733 . . . . . . . . . 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 3370 . . . . . . . . . . . 12  |-  ( t  =  x  ->  (
t  C.  u  <->  x  C.  u ) )
36 eqeq2 2389 . . . . . . . . . . . . . 14  |-  ( t  =  x  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  x ) )
3736anbi2d 685 . . . . . . . . . . . . 13  |-  ( t  =  x  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  x ) ) )
3837rexbidv 2663 . . . . . . . . . . . 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 ) ) )
3935, 38imbi12d 312 . . . . . . . . . . 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 ) ) ) )
4039cbvralv 2868 . . . . . . . . . 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 ) ) )
4134, 40sylib 189 . . . . . . . . 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 ) ) )
423, 28, 29, 31, 32, 33, 41pgpfaclem3 15561 . . . . . . . 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 ) )
4342exp32 589 . . . . . . 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 ) ) ) )
4443a1i 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 ) ) ) ) )
4544a2d 24 . . . . 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 ) ) ) ) )
4627, 45syl5bi 209 . . . 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 ) ) ) ) )
4712, 18, 46findcard3 7279 . . 3  |-  ( B  e.  Fin  ->  ( ph  ->  ( B  e.  (SubGrp `  G )  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) ) ) )
486, 47mpcom 34 . 2  |-  ( ph  ->  ( B  e.  (SubGrp `  G )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) ) )
495, 48mpd 15 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   {crab 2646    i^i cin 3255    C. wpss 3257   class class class wbr 4146   dom cdm 4811   ran crn 4812   ` cfv 5387  (class class class)co 6013   Fincfn 7038  Word cword 11637   Basecbs 13389   ↾s cress 13390   Grpcgrp 14605  SubGrpcsubg 14858   pGrp cpgp 15085   Abelcabel 15333  CycGrpccyg 15407   DProd cdprd 15474
This theorem is referenced by:  ablfaclem3  15565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-disj 4117  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-tpos 6408  df-rpss 6451  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-omul 6658  df-er 6834  df-ec 6836  df-qs 6840  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-acn 7755  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-dvds 12773  df-gcd 12927  df-prm 13000  df-pc 13131  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-0g 13647  df-gsum 13648  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-mhm 14658  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-mulg 14735  df-subg 14861  df-eqg 14863  df-ghm 14924  df-gim 14966  df-ga 14987  df-cntz 15036  df-oppg 15062  df-od 15087  df-gex 15088  df-pgp 15089  df-lsm 15190  df-pj1 15191  df-cmn 15334  df-abl 15335  df-cyg 15408  df-dprd 15476
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