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Theorem pgpfac 17261
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 17257. 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 16929 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
3 pgpfac.b . . . 4  |-  B  =  ( Base `  G
)
43subgid 16329 . . 3  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
51, 2, 43syl 20 . 2  |-  ( ph  ->  B  e.  (SubGrp `  G ) )
6 pgpfac.f . . 3  |-  ( ph  ->  B  e.  Fin )
7 eleq1 2529 . . . . . 6  |-  ( t  =  u  ->  (
t  e.  (SubGrp `  G )  <->  u  e.  (SubGrp `  G ) ) )
8 eqeq2 2472 . . . . . . . 8  |-  ( t  =  u  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  u ) )
98anbi2d 703 . . . . . . 7  |-  ( t  =  u  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  u ) ) )
109rexbidv 2968 . . . . . 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 320 . . . . 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 316 . . . 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 2529 . . . . . 6  |-  ( t  =  B  ->  (
t  e.  (SubGrp `  G )  <->  B  e.  (SubGrp `  G ) ) )
14 eqeq2 2472 . . . . . . . 8  |-  ( t  =  B  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  B ) )
1514anbi2d 703 . . . . . . 7  |-  ( t  =  B  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  B ) ) )
1615rexbidv 2968 . . . . . 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 320 . . . . 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 316 . . . 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 bi2.04 361 . . . . . . . . 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 312 . . . . . . . 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 bi2.04 361 . . . . . . . 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 bi2.04 361 . . . . . . . 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 277 . . . . . . 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 1641 . . . . . 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 2812 . . . . . 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 2862 . . . . . 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 273 . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . 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 757 . . . . . . . . 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 756 . . . . . . . . . 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 3587 . . . . . . . . . . . 12  |-  ( t  =  x  ->  (
t  C.  u  <->  x  C.  u
) )
36 eqeq2 2472 . . . . . . . . . . . . . 14  |-  ( t  =  x  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  x ) )
3736anbi2d 703 . . . . . . . . . . . . 13  |-  ( t  =  x  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  x ) ) )
3837rexbidv 2968 . . . . . . . . . . . 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 320 . . . . . . . . . . 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 3084 . . . . . . . . . 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 196 . . . . . . . . 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 17260 . . . . . . . 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 605 . . . . . . 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 26 . . . . 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 217 . . . 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 7781 . . 3  |-  ( B  e.  Fin  ->  ( ph  ->  ( B  e.  (SubGrp `  G )  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) ) ) )
486, 47mpcom 36 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470    C. wpss 3472   class class class wbr 4456   dom cdm 5008   ran crn 5009   ` cfv 5594  (class class class)co 6296   Fincfn 7535  Word cword 12537   Basecbs 14643   ↾s cress 14644   Grpcgrp 16179  SubGrpcsubg 16321   pGrp cpgp 16677   Abelcabl 16925  CycGrpccyg 17006   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-rpss 6579  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-dvds 13998  df-gcd 14156  df-prm 14229  df-pc 14372  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-eqg 16326  df-ghm 16391  df-gim 16433  df-ga 16454  df-cntz 16481  df-oppg 16507  df-od 16679  df-gex 16680  df-pgp 16681  df-lsm 16782  df-pj1 16783  df-cmn 16926  df-abl 16927  df-cyg 17007  df-dprd 17152
This theorem is referenced by:  ablfaclem3  17264
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