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Theorem pgpfac 15319
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 15315. 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 15094 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
3 pgpfac.b . . . 4  |-  B  =  ( Base `  G
)
43subgid 14623 . . 3  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
51, 2, 43syl 18 . 2  |-  ( ph  ->  B  e.  (SubGrp `  G ) )
6 pgpfac.f . . 3  |-  ( ph  ->  B  e.  Fin )
7 eleq1 2343 . . . . . 6  |-  ( t  =  u  ->  (
t  e.  (SubGrp `  G )  <->  u  e.  (SubGrp `  G ) ) )
8 eqeq2 2292 . . . . . . . 8  |-  ( t  =  u  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  u ) )
98anbi2d 684 . . . . . . 7  |-  ( t  =  u  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  u ) ) )
109rexbidv 2564 . . . . . 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 311 . . . . 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 307 . . . 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 2343 . . . . . 6  |-  ( t  =  B  ->  (
t  e.  (SubGrp `  G )  <->  B  e.  (SubGrp `  G ) ) )
14 eqeq2 2292 . . . . . . . 8  |-  ( t  =  B  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  B ) )
1514anbi2d 684 . . . . . . 7  |-  ( t  =  B  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  B ) ) )
1615rexbidv 2564 . . . . . 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 311 . . . . 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 307 . . . 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 bi2.04 350 . . . . . . . . 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 303 . . . . . . . 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 bi2.04 350 . . . . . . . 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 bi2.04 350 . . . . . . . 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 268 . . . . . . 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 1553 . . . . . 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 2548 . . . . . 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 2630 . . . . . 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 264 . . . . 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 451 . . . . . . . . 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 451 . . . . . . . . 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 451 . . . . . . . . 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 733 . . . . . . . . 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 732 . . . . . . . . . 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 3263 . . . . . . . . . . . 12  |-  ( t  =  x  ->  (
t  C.  u  <->  x  C.  u ) )
36 eqeq2 2292 . . . . . . . . . . . . . 14  |-  ( t  =  x  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  x ) )
3736anbi2d 684 . . . . . . . . . . . . 13  |-  ( t  =  x  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  x ) ) )
3837rexbidv 2564 . . . . . . . . . . . 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 311 . . . . . . . . . . 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 2764 . . . . . . . . . 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 188 . . . . . . . . 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 15318 . . . . . . . 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 588 . . . . . . 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 10 . . . . . 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 23 . . . . 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 208 . . . 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 7100 . . 3  |-  ( B  e.  Fin  ->  ( ph  ->  ( B  e.  (SubGrp `  G )  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) ) ) )
486, 47mpcom 32 . 2  |-  ( ph  ->  ( B  e.  (SubGrp `  G )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) ) )
495, 48mpd 14 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 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C. wpss 3153   class class class wbr 4023   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   Fincfn 6863  Word cword 11403   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615   pGrp cpgp 14842   Abelcabel 15090  CycGrpccyg 15164   DProd cdprd 15231
This theorem is referenced by:  ablfaclem3  15322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233
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