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Theorem pgpfac 16597
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 16593. 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 16294 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
3 pgpfac.b . . . 4  |-  B  =  ( Base `  G
)
43subgid 15695 . . 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 2503 . . . . . 6  |-  ( t  =  u  ->  (
t  e.  (SubGrp `  G )  <->  u  e.  (SubGrp `  G ) ) )
8 eqeq2 2452 . . . . . . . 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 2748 . . . . . 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 2503 . . . . . 6  |-  ( t  =  B  ->  (
t  e.  (SubGrp `  G )  <->  B  e.  (SubGrp `  G ) ) )
14 eqeq2 2452 . . . . . . . 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 2748 . . . . . 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 1610 . . . . . 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 2732 . . . . . 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 2815 . . . . . 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 756 . . . . . . . . 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 755 . . . . . . . . . 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 3455 . . . . . . . . . . . 12  |-  ( t  =  x  ->  (
t  C.  u  <->  x  C.  u
) )
36 eqeq2 2452 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . 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 2959 . . . . . . . . . 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 16596 . . . . . . . 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 7567 . . 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 1367    = wceq 1369    e. wcel 1756   A.wral 2727   E.wrex 2728   {crab 2731    i^i cin 3339    C. wpss 3341   class class class wbr 4304   dom cdm 4852   ran crn 4853   ` cfv 5430  (class class class)co 6103   Fincfn 7322  Word cword 12233   Basecbs 14186   ↾s cress 14187   Grpcgrp 15422  SubGrpcsubg 15687   pGrp cpgp 16042   Abelcabel 16290  CycGrpccyg 16366   DProd cdprd 16487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-disj 4275  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-rpss 6372  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-omul 6937  df-er 7113  df-ec 7115  df-qs 7119  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-oi 7736  df-card 8121  df-acn 8124  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-word 12241  df-concat 12243  df-s1 12244  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-sum 13176  df-dvds 13548  df-gcd 13703  df-prm 13776  df-pc 13916  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-gsum 14393  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-eqg 15692  df-ghm 15757  df-gim 15799  df-ga 15820  df-cntz 15847  df-oppg 15873  df-od 16044  df-gex 16045  df-pgp 16046  df-lsm 16147  df-pj1 16148  df-cmn 16291  df-abl 16292  df-cyg 16367  df-dprd 16489
This theorem is referenced by:  ablfaclem3  16600
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