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Theorem pgpfac 15335
Description: Full factorization of a finite abelian p-group, by iterating pgpfac1 15331. 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 15110 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
3 pgpfac.b . . . 4  |-  B  =  ( Base `  G
)
43subgid 14639 . . 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 2356 . . . . . 6  |-  ( t  =  u  ->  (
t  e.  (SubGrp `  G )  <->  u  e.  (SubGrp `  G ) ) )
8 eqeq2 2305 . . . . . . . 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 2577 . . . . . 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 2356 . . . . . 6  |-  ( t  =  B  ->  (
t  e.  (SubGrp `  G )  <->  B  e.  (SubGrp `  G ) ) )
14 eqeq2 2305 . . . . . . . 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 2577 . . . . . 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 1556 . . . . . 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 2561 . . . . . 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 2643 . . . . . 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 3276 . . . . . . . . . . . 12  |-  ( t  =  x  ->  (
t  C.  u  <->  x  C.  u ) )
36 eqeq2 2305 . . . . . . . . . . . . . 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 2577 . . . . . . . . . . . 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 2777 . . . . . . . . . 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 15334 . . . . . . . 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 7116 . . 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 1530    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C. wpss 3166   class class class wbr 4039   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874   Fincfn 6879  Word cword 11419   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631   pGrp cpgp 14858   Abelcabel 15106  CycGrpccyg 15180   DProd cdprd 15247
This theorem is referenced by:  ablfaclem3  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-rpss 6293  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ghm 14697  df-gim 14739  df-ga 14760  df-cntz 14809  df-oppg 14835  df-od 14860  df-gex 14861  df-pgp 14862  df-lsm 14963  df-pj1 14964  df-cmn 15107  df-abl 15108  df-cyg 15181  df-dprd 15249
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