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Theorem ablfaclem1 15320
Description: Lemma for ablfac 15323. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
ablfac.b  |-  B  =  ( Base `  G
)
ablfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
ablfac.1  |-  ( ph  ->  G  e.  Abel )
ablfac.2  |-  ( ph  ->  B  e.  Fin )
ablfac.o  |-  O  =  ( od `  G
)
ablfac.a  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
ablfac.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac.w  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
Assertion
Ref Expression
ablfaclem1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
Distinct variable groups:    s, p, x, A    g, r, s, S    g, p, w, x, B, r, s    O, p, x    C, g, p, s, w, x    W, p, w, x    ph, p, s, w, x    U, g, s    g, G, p, r, s, w, x
Allowed substitution hints:    ph( g, r)    A( w, g, r)    C( r)    S( x, w, p)    U( x, w, r, p)    O( w, g, s, r)    W( g, s, r)

Proof of Theorem ablfaclem1
StepHypRef Expression
1 eqeq2 2292 . . . 4  |-  ( g  =  U  ->  (
( G DProd  s )  =  g  <->  ( G DProd  s
)  =  U ) )
21anbi2d 684 . . 3  |-  ( g  =  U  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) ) )
32rabbidv 2780 . 2  |-  ( g  =  U  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) } )
4 ablfac.w . 2  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
5 ablfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
6 fvex 5539 . . . . . 6  |-  (SubGrp `  G )  e.  _V
76rabex 4165 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  e.  _V
85, 7eqeltri 2353 . . . 4  |-  C  e. 
_V
9 wrdexg 11425 . . . 4  |-  ( C  e.  _V  -> Word  C  e. 
_V )
108, 9ax-mp 8 . . 3  |- Word  C  e. 
_V
1110rabex 4165 . 2  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  U ) }  e.  _V
123, 4, 11fvmpt 5602 1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   Fincfn 6863   ^cexp 11104   #chash 11337  Word cword 11403    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   ↾s cress 13149  SubGrpcsubg 14615   odcod 14840   pGrp cpgp 14842   Abelcabel 15090  CycGrpccyg 15164   DProd cdprd 15231
This theorem is referenced by:  ablfaclem2  15321  ablfaclem3  15322  ablfac  15323
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-pm 6775  df-neg 9040  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-word 11409
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