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Theorem ablfaclem1 15645
Description: Lemma for ablfac 15648. (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 2447 . . . 4  |-  ( g  =  U  ->  (
( G DProd  s )  =  g  <->  ( G DProd  s
)  =  U ) )
21anbi2d 686 . . 3  |-  ( g  =  U  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) ) )
32rabbidv 2950 . 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 5744 . . . . . 6  |-  (SubGrp `  G )  e.  _V
76rabex 4356 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  e.  _V
85, 7eqeltri 2508 . . . 4  |-  C  e. 
_V
9 wrdexg 11741 . . . 4  |-  ( C  e.  _V  -> Word  C  e. 
_V )
108, 9ax-mp 8 . . 3  |- Word  C  e. 
_V
1110rabex 4356 . 2  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  U ) }  e.  _V
123, 4, 11fvmpt 5808 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 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    i^i cin 3321   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   ran crn 4881   ` cfv 5456  (class class class)co 6083   Fincfn 7111   ^cexp 11384   #chash 11620  Word cword 11719    || cdivides 12854   Primecprime 13081    pCnt cpc 13212   Basecbs 13471   ↾s cress 13472  SubGrpcsubg 14940   odcod 15165   pGrp cpgp 15167   Abelcabel 15415  CycGrpccyg 15489   DProd cdprd 15556
This theorem is referenced by:  ablfaclem2  15646  ablfaclem3  15647  ablfac  15648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-pm 7023  df-neg 9296  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-word 11725
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