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Theorem pgpfaclem2 17793
Description: Lemma for pgpfac 17795. (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 )
pgpfac.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac.a  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
pgpfac.h  |-  H  =  ( Gs  U )
pgpfac.k  |-  K  =  (mrCls `  (SubGrp `  H
) )
pgpfac.o  |-  O  =  ( od `  H
)
pgpfac.e  |-  E  =  (gEx `  H )
pgpfac.0  |-  .0.  =  ( 0g `  H )
pgpfac.l  |-  .(+)  =  (
LSSum `  H )
pgpfac.1  |-  ( ph  ->  E  =/=  1 )
pgpfac.x  |-  ( ph  ->  X  e.  U )
pgpfac.oe  |-  ( ph  ->  ( O `  X
)  =  E )
pgpfac.w  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
pgpfac.i  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
pgpfac.s  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
Assertion
Ref Expression
pgpfaclem2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    K, r, s    ph, t    B, s, t    U, r, s, t    W, s, t    X, r, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    E( t, s, r)    H( t, s, r)    K( t)    O( t, s, r)    W( r)    X( t)    .0. ( t, s, r)

Proof of Theorem pgpfaclem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
2 pgpfac.u . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 pgpfac.h . . . . . . . 8  |-  H  =  ( Gs  U )
43subsubg 16918 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
52, 4syl 17 . . . . . 6  |-  ( ph  ->  ( W  e.  (SubGrp `  H )  <->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) ) )
61, 5mpbid 215 . . . . 5  |-  ( ph  ->  ( W  e.  (SubGrp `  G )  /\  W  C_  U ) )
76simpld 466 . . . 4  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
8 pgpfac.a . . . 4  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
96simprd 470 . . . . 5  |-  ( ph  ->  W  C_  U )
10 pgpfac.f . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
11 pgpfac.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
1211subgss 16896 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
132, 12syl 17 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
14 ssfi 7810 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
1510, 13, 14syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  U  e.  Fin )
16 ssfi 7810 . . . . . . . . . 10  |-  ( ( U  e.  Fin  /\  W  C_  U )  ->  W  e.  Fin )
1715, 9, 16syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  W  e.  Fin )
18 hashcl 12576 . . . . . . . . 9  |-  ( W  e.  Fin  ->  ( # `
 W )  e. 
NN0 )
1917, 18syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
2019nn0red 10950 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  e.  RR )
21 pgpfac.0 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  H )
22 fvex 5889 . . . . . . . . . . . 12  |-  ( 0g
`  H )  e. 
_V
2321, 22eqeltri 2545 . . . . . . . . . . 11  |-  .0.  e.  _V
24 hashsng 12587 . . . . . . . . . . 11  |-  (  .0. 
e.  _V  ->  ( # `  {  .0.  } )  =  1 )
2523, 24ax-mp 5 . . . . . . . . . 10  |-  ( # `  {  .0.  } )  =  1
26 subgrcl 16900 . . . . . . . . . . . . . . . 16  |-  ( W  e.  (SubGrp `  H
)  ->  H  e.  Grp )
27 eqid 2471 . . . . . . . . . . . . . . . . 17  |-  ( Base `  H )  =  (
Base `  H )
2827subgacs 16930 . . . . . . . . . . . . . . . 16  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
29 acsmre 15636 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
301, 26, 28, 294syl 19 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
31 pgpfac.k . . . . . . . . . . . . . . 15  |-  K  =  (mrCls `  (SubGrp `  H
) )
3230, 31mrcssvd 15607 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( K `  { X } )  C_  ( Base `  H ) )
333subgbas 16899 . . . . . . . . . . . . . . 15  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
342, 33syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  =  ( Base `  H ) )
3532, 34sseqtr4d 3455 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K `  { X } )  C_  U
)
36 ssfi 7810 . . . . . . . . . . . . 13  |-  ( ( U  e.  Fin  /\  ( K `  { X } )  C_  U
)  ->  ( K `  { X } )  e.  Fin )
3715, 35, 36syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( K `  { X } )  e.  Fin )
38 pgpfac.x . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  U )
3938, 34eleqtrd 2551 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  e.  ( Base `  H ) )
4031mrcsncl 15596 . . . . . . . . . . . . . . . 16  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
4130, 39, 40syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
4221subg0cl 16903 . . . . . . . . . . . . . . 15  |-  ( ( K `  { X } )  e.  (SubGrp `  H )  ->  .0.  e.  ( K `  { X } ) )
4341, 42syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  .0.  e.  ( K `
 { X }
) )
4443snssd 4108 . . . . . . . . . . . . 13  |-  ( ph  ->  {  .0.  }  C_  ( K `  { X } ) )
4539snssd 4108 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { X }  C_  ( Base `  H )
)
4630, 31, 45mrcssidd 15609 . . . . . . . . . . . . . 14  |-  ( ph  ->  { X }  C_  ( K `  { X } ) )
47 snssg 4096 . . . . . . . . . . . . . . 15  |-  ( X  e.  U  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4838, 47syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
4946, 48mpbird 240 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  ( K `
 { X }
) )
50 pgpfac.oe . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( O `  X
)  =  E )
51 pgpfac.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E  =/=  1 )
5250, 51eqnetrd 2710 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  X
)  =/=  1 )
53 pgpfac.o . . . . . . . . . . . . . . . . . 18  |-  O  =  ( od `  H
)
5453, 21od1 17288 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  ( O `  .0.  )  =  1 )
551, 26, 543syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( O `  .0.  )  =  1 )
56 elsni 3985 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  {  .0.  }  ->  X  =  .0.  )
5756fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  {  .0.  }  ->  ( O `  X
)  =  ( O `
 .0.  ) )
5857eqeq1d 2473 . . . . . . . . . . . . . . . 16  |-  ( X  e.  {  .0.  }  ->  ( ( O `  X )  =  1  <-> 
( O `  .0.  )  =  1 ) )
5955, 58syl5ibrcom 230 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  {  .0.  }  ->  ( O `  X )  =  1 ) )
6059necon3ad 2656 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( O `  X )  =/=  1  ->  -.  X  e.  {  .0.  } ) )
6152, 60mpd 15 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  {  .0.  } )
6244, 49, 61ssnelpssd 3531 . . . . . . . . . . . 12  |-  ( ph  ->  {  .0.  }  C.  ( K `  { X } ) )
63 php3 7776 . . . . . . . . . . . 12  |-  ( ( ( K `  { X } )  e.  Fin  /\ 
{  .0.  }  C.  ( K `  { X } ) )  ->  {  .0.  }  ~<  ( K `  { X } ) )
6437, 62, 63syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  {  .0.  }  ~<  ( K `  { X } ) )
65 snfi 7668 . . . . . . . . . . . 12  |-  {  .0.  }  e.  Fin
66 hashsdom 12598 . . . . . . . . . . . 12  |-  ( ( {  .0.  }  e.  Fin  /\  ( K `  { X } )  e. 
Fin )  ->  (
( # `  {  .0.  } )  <  ( # `  ( K `  { X } ) )  <->  {  .0.  } 
~<  ( K `  { X } ) ) )
6765, 37, 66sylancr 676 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) )  <->  {  .0.  }  ~<  ( K `  { X } ) ) )
6864, 67mpbird 240 . . . . . . . . . 10  |-  ( ph  ->  ( # `  {  .0.  } )  <  ( # `
 ( K `  { X } ) ) )
6925, 68syl5eqbrr 4430 . . . . . . . . 9  |-  ( ph  ->  1  <  ( # `  ( K `  { X } ) ) )
70 1red 9676 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
71 hashcl 12576 . . . . . . . . . . . 12  |-  ( ( K `  { X } )  e.  Fin  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7237, 71syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e. 
NN0 )
7372nn0red 10950 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( K `  { X } ) )  e.  RR )
7421subg0cl 16903 . . . . . . . . . . . . 13  |-  ( W  e.  (SubGrp `  H
)  ->  .0.  e.  W )
75 ne0i 3728 . . . . . . . . . . . . 13  |-  (  .0. 
e.  W  ->  W  =/=  (/) )
761, 74, 753syl 18 . . . . . . . . . . . 12  |-  ( ph  ->  W  =/=  (/) )
77 hashnncl 12585 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
7817, 77syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
7976, 78mpbird 240 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  NN )
8079nngt0d 10675 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( # `  W ) )
81 ltmul1 10477 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( # `  ( K `
 { X }
) )  e.  RR  /\  ( ( # `  W
)  e.  RR  /\  0  <  ( # `  W
) ) )  -> 
( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8270, 73, 20, 80, 81syl112anc 1296 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  ( # `
 ( K `  { X } ) )  <-> 
( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) ) )
8369, 82mpbid 215 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  <  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) ) )
8420recnd 9687 . . . . . . . . 9  |-  ( ph  ->  ( # `  W
)  e.  CC )
8584mulid2d 9679 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  ( # `
 W ) )  =  ( # `  W
) )
86 pgpfac.l . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  H )
87 eqid 2471 . . . . . . . . . 10  |-  (Cntz `  H )  =  (Cntz `  H )
88 pgpfac.i . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
89 pgpfac.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  Abel )
903subgabl 17554 . . . . . . . . . . . 12  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
9189, 2, 90syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  Abel )
9287, 91, 41, 1ablcntzd 17573 . . . . . . . . . 10  |-  ( ph  ->  ( K `  { X } )  C_  (
(Cntz `  H ) `  W ) )
9386, 21, 87, 41, 1, 88, 92, 37, 17lsmhash 17433 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( (
# `  ( K `  { X } ) )  x.  ( # `  W ) ) )
94 pgpfac.s . . . . . . . . . 10  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
9594fveq2d 5883 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
( K `  { X } )  .(+)  W ) )  =  ( # `  U ) )
9693, 95eqtr3d 2507 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( K `  { X } ) )  x.  ( # `  W
) )  =  (
# `  U )
)
9783, 85, 963brtr3d 4425 . . . . . . 7  |-  ( ph  ->  ( # `  W
)  <  ( # `  U
) )
9820, 97ltned 9788 . . . . . 6  |-  ( ph  ->  ( # `  W
)  =/=  ( # `  U ) )
99 fveq2 5879 . . . . . . 7  |-  ( W  =  U  ->  ( # `
 W )  =  ( # `  U
) )
10099necon3i 2675 . . . . . 6  |-  ( (
# `  W )  =/=  ( # `  U
)  ->  W  =/=  U )
10198, 100syl 17 . . . . 5  |-  ( ph  ->  W  =/=  U )
102 df-pss 3406 . . . . 5  |-  ( W 
C.  U  <->  ( W  C_  U  /\  W  =/= 
U ) )
1039, 101, 102sylanbrc 677 . . . 4  |-  ( ph  ->  W  C.  U )
104 psseq1 3506 . . . . . 6  |-  ( t  =  W  ->  (
t  C.  U  <->  W  C.  U
) )
105 eqeq2 2482 . . . . . . . 8  |-  ( t  =  W  ->  (
( G DProd  s )  =  t  <->  ( G DProd  s
)  =  W ) )
106105anbi2d 718 . . . . . . 7  |-  ( t  =  W  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  t )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  W ) ) )
107106rexbidv 2892 . . . . . 6  |-  ( t  =  W  ->  ( 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
)  =  W ) ) )
108104, 107imbi12d 327 . . . . 5  |-  ( t  =  W  ->  (
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  <-> 
( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
109108rspcv 3132 . . . 4  |-  ( W  e.  (SubGrp `  G
)  ->  ( A. t  e.  (SubGrp `  G
) ( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) )  ->  ( W  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) ) ) )
1107, 8, 103, 109syl3c 62 . . 3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  W ) )
111 breq2 4399 . . . . 5  |-  ( s  =  a  ->  ( G dom DProd  s  <->  G dom DProd  a ) )
112 oveq2 6316 . . . . . 6  |-  ( s  =  a  ->  ( G DProd  s )  =  ( G DProd  a ) )
113112eqeq1d 2473 . . . . 5  |-  ( s  =  a  ->  (
( G DProd  s )  =  W  <->  ( G DProd  a
)  =  W ) )
114111, 113anbi12d 725 . . . 4  |-  ( s  =  a  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  ( G dom DProd  a  /\  ( G DProd 
a )  =  W ) ) )
115114cbvrexv 3006 . . 3  |-  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  W )  <->  E. a  e. Word  C ( G dom DProd  a  /\  ( G DProd  a
)  =  W ) )
116110, 115sylib 201 . 2  |-  ( ph  ->  E. a  e. Word  C
( G dom DProd  a  /\  ( G DProd  a )  =  W ) )
117 pgpfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
11889adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G  e.  Abel )
119 pgpfac.p . . . 4  |-  ( ph  ->  P pGrp  G )
120119adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  P pGrp  G )
12110adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  B  e.  Fin )
1222adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  U  e.  (SubGrp `  G
) )
1238adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
124 pgpfac.e . . 3  |-  E  =  (gEx `  H )
12551adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E  =/=  1 )
12638adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  X  e.  U )
12750adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( O `  X
)  =  E )
1281adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  W  e.  (SubGrp `  H
) )
12988adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
13094adantr 472 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( ( K `  { X } )  .(+)  W )  =  U )
131 simprl 772 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
a  e. Word  C )
132 simprrl 782 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  G dom DProd  a )
133 simprrr 783 . . 3  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  -> 
( G DProd  a )  =  W )
134 eqid 2471 . . 3  |-  ( a ++ 
<" ( K `  { X } ) "> )  =  ( a ++  <" ( K `
 { X }
) "> )
13511, 117, 118, 120, 121, 122, 123, 3, 31, 53, 124, 21, 86, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134pgpfaclem1 17792 . 2  |-  ( (
ph  /\  ( a  e. Word  C  /\  ( G dom DProd  a  /\  ( G DProd  a )  =  W ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
136116, 135rexlimddv 2875 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390    C. wpss 3391   (/)c0 3722   {csn 3959   class class class wbr 4395   dom cdm 4839   ran crn 4840   ` cfv 5589  (class class class)co 6308    ~< csdm 7586   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693   NNcn 10631   NN0cn0 10893   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706   Basecbs 15199   ↾s cress 15200   0gc0g 15416  Moorecmre 15566  mrClscmrc 15567  ACScacs 15569   Grpcgrp 16747  SubGrpcsubg 16889  Cntzccntz 17047   odcod 17243  gExcgex 17245   pGrp cpgp 17247   LSSumclsm 17364   Abelcabl 17509  CycGrpccyg 17590   DProd cdprd 17703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-od 17250  df-pgp 17254  df-lsm 17366  df-pj1 17367  df-cmn 17510  df-abl 17511  df-cyg 17591  df-dprd 17705
This theorem is referenced by:  pgpfaclem3  17794
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