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Theorem pgpfaclem3 15641
Description: Lemma for pgpfac 15642. (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 ) ) )
Assertion
Ref Expression
pgpfaclem3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    ph, t    B, s, t    U, r, s, t
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)

Proof of Theorem pgpfaclem3
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd0 11732 . . 3  |-  (/)  e. Word  C
2 pgpfac.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
3 ablgrp 15417 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 eqid 2436 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
54dprd0 15589 . . . . . 6  |-  ( G  e.  Grp  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
62, 3, 53syl 19 . . . . 5  |-  ( ph  ->  ( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  {
( 0g `  G
) } ) )
76adantr 452 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
8 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
94subg0cl 14952 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  U
)
108, 9syl 16 . . . . . . . 8  |-  ( ph  ->  ( 0g `  G
)  e.  U )
1110adantr 452 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( 0g `  G )  e.  U )
12 eqid 2436 . . . . . . . . . . 11  |-  ( Gs  U )  =  ( Gs  U )
1312subgbas 14948 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  ( Gs  U
) ) )
148, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  ( Gs  U ) ) )
1514adantr 452 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  ( Base `  ( Gs  U ) ) )
1612subggrp 14947 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  ( Gs  U
)  e.  Grp )
178, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Gs  U )  e.  Grp )
18 grpmnd 14817 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Grp  ->  ( Gs  U )  e.  Mnd )
19 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  ( Gs  U ) )  =  ( Base `  ( Gs  U ) )
20 eqid 2436 . . . . . . . . . . 11  |-  (gEx `  ( Gs  U ) )  =  (gEx `  ( Gs  U
) )
2119, 20gex1 15225 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Mnd  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2217, 18, 213syl 19 . . . . . . . . 9  |-  ( ph  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2322biimpa 471 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( Base `  ( Gs  U ) )  ~~  1o )
2415, 23eqbrtrd 4232 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  ~~  1o )
25 en1eqsn 7338 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  U  /\  U  ~~  1o )  ->  U  =  { ( 0g `  G ) } )
2611, 24, 25syl2anc 643 . . . . . 6  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  { ( 0g `  G ) } )
2726eqeq2d 2447 . . . . 5  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G DProd  (/) )  =  U  <->  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
2827anbi2d 685 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) ) )
297, 28mpbird 224 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )
30 breq2 4216 . . . . 5  |-  ( s  =  (/)  ->  ( G dom DProd  s  <->  G dom DProd  (/) ) )
31 oveq2 6089 . . . . . 6  |-  ( s  =  (/)  ->  ( G DProd 
s )  =  ( G DProd  (/) ) )
3231eqeq1d 2444 . . . . 5  |-  ( s  =  (/)  ->  ( ( G DProd  s )  =  U  <->  ( G DProd  (/) )  =  U ) )
3330, 32anbi12d 692 . . . 4  |-  ( s  =  (/)  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) ) )
3433rspcev 3052 . . 3  |-  ( (
(/)  e. Word  C  /\  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
351, 29, 34sylancr 645 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
3612subgabl 15455 . . . . . 6  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  ( Gs  U
)  e.  Abel )
372, 8, 36syl2anc 643 . . . . 5  |-  ( ph  ->  ( Gs  U )  e.  Abel )
38 pgpfac.f . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
39 pgpfac.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
4039subgss 14945 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
418, 40syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  B )
42 ssfi 7329 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
4338, 41, 42syl2anc 643 . . . . . . 7  |-  ( ph  ->  U  e.  Fin )
4414, 43eqeltrrd 2511 . . . . . 6  |-  ( ph  ->  ( Base `  ( Gs  U ) )  e. 
Fin )
4519, 20gexcl2 15223 . . . . . 6  |-  ( ( ( Gs  U )  e.  Grp  /\  ( Base `  ( Gs  U ) )  e. 
Fin )  ->  (gEx `  ( Gs  U ) )  e.  NN )
4617, 44, 45syl2anc 643 . . . . 5  |-  ( ph  ->  (gEx `  ( Gs  U
) )  e.  NN )
47 eqid 2436 . . . . . 6  |-  ( od
`  ( Gs  U ) )  =  ( od
`  ( Gs  U ) )
4819, 20, 47gexex 15468 . . . . 5  |-  ( ( ( Gs  U )  e.  Abel  /\  (gEx `  ( Gs  U
) )  e.  NN )  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
4937, 46, 48syl2anc 643 . . . 4  |-  ( ph  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
5049adantr 452 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. x  e.  ( Base `  ( Gs  U ) ) ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
51 eqid 2436 . . . . 5  |-  (mrCls `  (SubGrp `  ( Gs  U ) ) )  =  (mrCls `  (SubGrp `  ( Gs  U
) ) )
52 eqid 2436 . . . . 5  |-  ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  =  ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )
53 eqid 2436 . . . . 5  |-  ( 0g
`  ( Gs  U ) )  =  ( 0g
`  ( Gs  U ) )
54 eqid 2436 . . . . 5  |-  ( LSSum `  ( Gs  U ) )  =  ( LSSum `  ( Gs  U
) )
55 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
56 subgpgp 15231 . . . . . . 7  |-  ( ( P pGrp  G  /\  U  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  U ) )
5755, 8, 56syl2anc 643 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  U ) )
5857ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  P pGrp  ( Gs  U ) )
5937ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Gs  U )  e.  Abel )
6044ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Base `  ( Gs  U
) )  e.  Fin )
61 simprr 734 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
62 simprl 733 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
6351, 52, 19, 47, 20, 53, 54, 58, 59, 60, 61, 62pgpfac1 15638 . . . 4  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. w  e.  (SubGrp `  ( Gs  U ) ) ( ( ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )  i^i  w )  =  {
( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) )
64 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
652ad3antrrr 711 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  G  e.  Abel )
6655ad3antrrr 711 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  P pGrp  G )
6738ad3antrrr 711 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  B  e.  Fin )
688ad3antrrr 711 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  e.  (SubGrp `  G )
)
69 pgpfac.a . . . . . 6  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
7069ad3antrrr 711 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  A. t  e.  (SubGrp `  G )
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) ) )
71 simpllr 736 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (gEx `  ( Gs  U ) )  =/=  1 )
72 simplrl 737 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
7368, 13syl 16 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  =  ( Base `  ( Gs  U ) ) )
7472, 73eleqtrrd 2513 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  U )
75 simplrr 738 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
76 simprl 733 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  w  e.  (SubGrp `  ( Gs  U
) ) )
77 simprrl 741 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
)  i^i  w )  =  { ( 0g `  ( Gs  U ) ) } )
78 simprrr 742 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  ( Base `  ( Gs  U ) ) )
7978, 73eqtr4d 2471 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  U )
8039, 64, 65, 66, 67, 68, 70, 12, 51, 47, 20, 53, 54, 71, 74, 75, 76, 77, 79pgpfaclem2 15640 . . . 4  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8163, 80rexlimddv 2834 . . 3  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
8250, 81rexlimddv 2834 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8335, 82pm2.61dane 2682 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    i^i cin 3319    C_ wss 3320    C. wpss 3321   (/)c0 3628   {csn 3814   class class class wbr 4212   dom cdm 4878   ran crn 4879   ` cfv 5454  (class class class)co 6081   1oc1o 6717    ~~ cen 7106   Fincfn 7109   1c1 8991   NNcn 10000  Word cword 11717   Basecbs 13469   ↾s cress 13470   0gc0g 13723  mrClscmrc 13808   Mndcmnd 14684   Grpcgrp 14685  SubGrpcsubg 14938   odcod 15163  gExcgex 15164   pGrp cpgp 15165   LSSumclsm 15268   Abelcabel 15413  CycGrpccyg 15487   DProd cdprd 15554
This theorem is referenced by:  pgpfac  15642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-rpss 6522  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-eqg 14943  df-ghm 15004  df-gim 15046  df-ga 15067  df-cntz 15116  df-oppg 15142  df-od 15167  df-gex 15168  df-pgp 15169  df-lsm 15270  df-pj1 15271  df-cmn 15414  df-abl 15415  df-cyg 15488  df-dprd 15556
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