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Theorem pgpfaclem1 15332
Description: Lemma for pgpfac 15335. (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 )
pgpfac.2  |-  ( ph  ->  S  e. Word  C )
pgpfac.4  |-  ( ph  ->  G dom DProd  S )
pgpfac.5  |-  ( ph  ->  ( G DProd  S )  =  W )
pgpfac.t  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
Assertion
Ref Expression
pgpfaclem1  |-  ( 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    T, s
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)    .(+) ( t, s, r)    S( t, s, r)    T( t, 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 pgpfaclem1
StepHypRef Expression
1 pgpfac.t . . 3  |-  T  =  ( S concat  <" ( K `  { X } ) "> )
2 pgpfac.2 . . 3  |-  ( ph  ->  S  e. Word  C )
3 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 pgpfac.h . . . . . . . . . 10  |-  H  =  ( Gs  U )
54subggrp 14640 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  H  e.  Grp )
63, 5syl 15 . . . . . . . 8  |-  ( ph  ->  H  e.  Grp )
7 eqid 2296 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
87subgacs 14668 . . . . . . . 8  |-  ( H  e.  Grp  ->  (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) ) )
9 acsmre 13570 . . . . . . . 8  |-  ( (SubGrp `  H )  e.  (ACS
`  ( Base `  H
) )  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H
) ) )
106, 8, 93syl 18 . . . . . . 7  |-  ( ph  ->  (SubGrp `  H )  e.  (Moore `  ( Base `  H ) ) )
11 pgpfac.x . . . . . . . 8  |-  ( ph  ->  X  e.  U )
124subgbas 14641 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  H )
)
133, 12syl 15 . . . . . . . 8  |-  ( ph  ->  U  =  ( Base `  H ) )
1411, 13eleqtrd 2372 . . . . . . 7  |-  ( ph  ->  X  e.  ( Base `  H ) )
15 pgpfac.k . . . . . . . 8  |-  K  =  (mrCls `  (SubGrp `  H
) )
1615mrcsncl 13530 . . . . . . 7  |-  ( ( (SubGrp `  H )  e.  (Moore `  ( Base `  H ) )  /\  X  e.  ( Base `  H ) )  -> 
( K `  { X } )  e.  (SubGrp `  H ) )
1710, 14, 16syl2anc 642 . . . . . 6  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  H ) )
184subsubg 14656 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
193, 18syl 15 . . . . . 6  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  H )  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) ) )
2017, 19mpbid 201 . . . . 5  |-  ( ph  ->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
) )
2120simpld 445 . . . 4  |-  ( ph  ->  ( K `  { X } )  e.  (SubGrp `  G ) )
224oveq1i 5884 . . . . . . 7  |-  ( Hs  ( K `  { X } ) )  =  ( ( Gs  U )s  ( K `  { X } ) )
2320simprd 449 . . . . . . . 8  |-  ( ph  ->  ( K `  { X } )  C_  U
)
24 ressabs 13222 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U
)  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
253, 23, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( Gs  U )s  ( K `  { X } ) )  =  ( Gs  ( K `  { X } ) ) )
2622, 25syl5eq 2340 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  =  ( Gs  ( K `
 { X }
) ) )
277, 15cycsubgcyg2 15204 . . . . . . 7  |-  ( ( H  e.  Grp  /\  X  e.  ( Base `  H ) )  -> 
( Hs  ( K `  { X } ) )  e. CycGrp )
286, 14, 27syl2anc 642 . . . . . 6  |-  ( ph  ->  ( Hs  ( K `  { X } ) )  e. CycGrp )
2926, 28eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e. CycGrp )
30 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
31 pgpprm 14920 . . . . . . 7  |-  ( P pGrp 
G  ->  P  e.  Prime )
3230, 31syl 15 . . . . . 6  |-  ( ph  ->  P  e.  Prime )
33 subgpgp 14924 . . . . . . 7  |-  ( ( P pGrp  G  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  ( K `  { X } ) ) )
3430, 21, 33syl2anc 642 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  ( K `
 { X }
) ) )
35 brelrng 4924 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( Gs  ( K `  { X } ) )  e. CycGrp  /\  P pGrp  ( Gs  ( K `  { X } ) ) )  ->  ( Gs  ( K `
 { X }
) )  e.  ran pGrp  )
3632, 29, 34, 35syl3anc 1182 . . . . 5  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  ran pGrp  )
37 elin 3371 . . . . 5  |-  ( ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( ( Gs  ( K `  { X } ) )  e. CycGrp  /\  ( Gs  ( K `  { X } ) )  e.  ran pGrp  ) )
3829, 36, 37sylanbrc 645 . . . 4  |-  ( ph  ->  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) )
39 oveq2 5882 . . . . . 6  |-  ( r  =  ( K `  { X } )  -> 
( Gs  r )  =  ( Gs  ( K `  { X } ) ) )
4039eleq1d 2362 . . . . 5  |-  ( r  =  ( K `  { X } )  -> 
( ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <-> 
( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
41 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
4240, 41elrab2 2938 . . . 4  |-  ( ( K `  { X } )  e.  C  <->  ( ( K `  { X } )  e.  (SubGrp `  G )  /\  ( Gs  ( K `  { X } ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
4321, 38, 42sylanbrc 645 . . 3  |-  ( ph  ->  ( K `  { X } )  e.  C
)
441, 2, 43cats1cld 11521 . 2  |-  ( ph  ->  T  e. Word  C )
45 wrdf 11435 . . . . 5  |-  ( T  e. Word  C  ->  T : ( 0..^ (
# `  T )
) --> C )
4644, 45syl 15 . . . 4  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> C )
47 ssrab2 3271 . . . . 5  |-  { r  e.  (SubGrp `  G
)  |  ( Gs  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  G )
4841, 47eqsstri 3221 . . . 4  |-  C  C_  (SubGrp `  G )
49 fss 5413 . . . 4  |-  ( ( T : ( 0..^ ( # `  T
) ) --> C  /\  C  C_  (SubGrp `  G
) )  ->  T : ( 0..^ (
# `  T )
) --> (SubGrp `  G )
)
5046, 48, 49sylancl 643 . . 3  |-  ( ph  ->  T : ( 0..^ ( # `  T
) ) --> (SubGrp `  G ) )
51 fzodisj 10916 . . . 4  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  1 ) ) )  =  (/)
52 lencl 11437 . . . . . . . 8  |-  ( S  e. Word  C  ->  ( # `
 S )  e. 
NN0 )
532, 52syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  S
)  e.  NN0 )
5453nn0zd 10131 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ZZ )
55 fzosn 10928 . . . . . 6  |-  ( (
# `  S )  e.  ZZ  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) )  =  { (
# `  S ) } )
5654, 55syl 15 . . . . 5  |-  ( ph  ->  ( ( # `  S
)..^ ( ( # `  S )  +  1 ) )  =  {
( # `  S ) } )
5756ineq2d 3383 . . . 4  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  ( ( # `
 S )..^ ( ( # `  S
)  +  1 ) ) )  =  ( ( 0..^ ( # `  S ) )  i^i 
{ ( # `  S
) } ) )
5851, 57syl5reqr 2343 . . 3  |-  ( ph  ->  ( ( 0..^ (
# `  S )
)  i^i  { ( # `
 S ) } )  =  (/) )
591fveq2i 5544 . . . . . . 7  |-  ( # `  T )  =  (
# `  ( S concat  <" ( K `  { X } ) "> ) )
6043s1cld 11458 . . . . . . . 8  |-  ( ph  ->  <" ( K `
 { X }
) ">  e. Word  C )
61 ccatlen 11446 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C )  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
622, 60, 61syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( # `  ( S concat  <" ( K `
 { X }
) "> )
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
6359, 62syl5eq 2340 . . . . . 6  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) ) )
64 s1len 11460 . . . . . . 7  |-  ( # `  <" ( K `
 { X }
) "> )  =  1
6564oveq2i 5885 . . . . . 6  |-  ( (
# `  S )  +  ( # `  <" ( K `  { X } ) "> ) )  =  ( ( # `  S
)  +  1 )
6663, 65syl6eq 2344 . . . . 5  |-  ( ph  ->  ( # `  T
)  =  ( (
# `  S )  +  1 ) )
6766oveq2d 5890 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( 0..^ ( (
# `  S )  +  1 ) ) )
68 nn0uz 10278 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
6953, 68syl6eleq 2386 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
70 fzosplitsn 10936 . . . . 5  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7169, 70syl 15 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  S )  +  1 ) )  =  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
7267, 71eqtrd 2328 . . 3  |-  ( ph  ->  ( 0..^ ( # `  T ) )  =  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } ) )
73 eqid 2296 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
74 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
75 pgpfac.4 . . . 4  |-  ( ph  ->  G dom DProd  S )
76 cats1un 11492 . . . . . . . 8  |-  ( ( S  e. Word  C  /\  ( K `  { X } )  e.  C
)  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
772, 43, 76syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( S concat  <" ( K `  { X } ) "> )  =  ( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } ) )
781, 77syl5eq 2340 . . . . . 6  |-  ( ph  ->  T  =  ( S  u.  { <. ( # `
 S ) ,  ( K `  { X } ) >. } ) )
7978reseq1d 4970 . . . . 5  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) ) )
80 wrdf 11435 . . . . . . 7  |-  ( S  e. Word  C  ->  S : ( 0..^ (
# `  S )
) --> C )
81 ffn 5405 . . . . . . 7  |-  ( S : ( 0..^ (
# `  S )
) --> C  ->  S  Fn  ( 0..^ ( # `  S ) ) )
822, 80, 813syl 18 . . . . . 6  |-  ( ph  ->  S  Fn  ( 0..^ ( # `  S
) ) )
83 fzonel 10903 . . . . . 6  |-  -.  ( # `
 S )  e.  ( 0..^ ( # `  S ) )
84 fsnunres 5737 . . . . . 6  |-  ( ( S  Fn  ( 0..^ ( # `  S
) )  /\  -.  ( # `  S )  e.  ( 0..^ (
# `  S )
) )  ->  (
( S  u.  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8582, 83, 84sylancl 643 . . . . 5  |-  ( ph  ->  ( ( S  u.  {
<. ( # `  S
) ,  ( K `
 { X }
) >. } )  |`  ( 0..^ ( # `  S
) ) )  =  S )
8679, 85eqtrd 2328 . . . 4  |-  ( ph  ->  ( T  |`  (
0..^ ( # `  S
) ) )  =  S )
8775, 86breqtrrd 4065 . . 3  |-  ( ph  ->  G dom DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )
88 fvex 5555 . . . . . 6  |-  ( # `  S )  e.  _V
89 dprdsn 15287 . . . . . 6  |-  ( ( ( # `  S
)  e.  _V  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9088, 21, 89sylancr 644 . . . . 5  |-  ( ph  ->  ( G dom DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. }  /\  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) ) )
9190simpld 445 . . . 4  |-  ( ph  ->  G dom DProd  { <. ( # `
 S ) ,  ( K `  { X } ) >. } )
92 ffn 5405 . . . . . . 7  |-  ( T : ( 0..^ (
# `  T )
) --> C  ->  T  Fn  ( 0..^ ( # `  T ) ) )
9344, 45, 923syl 18 . . . . . 6  |-  ( ph  ->  T  Fn  ( 0..^ ( # `  T
) ) )
94 ssun2 3352 . . . . . . . 8  |-  { (
# `  S ) }  C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } )
9588snss 3761 . . . . . . . 8  |-  ( (
# `  S )  e.  ( ( 0..^ (
# `  S )
)  u.  { (
# `  S ) } )  <->  { ( # `
 S ) } 
C_  ( ( 0..^ ( # `  S
) )  u.  {
( # `  S ) } ) )
9694, 95mpbir 200 . . . . . . 7  |-  ( # `  S )  e.  ( ( 0..^ ( # `  S ) )  u. 
{ ( # `  S
) } )
9796, 72syl5eleqr 2383 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ( 0..^ ( # `  T
) ) )
98 fnressn 5721 . . . . . 6  |-  ( ( T  Fn  ( 0..^ ( # `  T
) )  /\  ( # `
 S )  e.  ( 0..^ ( # `  T ) ) )  ->  ( T  |`  { ( # `  S
) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
9993, 97, 98syl2anc 642 . . . . 5  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. } )
1001fveq1i 5542 . . . . . . . . 9  |-  ( T `
 ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )
10153nn0cnd 10036 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  S
)  e.  CC )
102101addid2d 9029 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  +  (
# `  S )
)  =  ( # `  S ) )
103102eqcomd 2301 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  =  ( 0  +  ( # `  S
) ) )
104103fveq2d 5545 . . . . . . . . 9  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( # `  S
) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) ) )
105100, 104syl5eq 2340 . . . . . . . 8  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( ( S concat  <" ( K `  { X } ) "> ) `  (
0  +  ( # `  S ) ) ) )
106 1nn 9773 . . . . . . . . . . . 12  |-  1  e.  NN
107106a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  NN )
10864, 107syl5eqel 2380 . . . . . . . . . 10  |-  ( ph  ->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
109 lbfzo0 10919 . . . . . . . . . 10  |-  ( 0  e.  ( 0..^ (
# `  <" ( K `  { X } ) "> ) )  <->  ( # `  <" ( K `  { X } ) "> )  e.  NN )
110108, 109sylibr 203 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )
111 ccatval3 11449 . . . . . . . . 9  |-  ( ( S  e. Word  C  /\  <" ( K `  { X } ) ">  e. Word  C  /\  0  e.  ( 0..^ ( # `  <" ( K `  { X } ) "> ) ) )  -> 
( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
1122, 60, 110, 111syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( S concat  <" ( K `  { X } ) "> ) `  ( 0  +  ( # `  S
) ) )  =  ( <" ( K `  { X } ) "> `  0 ) )
113 fvex 5555 . . . . . . . . 9  |-  ( K `
 { X }
)  e.  _V
114 s1fv 11462 . . . . . . . . 9  |-  ( ( K `  { X } )  e.  _V  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
115113, 114mp1i 11 . . . . . . . 8  |-  ( ph  ->  ( <" ( K `  { X } ) "> `  0 )  =  ( K `  { X } ) )
116105, 112, 1153eqtrd 2332 . . . . . . 7  |-  ( ph  ->  ( T `  ( # `
 S ) )  =  ( K `  { X } ) )
117116opeq2d 3819 . . . . . 6  |-  ( ph  -> 
<. ( # `  S
) ,  ( T `
 ( # `  S
) ) >.  =  <. (
# `  S ) ,  ( K `  { X } ) >.
)
118117sneqd 3666 . . . . 5  |-  ( ph  ->  { <. ( # `  S
) ,  ( T `
 ( # `  S
) ) >. }  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
11999, 118eqtrd 2328 . . . 4  |-  ( ph  ->  ( T  |`  { (
# `  S ) } )  =  { <. ( # `  S
) ,  ( K `
 { X }
) >. } )
12091, 119breqtrrd 4065 . . 3  |-  ( ph  ->  G dom DProd  ( T  |` 
{ ( # `  S
) } ) )
121 pgpfac.g . . . 4  |-  ( ph  ->  G  e.  Abel )
122 dprdsubg 15275 . . . . 5  |-  ( G dom DProd  ( T  |`  ( 0..^ ( # `  S
) ) )  -> 
( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
12387, 122syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  e.  (SubGrp `  G ) )
124 dprdsubg 15275 . . . . 5  |-  ( G dom DProd  ( T  |`  { ( # `  S
) } )  -> 
( G DProd  ( T  |` 
{ ( # `  S
) } ) )  e.  (SubGrp `  G
) )
125120, 124syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  e.  (SubGrp `  G
) )
12673, 121, 123, 125ablcntzd 15165 . . 3  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( T  |`  { (
# `  S ) } ) ) ) )
127 pgpfac.i . . . 4  |-  ( ph  ->  ( ( K `  { X } )  i^i 
W )  =  {  .0.  } )
12886oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  ( G DProd 
S ) )
129 pgpfac.5 . . . . . . 7  |-  ( ph  ->  ( G DProd  S )  =  W )
130128, 129eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  ( 0..^ ( # `  S ) ) ) )  =  W )
131119oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } ) )
13290simprd 449 . . . . . . 7  |-  ( ph  ->  ( G DProd  { <. (
# `  S ) ,  ( K `  { X } ) >. } )  =  ( K `  { X } ) )
133131, 132eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( G DProd  ( T  |`  { ( # `  S
) } ) )  =  ( K `  { X } ) )
134130, 133ineq12d 3384 . . . . 5  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W  i^i  ( K `  { X } ) ) )
135 incom 3374 . . . . 5  |-  ( W  i^i  ( K `  { X } ) )  =  ( ( K `
 { X }
)  i^i  W )
136134, 135syl6eq 2344 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( ( K `  { X } )  i^i  W
) )
1374, 74subg0 14643 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
1383, 137syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
139 pgpfac.0 . . . . . 6  |-  .0.  =  ( 0g `  H )
140138, 139syl6eqr 2346 . . . . 5  |-  ( ph  ->  ( 0g `  G
)  =  .0.  )
141140sneqd 3666 . . . 4  |-  ( ph  ->  { ( 0g `  G ) }  =  {  .0.  } )
142127, 136, 1413eqtr4d 2338 . . 3  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) )  i^i  ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  { ( 0g `  G ) } )
14350, 58, 72, 73, 74, 87, 120, 126, 142dmdprdsplit2 15297 . 2  |-  ( ph  ->  G dom DProd  T )
144 eqid 2296 . . . . 5  |-  ( LSSum `  G )  =  (
LSSum `  G )
14550, 58, 72, 144, 143dprdsplit 15299 . . . 4  |-  ( ph  ->  ( G DProd  T )  =  ( ( G DProd 
( T  |`  (
0..^ ( # `  S
) ) ) ) ( LSSum `  G )
( G DProd  ( T  |` 
{ ( # `  S
) } ) ) ) )
146130, 133oveq12d 5892 . . . 4  |-  ( ph  ->  ( ( G DProd  ( T  |`  ( 0..^ (
# `  S )
) ) ) (
LSSum `  G ) ( G DProd  ( T  |`  { ( # `  S
) } ) ) )  =  ( W ( LSSum `  G )
( K `  { X } ) ) )
147130, 123eqeltrrd 2371 . . . . 5  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
148144lsmcom 15166 . . . . 5  |-  ( ( G  e.  Abel  /\  W  e.  (SubGrp `  G )  /\  ( K `  { X } )  e.  (SubGrp `  G ) )  -> 
( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
149121, 147, 21, 148syl3anc 1182 . . . 4  |-  ( ph  ->  ( W ( LSSum `  G ) ( K `
 { X }
) )  =  ( ( K `  { X } ) ( LSSum `  G ) W ) )
150145, 146, 1493eqtrd 2332 . . 3  |-  ( ph  ->  ( G DProd  T )  =  ( ( K `
 { X }
) ( LSSum `  G
) W ) )
151 pgpfac.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  H ) )
1527subgss 14638 . . . . . 6  |-  ( W  e.  (SubGrp `  H
)  ->  W  C_  ( Base `  H ) )
153151, 152syl 15 . . . . 5  |-  ( ph  ->  W  C_  ( Base `  H ) )
154153, 13sseqtr4d 3228 . . . 4  |-  ( ph  ->  W  C_  U )
155 pgpfac.l . . . . 5  |-  .(+)  =  (
LSSum `  H )
1564, 144, 155subglsm 14998 . . . 4  |-  ( ( U  e.  (SubGrp `  G )  /\  ( K `  { X } )  C_  U  /\  W  C_  U )  ->  ( ( K `
 { X }
) ( LSSum `  G
) W )  =  ( ( K `  { X } )  .(+)  W ) )
1573, 23, 154, 156syl3anc 1182 . . 3  |-  ( ph  ->  ( ( K `  { X } ) (
LSSum `  G ) W )  =  ( ( K `  { X } )  .(+)  W ) )
158 pgpfac.s . . 3  |-  ( ph  ->  ( ( K `  { X } )  .(+)  W )  =  U )
159150, 157, 1583eqtrd 2332 . 2  |-  ( ph  ->  ( G DProd  T )  =  U )
160 breq2 4043 . . . 4  |-  ( s  =  T  ->  ( G dom DProd  s  <->  G dom DProd  T ) )
161 oveq2 5882 . . . . 5  |-  ( s  =  T  ->  ( G DProd  s )  =  ( G DProd  T ) )
162161eqeq1d 2304 . . . 4  |-  ( s  =  T  ->  (
( G DProd  s )  =  U  <->  ( G DProd  T
)  =  U ) )
163160, 162anbi12d 691 . . 3  |-  ( s  =  T  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  T  /\  ( G DProd 
T )  =  U ) ) )
164163rspcev 2897 . 2  |-  ( ( T  e. Word  C  /\  ( G dom DProd  T  /\  ( G DProd  T )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
16544, 143, 159, 164syl12anc 1180 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165    C. wpss 3166   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039   dom cdm 4705   ran crn 4706    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   0cc0 8753   1c1 8754    + caddc 8756   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   <"cs1 11421   Primecprime 12774   Basecbs 13164   ↾s cress 13165   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   odcod 14856  gExcgex 14857   pGrp cpgp 14858   LSSumclsm 14961   Abelcabel 15106  CycGrpccyg 15180   DProd cdprd 15247
This theorem is referenced by:  pgpfaclem2  15333
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
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-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-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  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-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  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-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-od 14860  df-pgp 14862  df-lsm 14963  df-cmn 15107  df-abl 15108  df-cyg 15181  df-dprd 15249
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