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Theorem dprd2dlem1 15292
Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dprd2d.1  |-  ( ph  ->  Rel  A )
dprd2d.2  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
dprd2d.3  |-  ( ph  ->  dom  A  C_  I
)
dprd2d.4  |-  ( (
ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
dprd2d.5  |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
dprd2d.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
dprd2d.6  |-  ( ph  ->  C  C_  I )
Assertion
Ref Expression
dprd2dlem1  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( G DProd  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) ) )
Distinct variable groups:    i, j, A    C, i    i, G, j    i, I    i, K    ph, i, j    S, i, j
Allowed substitution hints:    C( j)    I(
j)    K( j)

Proof of Theorem dprd2dlem1
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprd2d.5 . . . . . 6  |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
2 dprdgrp 15256 . . . . . 6  |-  ( G dom DProd  ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  ->  G  e.  Grp )
31, 2syl 15 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
54subgacs 14668 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
6 acsmre 13570 . . . . 5  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
73, 5, 63syl 18 . . . 4  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
8 dprd2d.2 . . . . . 6  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
9 ffun 5407 . . . . . 6  |-  ( S : A --> (SubGrp `  G )  ->  Fun  S )
10 funiunfv 5790 . . . . . 6  |-  ( Fun 
S  ->  U_ x  e.  ( A  |`  C ) ( S `  x
)  =  U. ( S " ( A  |`  C ) ) )
118, 9, 103syl 18 . . . . 5  |-  ( ph  ->  U_ x  e.  ( A  |`  C )
( S `  x
)  =  U. ( S " ( A  |`  C ) ) )
12 resss 4995 . . . . . . . . . 10  |-  ( A  |`  C )  C_  A
1312sseli 3189 . . . . . . . . 9  |-  ( x  e.  ( A  |`  C )  ->  x  e.  A )
14 dprd2d.1 . . . . . . . . . 10  |-  ( ph  ->  Rel  A )
15 dprd2d.3 . . . . . . . . . 10  |-  ( ph  ->  dom  A  C_  I
)
16 dprd2d.4 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
17 dprd2d.k . . . . . . . . . 10  |-  K  =  (mrCls `  (SubGrp `  G
) )
1814, 8, 15, 16, 1, 17dprd2dlem2 15291 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) ) )
1913, 18sylan2 460 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) ) )
20 1st2nd 6182 . . . . . . . . . . . . 13  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2114, 13, 20syl2an 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
22 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  e.  ( A  |`  C ) )
2321, 22eqeltrrd 2371 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  ( A  |`  C ) )
24 fvex 5555 . . . . . . . . . . . . 13  |-  ( 2nd `  x )  e.  _V
2524opelres 4976 . . . . . . . . . . . 12  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( A  |`  C )  <->  (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A  /\  ( 1st `  x
)  e.  C ) )
2625simprbi 450 . . . . . . . . . . 11  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( A  |`  C )  ->  ( 1st `  x
)  e.  C )
2723, 26syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( 1st `  x )  e.  C
)
28 ovex 5899 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) )  e.  _V
29 eqid 2296 . . . . . . . . . . 11  |-  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  =  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )
30 sneq 3664 . . . . . . . . . . . . . 14  |-  ( i  =  ( 1st `  x
)  ->  { i }  =  { ( 1st `  x ) } )
3130imaeq2d 5028 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( A " { i } )  =  ( A " { ( 1st `  x
) } ) )
32 oveq1 5881 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( i S j )  =  ( ( 1st `  x
) S j ) )
3331, 32mpteq12dv 4114 . . . . . . . . . . . 12  |-  ( i  =  ( 1st `  x
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )
3433oveq2d 5890 . . . . . . . . . . 11  |-  ( i  =  ( 1st `  x
)  ->  ( G DProd  ( j  e.  ( A
" { i } )  |->  ( i S j ) ) )  =  ( G DProd  (
j  e.  ( A
" { ( 1st `  x ) } ) 
|->  ( ( 1st `  x
) S j ) ) ) )
3529, 34elrnmpt1s 4943 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  e.  C  /\  ( G DProd  ( j  e.  ( A " {
( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )  e.  _V )  ->  ( G DProd  (
j  e.  ( A
" { ( 1st `  x ) } ) 
|->  ( ( 1st `  x
) S j ) ) )  e.  ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
3627, 28, 35sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( G DProd  ( j  e.  ( A
" { ( 1st `  x ) } ) 
|->  ( ( 1st `  x
) S j ) ) )  e.  ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
37 elssuni 3871 . . . . . . . . 9  |-  ( ( G DProd  ( j  e.  ( A " {
( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )  e.  ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) )  ->  ( G DProd  (
j  e.  ( A
" { ( 1st `  x ) } ) 
|->  ( ( 1st `  x
) S j ) ) )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
3836, 37syl 15 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( G DProd  ( j  e.  ( A
" { ( 1st `  x ) } ) 
|->  ( ( 1st `  x
) S j ) ) )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
3919, 38sstrd 3202 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
4039ralrimiva 2639 . . . . . 6  |-  ( ph  ->  A. x  e.  ( A  |`  C )
( S `  x
)  C_  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
41 iunss 3959 . . . . . 6  |-  ( U_ x  e.  ( A  |`  C ) ( S `
 x )  C_  U.
ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  <->  A. x  e.  ( A  |`  C )
( S `  x
)  C_  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
4240, 41sylibr 203 . . . . 5  |-  ( ph  ->  U_ x  e.  ( A  |`  C )
( S `  x
)  C_  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
4311, 42eqsstr3d 3226 . . . 4  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
44 dprd2d.6 . . . . . . . . . . . 12  |-  ( ph  ->  C  C_  I )
4544sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  C )  ->  i  e.  I )
4645, 16syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
47 ovex 5899 . . . . . . . . . . . 12  |-  ( i S j )  e. 
_V
48 eqid 2296 . . . . . . . . . . . 12  |-  ( j  e.  ( A " { i } ) 
|->  ( i S j ) )  =  ( j  e.  ( A
" { i } )  |->  ( i S j ) )
4947, 48dmmpti 5389 . . . . . . . . . . 11  |-  dom  (
j  e.  ( A
" { i } )  |->  ( i S j ) )  =  ( A " {
i } )
5049a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  dom  ( j  e.  ( A " { i } )  |->  ( i S j ) )  =  ( A " { i } ) )
51 imassrn 5041 . . . . . . . . . . . . . 14  |-  ( S
" ( A  |`  C ) )  C_  ran  S
52 frn 5411 . . . . . . . . . . . . . . . 16  |-  ( S : A --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
538, 52syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
54 mresspw 13510 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
557, 54syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
5653, 55sstrd 3202 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5751, 56syl5ss 3203 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S " ( A  |`  C ) ) 
C_  ~P ( Base `  G
) )
58 sspwuni 4003 . . . . . . . . . . . . 13  |-  ( ( S " ( A  |`  C ) )  C_  ~P ( Base `  G
)  <->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
5957, 58sylib 188 . . . . . . . . . . . 12  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
6017mrccl 13529 . . . . . . . . . . . 12  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( A  |`  C ) ) 
C_  ( Base `  G
) )  ->  ( K `  U. ( S
" ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
617, 59, 60syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
6261adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  ( K `  U. ( S
" ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
63 oveq2 5882 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
i S j )  =  ( i S k ) )
6463, 48, 47fvmpt3i 5621 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { i } )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
6564adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
66 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( i S k )  =  ( S `  <. i ,  k >. )
67 ffn 5405 . . . . . . . . . . . . . . . . 17  |-  ( S : A --> (SubGrp `  G )  ->  S  Fn  A )
688, 67syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  Fn  A )
6968ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  S  Fn  A
)
7012a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( A  |`  C )  C_  A
)
71 elrelimasn 5053 . . . . . . . . . . . . . . . . . . . 20  |-  ( Rel 
A  ->  ( k  e.  ( A " {
i } )  <->  i A
k ) )
7214, 71syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( k  e.  ( A " { i } )  <->  i A
k ) )
7372adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  C )  ->  (
k  e.  ( A
" { i } )  <->  i A k ) )
7473biimpa 470 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i A k )
75 df-br 4040 . . . . . . . . . . . . . . . . 17  |-  ( i A k  <->  <. i ,  k >.  e.  A
)
7674, 75sylib 188 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  A )
77 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i  e.  C
)
78 vex 2804 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
7978opelres 4976 . . . . . . . . . . . . . . . 16  |-  ( <.
i ,  k >.  e.  ( A  |`  C )  <-> 
( <. i ,  k
>.  e.  A  /\  i  e.  C ) )
8076, 77, 79sylanbrc 645 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  ( A  |`  C ) )
81 fnfvima 5772 . . . . . . . . . . . . . . 15  |-  ( ( S  Fn  A  /\  ( A  |`  C ) 
C_  A  /\  <. i ,  k >.  e.  ( A  |`  C )
)  ->  ( S `  <. i ,  k
>. )  e.  ( S " ( A  |`  C ) ) )
8269, 70, 80, 81syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( S `  <. i ,  k >.
)  e.  ( S
" ( A  |`  C ) ) )
8366, 82syl5eqel 2380 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  e.  ( S " ( A  |`  C ) ) )
84 elssuni 3871 . . . . . . . . . . . . 13  |-  ( ( i S k )  e.  ( S "
( A  |`  C ) )  ->  ( i S k )  C_  U. ( S " ( A  |`  C ) ) )
8583, 84syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  U. ( S " ( A  |`  C ) ) )
8617mrcssid 13535 . . . . . . . . . . . . . 14  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( A  |`  C ) ) 
C_  ( Base `  G
) )  ->  U. ( S " ( A  |`  C ) )  C_  ( K `  U. ( S " ( A  |`  C ) ) ) )
877, 59, 86syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8887ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8985, 88sstrd 3202 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
9065, 89eqsstrd 3225 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
9146, 50, 62, 90dprdlub 15277 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
92 ovex 5899 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  _V
9392elpw 3644 . . . . . . . . 9  |-  ( ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) )  e.  ~P ( K `  U. ( S " ( A  |`  C ) ) )  <-> 
( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
9491, 93sylibr 203 . . . . . . . 8  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
9594, 29fmptd 5700 . . . . . . 7  |-  ( ph  ->  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) : C --> ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
96 frn 5411 . . . . . . 7  |-  ( ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) : C --> ~P ( K `
 U. ( S
" ( A  |`  C ) ) )  ->  ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ~P ( K `  U. ( S " ( A  |`  C ) ) ) )
9795, 96syl 15 . . . . . 6  |-  ( ph  ->  ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  C_  ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
98 sspwuni 4003 . . . . . 6  |-  ( ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) 
C_  ~P ( K `  U. ( S " ( A  |`  C ) ) )  <->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
9997, 98sylib 188 . . . . 5  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
1004subgss 14638 . . . . . 6  |-  ( ( K `  U. ( S " ( A  |`  C ) ) )  e.  (SubGrp `  G
)  ->  ( K `  U. ( S "
( A  |`  C ) ) )  C_  ( Base `  G ) )
10161, 100syl 15 . . . . 5  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( Base `  G
) )
10299, 101sstrd 3202 . . . 4  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( Base `  G ) )
10317mrcss 13534 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " ( A  |`  C ) ) 
C_  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  /\  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) 
C_  ( Base `  G
) )  ->  ( K `  U. ( S
" ( A  |`  C ) ) ) 
C_  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
1047, 43, 102, 103syl3anc 1182 . . 3  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
10517mrcsscl 13538 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U.
ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) )  /\  ( K `  U. ( S
" ( A  |`  C ) ) )  e.  (SubGrp `  G
) )  ->  ( K `  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) ) 
C_  ( K `  U. ( S " ( A  |`  C ) ) ) )
1067, 99, 61, 105syl3anc 1182 . . 3  |-  ( ph  ->  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
107104, 106eqssd 3209 . 2  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
108 eqid 2296 . . . . . . . 8  |-  ( i  e.  I  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  =  ( i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )
10992, 108dmmpti 5389 . . . . . . 7  |-  dom  (
i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  =  I
110109a1i 10 . . . . . 6  |-  ( ph  ->  dom  ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  =  I )
1111, 110, 44dprdres 15279 . . . . 5  |-  ( ph  ->  ( G dom DProd  ( ( i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  |`  C )  /\  ( G DProd  ( ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  |`  C )
)  C_  ( G DProd  ( i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) ) ) )
112111simpld 445 . . . 4  |-  ( ph  ->  G dom DProd  ( (
i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  |`  C ) )
113 resmpt 5016 . . . . 5  |-  ( C 
C_  I  ->  (
( i  e.  I  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) )  |`  C )  =  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
11444, 113syl 15 . . . 4  |-  ( ph  ->  ( ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  |`  C )  =  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) )
115112, 114breqtrd 4063 . . 3  |-  ( ph  ->  G dom DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
11617dprdspan 15278 . . 3  |-  ( G dom DProd  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  ->  ( G DProd  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
117115, 116syl 15 . 2  |-  ( ph  ->  ( G DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) ) )
118107, 117eqtr4d 2331 1  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( G DProd  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631   DProd cdprd 15247
This theorem is referenced by:  dprd2da  15293  dprd2db  15294
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-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-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-cmn 15107  df-dprd 15249
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