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Theorem dprd2dlem1 15604
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 15568 . . . . . 6  |-  ( G dom DProd  ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  ->  G  e.  Grp )
31, 2syl 16 . . . . 5  |-  ( ph  ->  G  e.  Grp )
4 eqid 2438 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
54subgacs 14980 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
6 acsmre 13882 . . . . 5  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
73, 5, 63syl 19 . . . 4  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
8 dprd2d.k . . . 4  |-  K  =  (mrCls `  (SubGrp `  G
) )
9 dprd2d.2 . . . . . 6  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
10 ffun 5596 . . . . . 6  |-  ( S : A --> (SubGrp `  G )  ->  Fun  S )
11 funiunfv 5998 . . . . . 6  |-  ( Fun 
S  ->  U_ x  e.  ( A  |`  C ) ( S `  x
)  =  U. ( S " ( A  |`  C ) ) )
129, 10, 113syl 19 . . . . 5  |-  ( ph  ->  U_ x  e.  ( A  |`  C )
( S `  x
)  =  U. ( S " ( A  |`  C ) ) )
13 resss 5173 . . . . . . . . . 10  |-  ( A  |`  C )  C_  A
1413sseli 3346 . . . . . . . . 9  |-  ( x  e.  ( A  |`  C )  ->  x  e.  A )
15 dprd2d.1 . . . . . . . . . 10  |-  ( ph  ->  Rel  A )
16 dprd2d.3 . . . . . . . . . 10  |-  ( ph  ->  dom  A  C_  I
)
17 dprd2d.4 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
1815, 9, 16, 17, 1, 8dprd2dlem2 15603 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) ) )
1914, 18sylan2 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) ) )
20 1st2nd 6396 . . . . . . . . . . . . 13  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2115, 14, 20syl2an 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
22 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  e.  ( A  |`  C ) )
2321, 22eqeltrrd 2513 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  ( A  |`  C ) )
24 fvex 5745 . . . . . . . . . . . . 13  |-  ( 2nd `  x )  e.  _V
2524opelres 5154 . . . . . . . . . . . 12  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( A  |`  C )  <->  (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A  /\  ( 1st `  x
)  e.  C ) )
2625simprbi 452 . . . . . . . . . . 11  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( A  |`  C )  ->  ( 1st `  x
)  e.  C )
2723, 26syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( 1st `  x )  e.  C
)
28 ovex 6109 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) )  e.  _V
29 eqid 2438 . . . . . . . . . . 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 3827 . . . . . . . . . . . . . 14  |-  ( i  =  ( 1st `  x
)  ->  { i }  =  { ( 1st `  x ) } )
3130imaeq2d 5206 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( A " { i } )  =  ( A " { ( 1st `  x
) } ) )
32 oveq1 6091 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( i S j )  =  ( ( 1st `  x
) S j ) )
3331, 32mpteq12dv 4290 . . . . . . . . . . . 12  |-  ( i  =  ( 1st `  x
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )
3433oveq2d 6100 . . . . . . . . . . 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 5121 . . . . . . . . . 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 645 . . . . . . . . 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 4045 . . . . . . . . 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 16 . . . . . . . 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 3360 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
4039ralrimiva 2791 . . . . . 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 4134 . . . . . 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 205 . . . . 5  |-  ( ph  ->  U_ x  e.  ( A  |`  C )
( S `  x
)  C_  U. ran  (
i  e.  C  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) ) )
4312, 42eqsstr3d 3385 . . . 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 3350 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  C )  ->  i  e.  I )
4645, 17syldan 458 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
47 ovex 6109 . . . . . . . . . . . 12  |-  ( i S j )  e. 
_V
48 eqid 2438 . . . . . . . . . . . 12  |-  ( j  e.  ( A " { i } ) 
|->  ( i S j ) )  =  ( j  e.  ( A
" { i } )  |->  ( i S j ) )
4947, 48dmmpti 5577 . . . . . . . . . . 11  |-  dom  (
j  e.  ( A
" { i } )  |->  ( i S j ) )  =  ( A " {
i } )
5049a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  dom  ( j  e.  ( A " { i } )  |->  ( i S j ) )  =  ( A " { i } ) )
51 imassrn 5219 . . . . . . . . . . . . . 14  |-  ( S
" ( A  |`  C ) )  C_  ran  S
52 frn 5600 . . . . . . . . . . . . . . . 16  |-  ( S : A --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
539, 52syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
54 mresspw 13822 . . . . . . . . . . . . . . . 16  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
557, 54syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (SubGrp `  G )  C_ 
~P ( Base `  G
) )
5653, 55sstrd 3360 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5751, 56syl5ss 3361 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S " ( A  |`  C ) ) 
C_  ~P ( Base `  G
) )
58 sspwuni 4179 . . . . . . . . . . . . 13  |-  ( ( S " ( A  |`  C ) )  C_  ~P ( Base `  G
)  <->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
5957, 58sylib 190 . . . . . . . . . . . 12  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
608mrccl 13841 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
6261adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  ( K `  U. ( S
" ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
63 oveq2 6092 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
i S j )  =  ( i S k ) )
6463, 48, 47fvmpt3i 5812 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { i } )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
6564adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
66 df-ov 6087 . . . . . . . . . . . . . 14  |-  ( i S k )  =  ( S `  <. i ,  k >. )
67 ffn 5594 . . . . . . . . . . . . . . . . 17  |-  ( S : A --> (SubGrp `  G )  ->  S  Fn  A )
689, 67syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  Fn  A )
6968ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  S  Fn  A
)
7013a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( A  |`  C )  C_  A
)
71 elrelimasn 5231 . . . . . . . . . . . . . . . . . . . 20  |-  ( Rel 
A  ->  ( k  e.  ( A " {
i } )  <->  i A
k ) )
7215, 71syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( k  e.  ( A " { i } )  <->  i A
k ) )
7372adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  C )  ->  (
k  e.  ( A
" { i } )  <->  i A k ) )
7473biimpa 472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i A k )
75 df-br 4216 . . . . . . . . . . . . . . . . 17  |-  ( i A k  <->  <. i ,  k >.  e.  A
)
7674, 75sylib 190 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  A )
77 simplr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i  e.  C
)
78 vex 2961 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
7978opelres 5154 . . . . . . . . . . . . . . . 16  |-  ( <.
i ,  k >.  e.  ( A  |`  C )  <-> 
( <. i ,  k
>.  e.  A  /\  i  e.  C ) )
8076, 77, 79sylanbrc 647 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  ( A  |`  C ) )
81 fnfvima 5979 . . . . . . . . . . . . . . 15  |-  ( ( S  Fn  A  /\  ( A  |`  C ) 
C_  A  /\  <. i ,  k >.  e.  ( A  |`  C )
)  ->  ( S `  <. i ,  k
>. )  e.  ( S " ( A  |`  C ) ) )
8269, 70, 80, 81syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( S `  <. i ,  k >.
)  e.  ( S
" ( A  |`  C ) ) )
8366, 82syl5eqel 2522 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  e.  ( S " ( A  |`  C ) ) )
84 elssuni 4045 . . . . . . . . . . . . 13  |-  ( ( i S k )  e.  ( S "
( A  |`  C ) )  ->  ( i S k )  C_  U. ( S " ( A  |`  C ) ) )
8583, 84syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  U. ( S " ( A  |`  C ) ) )
867, 8, 59mrcssidd 13855 . . . . . . . . . . . . 13  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8786ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8885, 87sstrd 3360 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
8965, 88eqsstrd 3384 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
9046, 50, 62, 89dprdlub 15589 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
91 ovex 6109 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  _V
9291elpw 3807 . . . . . . . . 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 ) ) ) )
9390, 92sylibr 205 . . . . . . . 8  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
9493, 29fmptd 5896 . . . . . . 7  |-  ( ph  ->  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) : C --> ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
95 frn 5600 . . . . . . 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 ) ) ) )
9694, 95syl 16 . . . . . 6  |-  ( ph  ->  ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  C_  ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
97 sspwuni 4179 . . . . . 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 ) ) ) )
9896, 97sylib 190 . . . . 5  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
997, 8mrcssvd 13853 . . . . 5  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( Base `  G
) )
10098, 99sstrd 3360 . . . 4  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( Base `  G ) )
1017, 8, 43, 100mrcssd 13854 . . 3  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
1028mrcsscl 13850 . . . 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 ) ) ) )
1037, 98, 61, 102syl3anc 1185 . . 3  |-  ( ph  ->  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
104101, 103eqssd 3367 . 2  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
105 eqid 2438 . . . . . . . 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 ) ) ) )
10691, 105dmmpti 5577 . . . . . . 7  |-  dom  (
i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  =  I
107106a1i 11 . . . . . 6  |-  ( ph  ->  dom  ( i  e.  I  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) )  =  I )
1081, 107, 44dprdres 15591 . . . . 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 ) ) ) ) ) ) )
109108simpld 447 . . . 4  |-  ( ph  ->  G dom DProd  ( (
i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  |`  C ) )
110 resmpt 5194 . . . . 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 ) ) ) ) )
11144, 110syl 16 . . . 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 ) ) ) ) )
112109, 111breqtrd 4239 . . 3  |-  ( ph  ->  G dom DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
1138dprdspan 15590 . . 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 ) ) ) ) ) )
114112, 113syl 16 . 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 ) ) ) ) ) )
115104, 114eqtr4d 2473 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   {csn 3816   <.cop 3819   U.cuni 4017   U_ciun 4095   class class class wbr 4215    e. cmpt 4269   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884   Rel wrel 4886   Fun wfun 5451    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   Basecbs 13474  Moorecmre 13812  mrClscmrc 13813  ACScacs 13815   Grpcgrp 14690  SubGrpcsubg 14943   DProd cdprd 15559
This theorem is referenced by:  dprd2da  15605  dprd2db  15606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-ghm 15009  df-gim 15051  df-cntz 15121  df-oppg 15147  df-cmn 15419  df-dprd 15561
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