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Theorem dprd2dlem1 15554
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 15518 . . . . . 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 2404 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
54subgacs 14930 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
6 acsmre 13832 . . . . 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 5552 . . . . . 6  |-  ( S : A --> (SubGrp `  G )  ->  Fun  S )
11 funiunfv 5954 . . . . . 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 5129 . . . . . . . . . 10  |-  ( A  |`  C )  C_  A
1413sseli 3304 . . . . . . . . 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 15553 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) ) )
1914, 18sylan2 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) ) )
20 1st2nd 6352 . . . . . . . . . . . . 13  |-  ( ( Rel  A  /\  x  e.  A )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2115, 14, 20syl2an 464 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
22 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  x  e.  ( A  |`  C ) )
2321, 22eqeltrrd 2479 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  ( A  |`  C ) )
24 fvex 5701 . . . . . . . . . . . . 13  |-  ( 2nd `  x )  e.  _V
2524opelres 5110 . . . . . . . . . . . 12  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( A  |`  C )  <->  (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A  /\  ( 1st `  x
)  e.  C ) )
2625simprbi 451 . . . . . . . . . . 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 6065 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) )  e.  _V
29 eqid 2404 . . . . . . . . . . 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 3785 . . . . . . . . . . . . . 14  |-  ( i  =  ( 1st `  x
)  ->  { i }  =  { ( 1st `  x ) } )
3130imaeq2d 5162 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( A " { i } )  =  ( A " { ( 1st `  x
) } ) )
32 oveq1 6047 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( i S j )  =  ( ( 1st `  x
) S j ) )
3331, 32mpteq12dv 4247 . . . . . . . . . . . 12  |-  ( i  =  ( 1st `  x
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )
3433oveq2d 6056 . . . . . . . . . . 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 5077 . . . . . . . . . 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 644 . . . . . . . . 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 4003 . . . . . . . . 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 3318 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
4039ralrimiva 2749 . . . . . 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 4092 . . . . . 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 204 . . . . 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 3343 . . . 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 3308 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  C )  ->  i  e.  I )
4645, 17syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
47 ovex 6065 . . . . . . . . . . . 12  |-  ( i S j )  e. 
_V
48 eqid 2404 . . . . . . . . . . . 12  |-  ( j  e.  ( A " { i } ) 
|->  ( i S j ) )  =  ( j  e.  ( A
" { i } )  |->  ( i S j ) )
4947, 48dmmpti 5533 . . . . . . . . . . 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 5175 . . . . . . . . . . . . . 14  |-  ( S
" ( A  |`  C ) )  C_  ran  S
52 frn 5556 . . . . . . . . . . . . . . . 16  |-  ( S : A --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
539, 52syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
54 mresspw 13772 . . . . . . . . . . . . . . . 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 3318 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5751, 56syl5ss 3319 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S " ( A  |`  C ) ) 
C_  ~P ( Base `  G
) )
58 sspwuni 4136 . . . . . . . . . . . . 13  |-  ( ( S " ( A  |`  C ) )  C_  ~P ( Base `  G
)  <->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
5957, 58sylib 189 . . . . . . . . . . . 12  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( Base `  G ) )
608mrccl 13791 . . . . . . . . . . . 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 643 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
6261adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  C )  ->  ( K `  U. ( S
" ( A  |`  C ) ) )  e.  (SubGrp `  G
) )
63 oveq2 6048 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
i S j )  =  ( i S k ) )
6463, 48, 47fvmpt3i 5768 . . . . . . . . . . . 12  |-  ( k  e.  ( A " { i } )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
6564adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) `  k
)  =  ( i S k ) )
66 df-ov 6043 . . . . . . . . . . . . . 14  |-  ( i S k )  =  ( S `  <. i ,  k >. )
67 ffn 5550 . . . . . . . . . . . . . . . . 17  |-  ( S : A --> (SubGrp `  G )  ->  S  Fn  A )
689, 67syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  Fn  A )
6968ad2antrr 707 . . . . . . . . . . . . . . 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 5187 . . . . . . . . . . . . . . . . . . . 20  |-  ( Rel 
A  ->  ( k  e.  ( A " {
i } )  <->  i A
k ) )
7215, 71syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( k  e.  ( A " { i } )  <->  i A
k ) )
7372adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  C )  ->  (
k  e.  ( A
" { i } )  <->  i A k ) )
7473biimpa 471 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i A k )
75 df-br 4173 . . . . . . . . . . . . . . . . 17  |-  ( i A k  <->  <. i ,  k >.  e.  A
)
7674, 75sylib 189 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  A )
77 simplr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  i  e.  C
)
78 vex 2919 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
7978opelres 5110 . . . . . . . . . . . . . . . 16  |-  ( <.
i ,  k >.  e.  ( A  |`  C )  <-> 
( <. i ,  k
>.  e.  A  /\  i  e.  C ) )
8076, 77, 79sylanbrc 646 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  <. i ,  k
>.  e.  ( A  |`  C ) )
81 fnfvima 5935 . . . . . . . . . . . . . . 15  |-  ( ( S  Fn  A  /\  ( A  |`  C ) 
C_  A  /\  <. i ,  k >.  e.  ( A  |`  C )
)  ->  ( S `  <. i ,  k
>. )  e.  ( S " ( A  |`  C ) ) )
8269, 70, 80, 81syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( S `  <. i ,  k >.
)  e.  ( S
" ( A  |`  C ) ) )
8366, 82syl5eqel 2488 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  e.  ( S " ( A  |`  C ) ) )
84 elssuni 4003 . . . . . . . . . . . . 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 13805 . . . . . . . . . . . . 13  |-  ( ph  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8786ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  U. ( S "
( A  |`  C ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
8885, 87sstrd 3318 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
8965, 88eqsstrd 3342 . . . . . . . . . 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 15539 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
91 ovex 6065 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  _V
9291elpw 3765 . . . . . . . . 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 204 . . . . . . . 8  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
9493, 29fmptd 5852 . . . . . . 7  |-  ( ph  ->  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) : C --> ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
95 frn 5556 . . . . . . 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 4136 . . . . . 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 189 . . . . 5  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
997, 8mrcssvd 13803 . . . . 5  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( Base `  G
) )
10098, 99sstrd 3318 . . . 4  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( Base `  G ) )
1017, 8, 43, 100mrcssd 13804 . . 3  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
1028mrcsscl 13800 . . . 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 1184 . . 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 3325 . 2  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
105 eqid 2404 . . . . . . . 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 5533 . . . . . . 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 15541 . . . . 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 446 . . . 4  |-  ( ph  ->  G dom DProd  ( (
i  e.  I  |->  ( G DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) ) )  |`  C ) )
110 resmpt 5150 . . . . 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 4196 . . 3  |-  ( ph  ->  G dom DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
1138dprdspan 15540 . . 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 2439 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   {csn 3774   <.cop 3777   U.cuni 3975   U_ciun 4053   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Basecbs 13424  Moorecmre 13762  mrClscmrc 13763  ACScacs 13765   Grpcgrp 14640  SubGrpcsubg 14893   DProd cdprd 15509
This theorem is referenced by:  dprd2da  15555  dprd2db  15556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-gim 15001  df-cntz 15071  df-oppg 15097  df-cmn 15369  df-dprd 15511
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