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Theorem dprd2dlem1 15526
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 15490 . . . . . 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 2387 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
54subgacs 14902 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
6 acsmre 13804 . . . . 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 5533 . . . . . 6  |-  ( S : A --> (SubGrp `  G )  ->  Fun  S )
11 funiunfv 5934 . . . . . 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 5110 . . . . . . . . . 10  |-  ( A  |`  C )  C_  A
1413sseli 3287 . . . . . . . . 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 15525 . . . . . . . . 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 6332 . . . . . . . . . . . . 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 2462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  ( A  |`  C ) )
24 fvex 5682 . . . . . . . . . . . . 13  |-  ( 2nd `  x )  e.  _V
2524opelres 5091 . . . . . . . . . . . 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 6045 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) )  e.  _V
29 eqid 2387 . . . . . . . . . . 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 3768 . . . . . . . . . . . . . 14  |-  ( i  =  ( 1st `  x
)  ->  { i }  =  { ( 1st `  x ) } )
3130imaeq2d 5143 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( A " { i } )  =  ( A " { ( 1st `  x
) } ) )
32 oveq1 6027 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( i S j )  =  ( ( 1st `  x
) S j ) )
3331, 32mpteq12dv 4228 . . . . . . . . . . . 12  |-  ( i  =  ( 1st `  x
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )
3433oveq2d 6036 . . . . . . . . . . 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 5058 . . . . . . . . . 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 3985 . . . . . . . . 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 3301 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
4039ralrimiva 2732 . . . . . 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 4073 . . . . . 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 3326 . . . 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 3291 . . . . . . . . . . 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 6045 . . . . . . . . . . . 12  |-  ( i S j )  e. 
_V
48 eqid 2387 . . . . . . . . . . . 12  |-  ( j  e.  ( A " { i } ) 
|->  ( i S j ) )  =  ( j  e.  ( A
" { i } )  |->  ( i S j ) )
4947, 48dmmpti 5514 . . . . . . . . . . 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 5156 . . . . . . . . . . . . . 14  |-  ( S
" ( A  |`  C ) )  C_  ran  S
52 frn 5537 . . . . . . . . . . . . . . . 16  |-  ( S : A --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
539, 52syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
54 mresspw 13744 . . . . . . . . . . . . . . . 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 3301 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5751, 56syl5ss 3302 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S " ( A  |`  C ) ) 
C_  ~P ( Base `  G
) )
58 sspwuni 4117 . . . . . . . . . . . . 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 13763 . . . . . . . . . . . 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 6028 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
i S j )  =  ( i S k ) )
6463, 48, 47fvmpt3i 5748 . . . . . . . . . . . 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 6023 . . . . . . . . . . . . . 14  |-  ( i S k )  =  ( S `  <. i ,  k >. )
67 ffn 5531 . . . . . . . . . . . . . . . . 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 5168 . . . . . . . . . . . . . . . . . . . 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 4154 . . . . . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
7978opelres 5091 . . . . . . . . . . . . . . . 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 5915 . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  e.  ( S " ( A  |`  C ) ) )
84 elssuni 3985 . . . . . . . . . . . . 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 13777 . . . . . . . . . . . . 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 3301 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
8965, 88eqsstrd 3325 . . . . . . . . . 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 15511 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
91 ovex 6045 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  _V
9291elpw 3748 . . . . . . . . 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 5832 . . . . . . 7  |-  ( ph  ->  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) : C --> ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
95 frn 5537 . . . . . . 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 4117 . . . . . 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 13775 . . . . 5  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( Base `  G
) )
10098, 99sstrd 3301 . . . 4  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( Base `  G ) )
1017, 8, 43, 100mrcssd 13776 . . 3  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) ) 
C_  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
1028mrcsscl 13772 . . . 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 3308 . 2  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
105 eqid 2387 . . . . . . . 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 5514 . . . . . . 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 15513 . . . . 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 5131 . . . . 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 4177 . . 3  |-  ( ph  ->  G dom DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
1138dprdspan 15512 . . 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 2422 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 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   {csn 3757   <.cop 3760   U.cuni 3957   U_ciun 4035   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821   Rel wrel 4823   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   Basecbs 13396  Moorecmre 13734  mrClscmrc 13735  ACScacs 13737   Grpcgrp 14612  SubGrpcsubg 14865   DProd cdprd 15481
This theorem is referenced by:  dprd2da  15527  dprd2db  15528
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-gim 14973  df-cntz 15043  df-oppg 15069  df-cmn 15341  df-dprd 15483
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