MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprd2dlem1 Unicode version

Theorem dprd2dlem1 15276
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 15240 . . . . . 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 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
54subgacs 14652 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
6 acsmre 13554 . . . . 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 5391 . . . . . 6  |-  ( S : A --> (SubGrp `  G )  ->  Fun  S )
10 funiunfv 5774 . . . . . 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 4979 . . . . . . . . . 10  |-  ( A  |`  C )  C_  A
1312sseli 3176 . . . . . . . . 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 15275 . . . . . . . . 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 6166 . . . . . . . . . . . . 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 2358 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  ( A  |`  C ) )
24 fvex 5539 . . . . . . . . . . . . 13  |-  ( 2nd `  x )  e.  _V
2524opelres 4960 . . . . . . . . . . . 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 5883 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { ( 1st `  x ) } )  |->  ( ( 1st `  x ) S j ) ) )  e.  _V
29 eqid 2283 . . . . . . . . . . 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 3651 . . . . . . . . . . . . . 14  |-  ( i  =  ( 1st `  x
)  ->  { i }  =  { ( 1st `  x ) } )
3130imaeq2d 5012 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( A " { i } )  =  ( A " { ( 1st `  x
) } ) )
32 oveq1 5865 . . . . . . . . . . . . 13  |-  ( i  =  ( 1st `  x
)  ->  ( i S j )  =  ( ( 1st `  x
) S j ) )
3331, 32mpteq12dv 4098 . . . . . . . . . . . 12  |-  ( i  =  ( 1st `  x
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  x
) } )  |->  ( ( 1st `  x
) S j ) ) )
3433oveq2d 5874 . . . . . . . . . . 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 4927 . . . . . . . . . 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 3855 . . . . . . . . 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 3189 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A  |`  C ) )  ->  ( S `  x )  C_  U. ran  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) )
4039ralrimiva 2626 . . . . . 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 3943 . . . . . 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 3213 . . . 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 3180 . . . . . . . . . . 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 5883 . . . . . . . . . . . 12  |-  ( i S j )  e. 
_V
48 eqid 2283 . . . . . . . . . . . 12  |-  ( j  e.  ( A " { i } ) 
|->  ( i S j ) )  =  ( j  e.  ( A
" { i } )  |->  ( i S j ) )
4947, 48dmmpti 5373 . . . . . . . . . . 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 5025 . . . . . . . . . . . . . 14  |-  ( S
" ( A  |`  C ) )  C_  ran  S
52 frn 5395 . . . . . . . . . . . . . . . 16  |-  ( S : A --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
538, 52syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
54 mresspw 13494 . . . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  S  C_  ~P ( Base `  G )
)
5751, 56syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S " ( A  |`  C ) ) 
C_  ~P ( Base `  G
) )
58 sspwuni 3987 . . . . . . . . . . . . 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 13513 . . . . . . . . . . . 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 5866 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
i S j )  =  ( i S k ) )
6463, 48, 47fvmpt3i 5605 . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . 14  |-  ( i S k )  =  ( S `  <. i ,  k >. )
67 ffn 5389 . . . . . . . . . . . . . . . . 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 5037 . . . . . . . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . . . 17  |-  k  e. 
_V
7978opelres 4960 . . . . . . . . . . . . . . . 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 5756 . . . . . . . . . . . . . . 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 2367 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  e.  ( S " ( A  |`  C ) ) )
84 elssuni 3855 . . . . . . . . . . . . 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 13519 . . . . . . . . . . . . . 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 3189 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  C )  /\  k  e.  ( A " {
i } ) )  ->  ( i S k )  C_  ( K `  U. ( S
" ( A  |`  C ) ) ) )
9065, 89eqsstrd 3212 . . . . . . . . . 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 15261 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  C )  ->  ( G DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )  C_  ( K `  U. ( S "
( A  |`  C ) ) ) )
92 ovex 5883 . . . . . . . . . 10  |-  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) )  e.  _V
9392elpw 3631 . . . . . . . . 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 5684 . . . . . . 7  |-  ( ph  ->  ( i  e.  C  |->  ( G DProd  ( j  e.  ( A " { i } ) 
|->  ( i S j ) ) ) ) : C --> ~P ( K `  U. ( S
" ( A  |`  C ) ) ) )
96 frn 5395 . . . . . . 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 3987 . . . . . 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 14622 . . . . . 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 3189 . . . 4  |-  ( ph  ->  U. ran  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) )  C_  ( Base `  G ) )
10317mrcss 13518 . . . 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 13522 . . . 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 3196 . 2  |-  ( ph  ->  ( K `  U. ( S " ( A  |`  C ) ) )  =  ( K `  U. ran  ( i  e.  C  |->  ( G DProd  (
j  e.  ( A
" { i } )  |->  ( i S j ) ) ) ) ) )
108 eqid 2283 . . . . . . . 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 5373 . . . . . . 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 15263 . . . . 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 5000 . . . . 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 4047 . . 3  |-  ( ph  ->  G dom DProd  ( i  e.  C  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
11617dprdspan 15262 . . 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 2318 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615   DProd cdprd 15231
This theorem is referenced by:  dprd2da  15277  dprd2db  15278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-cmn 15091  df-dprd 15233
  Copyright terms: Public domain W3C validator