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Theorem mamufval 18756
Description: Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mamufval.f  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
mamufval.b  |-  B  =  ( Base `  R
)
mamufval.t  |-  .x.  =  ( .r `  R )
mamufval.r  |-  ( ph  ->  R  e.  V )
mamufval.m  |-  ( ph  ->  M  e.  Fin )
mamufval.n  |-  ( ph  ->  N  e.  Fin )
mamufval.p  |-  ( ph  ->  P  e.  Fin )
Assertion
Ref Expression
mamufval  |-  ( ph  ->  F  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  ( N  X.  P
) )  |->  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
Distinct variable groups:    i, j,
k, x, y, M   
i, N, j, k, x, y    P, i, j, k, x, y    R, i, j, k, x, y    ph, i, j, k, x, y    x, B, y    x,  .x. , y, i, k
Allowed substitution hints:    B( i, j, k)    .x. ( j)    F( x, y, i, j, k)    V( x, y, i, j, k)

Proof of Theorem mamufval
Dummy variables  m  n  o  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamufval.f . 2  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
2 df-mamu 18755 . . . 4  |- maMul  =  ( r  e.  _V , 
o  e.  _V  |->  [_ ( 1st `  ( 1st `  o ) )  /  m ]_ [_ ( 2nd `  ( 1st `  o
) )  /  n ]_ [_ ( 2nd `  o
)  /  p ]_ ( x  e.  (
( Base `  r )  ^m  ( m  X.  n
) ) ,  y  e.  ( ( Base `  r )  ^m  (
n  X.  p ) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) ) ) )
32a1i 11 . . 3  |-  ( ph  -> maMul  =  ( r  e. 
_V ,  o  e. 
_V  |->  [_ ( 1st `  ( 1st `  o ) )  /  m ]_ [_ ( 2nd `  ( 1st `  o
) )  /  n ]_ [_ ( 2nd `  o
)  /  p ]_ ( x  e.  (
( Base `  r )  ^m  ( m  X.  n
) ) ,  y  e.  ( ( Base `  r )  ^m  (
n  X.  p ) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) ) ) ) )
4 fvex 5882 . . . . 5  |-  ( 1st `  ( 1st `  o
) )  e.  _V
5 fvex 5882 . . . . 5  |-  ( 2nd `  ( 1st `  o
) )  e.  _V
6 fvex 5882 . . . . . . 7  |-  ( 2nd `  o )  e.  _V
7 nfcv 2629 . . . . . . 7  |-  F/_ p
( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  ( 2nd `  o ) ) )  |->  ( i  e.  m ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) ) )
8 eqidd 2468 . . . . . . . 8  |-  ( p  =  ( 2nd `  o
)  ->  ( ( Base `  r )  ^m  ( m  X.  n
) )  =  ( ( Base `  r
)  ^m  ( m  X.  n ) ) )
9 xpeq2 5020 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  ( n  X.  p )  =  ( n  X.  ( 2nd `  o ) ) )
109oveq2d 6311 . . . . . . . 8  |-  ( p  =  ( 2nd `  o
)  ->  ( ( Base `  r )  ^m  ( n  X.  p
) )  =  ( ( Base `  r
)  ^m  ( n  X.  ( 2nd `  o
) ) ) )
11 eqidd 2468 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  m  =  m )
12 id 22 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  p  =  ( 2nd `  o ) )
13 eqidd 2468 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) )  =  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) )
1411, 12, 13mpt2eq123dv 6354 . . . . . . . 8  |-  ( p  =  ( 2nd `  o
)  ->  ( i  e.  m ,  k  e.  p  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) )  =  ( i  e.  m ,  k  e.  ( 2nd `  o )  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )
158, 10, 14mpt2eq123dv 6354 . . . . . . 7  |-  ( p  =  ( 2nd `  o
)  ->  ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  p
) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r )  ^m  (
m  X.  n ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( n  X.  ( 2nd `  o
) ) )  |->  ( i  e.  m ,  k  e.  ( 2nd `  o )  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) ) )
166, 7, 15csbief 3465 . . . . . 6  |-  [_ ( 2nd `  o )  /  p ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  p
) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r )  ^m  (
m  X.  n ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( n  X.  ( 2nd `  o
) ) )  |->  ( i  e.  m ,  k  e.  ( 2nd `  o )  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )
17 xpeq12 5024 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( m  X.  n )  =  ( ( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) ) )
1817oveq2d 6311 . . . . . . 7  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( ( Base `  r )  ^m  ( m  X.  n
) )  =  ( ( Base `  r
)  ^m  ( ( 1st `  ( 1st `  o
) )  X.  ( 2nd `  ( 1st `  o
) ) ) ) )
19 simpr 461 . . . . . . . . 9  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  n  =  ( 2nd `  ( 1st `  o ) ) )
2019xpeq1d 5028 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( n  X.  ( 2nd `  o
) )  =  ( ( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) ) )
2120oveq2d 6311 . . . . . . 7  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( ( Base `  r )  ^m  ( n  X.  ( 2nd `  o ) ) )  =  ( (
Base `  r )  ^m  ( ( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) ) ) )
22 id 22 . . . . . . . . 9  |-  ( m  =  ( 1st `  ( 1st `  o ) )  ->  m  =  ( 1st `  ( 1st `  o ) ) )
2322adantr 465 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  m  =  ( 1st `  ( 1st `  o ) ) )
24 eqidd 2468 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( 2nd `  o )  =  ( 2nd `  o ) )
25 eqidd 2468 . . . . . . . . . 10  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( (
i x j ) ( .r `  r
) ( j y k ) )  =  ( ( i x j ) ( .r
`  r ) ( j y k ) ) )
2619, 25mpteq12dv 4531 . . . . . . . . 9  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) )  =  ( j  e.  ( 2nd `  ( 1st `  o ) ) 
|->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) )
2726oveq2d 6311 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) )  =  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) ) )
2823, 24, 27mpt2eq123dv 6354 . . . . . . 7  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( i  e.  m ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) )  =  ( i  e.  ( 1st `  ( 1st `  o
) ) ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) ) 
|->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )
2918, 21, 28mpt2eq123dv 6354 . . . . . 6  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  ( 2nd `  o ) ) )  |->  ( i  e.  m ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r
)  ^m  ( ( 1st `  ( 1st `  o
) )  X.  ( 2nd `  ( 1st `  o
) ) ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( ( 2nd `  ( 1st `  o
) )  X.  ( 2nd `  o ) ) )  |->  ( i  e.  ( 1st `  ( 1st `  o ) ) ,  k  e.  ( 2nd `  o ) 
|->  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) ) ) ) )
3016, 29syl5eq 2520 . . . . 5  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  [_ ( 2nd `  o )  /  p ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  p
) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r )  ^m  (
( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) ) ) ,  y  e.  ( ( Base `  r )  ^m  (
( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) ) )  |->  ( i  e.  ( 1st `  ( 1st `  o
) ) ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) ) 
|->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) ) )
314, 5, 30csbie2 3470 . . . 4  |-  [_ ( 1st `  ( 1st `  o
) )  /  m ]_ [_ ( 2nd `  ( 1st `  o ) )  /  n ]_ [_ ( 2nd `  o )  /  p ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  ( n  X.  p
) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r )  ^m  (
( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) ) ) ,  y  e.  ( ( Base `  r )  ^m  (
( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) ) )  |->  ( i  e.  ( 1st `  ( 1st `  o
) ) ,  k  e.  ( 2nd `  o
)  |->  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) ) 
|->  ( ( i x j ) ( .r
`  r ) ( j y k ) ) ) ) ) )
32 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
r  =  R )
3332fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( Base `  r )  =  ( Base `  R
) )
34 mamufval.b . . . . . . 7  |-  B  =  ( Base `  R
)
3533, 34syl6eqr 2526 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( Base `  r )  =  B )
36 fveq2 5872 . . . . . . . . . 10  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 1st `  o )  =  ( 1st `  <. M ,  N ,  P >. ) )
3736fveq2d 5876 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 1st `  ( 1st `  o
) )  =  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) ) )
3837ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 1st `  ( 1st `  o ) )  =  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) ) )
39 mamufval.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  Fin )
40 mamufval.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  Fin )
41 mamufval.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  Fin )
42 ot1stg 6809 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin )  ->  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4339, 40, 41, 42syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4443adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4538, 44eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 1st `  ( 1st `  o ) )  =  M )
4636fveq2d 5876 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 2nd `  ( 1st `  o
) )  =  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) ) )
4746ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  o ) )  =  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) ) )
48 ot2ndg 6810 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin )  ->  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
4939, 40, 41, 48syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
5049adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
5147, 50eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  o ) )  =  N )
5245, 51xpeq12d 5030 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) )  =  ( M  X.  N ) )
5335, 52oveq12d 6313 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( Base `  r
)  ^m  ( ( 1st `  ( 1st `  o
) )  X.  ( 2nd `  ( 1st `  o
) ) ) )  =  ( B  ^m  ( M  X.  N
) ) )
54 fveq2 5872 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 2nd `  o )  =  ( 2nd `  <. M ,  N ,  P >. ) )
5554ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  o
)  =  ( 2nd `  <. M ,  N ,  P >. ) )
56 ot3rdg 6811 . . . . . . . . . 10  |-  ( P  e.  Fin  ->  ( 2nd `  <. M ,  N ,  P >. )  =  P )
5741, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. M ,  N ,  P >. )  =  P )
5857adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  <. M ,  N ,  P >. )  =  P )
5955, 58eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  o
)  =  P )
6051, 59xpeq12d 5030 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) )  =  ( N  X.  P ) )
6135, 60oveq12d 6313 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( Base `  r
)  ^m  ( ( 2nd `  ( 1st `  o
) )  X.  ( 2nd `  o ) ) )  =  ( B  ^m  ( N  X.  P ) ) )
6232fveq2d 5876 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( .r `  r
)  =  ( .r
`  R ) )
63 mamufval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
6462, 63syl6eqr 2526 . . . . . . . . 9  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( .r `  r
)  =  .x.  )
6564oveqd 6312 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( i x j ) ( .r
`  r ) ( j y k ) )  =  ( ( i x j ) 
.x.  ( j y k ) ) )
6651, 65mpteq12dv 4531 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) )  =  ( j  e.  N  |->  ( ( i x j ) 
.x.  ( j y k ) ) ) )
6732, 66oveq12d 6313 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) )
6845, 59, 67mpt2eq123dv 6354 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( i  e.  ( 1st `  ( 1st `  o ) ) ,  k  e.  ( 2nd `  o )  |->  ( r 
gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) ) )  =  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) )
6953, 61, 68mpt2eq123dv 6354 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( x  e.  ( ( Base `  r
)  ^m  ( ( 1st `  ( 1st `  o
) )  X.  ( 2nd `  ( 1st `  o
) ) ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( ( 2nd `  ( 1st `  o
) )  X.  ( 2nd `  o ) ) )  |->  ( i  e.  ( 1st `  ( 1st `  o ) ) ,  k  e.  ( 2nd `  o ) 
|->  ( r  gsumg  ( j  e.  ( 2nd `  ( 1st `  o ) )  |->  ( ( i x j ) ( .r `  r ) ( j y k ) ) ) ) ) )  =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  ( N  X.  P ) ) 
|->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
7031, 69syl5eq 2520 . . 3  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  ->  [_ ( 1st `  ( 1st `  o ) )  /  m ]_ [_ ( 2nd `  ( 1st `  o
) )  /  n ]_ [_ ( 2nd `  o
)  /  p ]_ ( x  e.  (
( Base `  r )  ^m  ( m  X.  n
) ) ,  y  e.  ( ( Base `  r )  ^m  (
n  X.  p ) )  |->  ( i  e.  m ,  k  e.  p  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( j y k ) ) ) ) ) )  =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  ( N  X.  P ) ) 
|->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
71 mamufval.r . . . 4  |-  ( ph  ->  R  e.  V )
72 elex 3127 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
7371, 72syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
74 otex 4718 . . . 4  |-  <. M ,  N ,  P >.  e. 
_V
7574a1i 11 . . 3  |-  ( ph  -> 
<. M ,  N ,  P >.  e.  _V )
76 ovex 6320 . . . . 5  |-  ( B  ^m  ( M  X.  N ) )  e. 
_V
77 ovex 6320 . . . . 5  |-  ( B  ^m  ( N  X.  P ) )  e. 
_V
7876, 77mpt2ex 6872 . . . 4  |-  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  ( N  X.  P
) )  |->  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) )  e.  _V
7978a1i 11 . . 3  |-  ( ph  ->  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  ( N  X.  P ) ) 
|->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) )  e.  _V )
803, 70, 73, 75, 79ovmpt2d 6425 . 2  |-  ( ph  ->  ( R maMul  <. M ,  N ,  P >. )  =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  ( N  X.  P ) ) 
|->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
811, 80syl5eq 2520 1  |-  ( ph  ->  F  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  ( N  X.  P
) )  |->  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   [_csb 3440   <.cotp 4041    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794    ^m cmap 7432   Fincfn 7528   Basecbs 14507   .rcmulr 14573    gsumg cgsu 14713   maMul cmmul 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-mamu 18755
This theorem is referenced by:  mamuval  18757  mamudm  18759
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