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Theorem mamufval 27420
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 27418 . . . 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 5742 . . . . 5  |-  ( 1st `  ( 1st `  o
) )  e.  _V
5 fvex 5742 . . . . 5  |-  ( 2nd `  ( 1st `  o
) )  e.  _V
6 fvex 5742 . . . . . . 7  |-  ( 2nd `  o )  e.  _V
7 nfcv 2572 . . . . . . 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 2437 . . . . . . . 8  |-  ( p  =  ( 2nd `  o
)  ->  ( ( Base `  r )  ^m  ( m  X.  n
) )  =  ( ( Base `  r
)  ^m  ( m  X.  n ) ) )
9 xpeq2 4893 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  ( n  X.  p )  =  ( n  X.  ( 2nd `  o ) ) )
109oveq2d 6097 . . . . . . . 8  |-  ( p  =  ( 2nd `  o
)  ->  ( ( Base `  r )  ^m  ( n  X.  p
) )  =  ( ( Base `  r
)  ^m  ( n  X.  ( 2nd `  o
) ) ) )
11 eqidd 2437 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  m  =  m )
12 id 20 . . . . . . . . 9  |-  ( p  =  ( 2nd `  o
)  ->  p  =  ( 2nd `  o ) )
13 eqidd 2437 . . . . . . . . 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 6136 . . . . . . . 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 6136 . . . . . . 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 3292 . . . . . 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 4897 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( m  X.  n )  =  ( ( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) ) )
1817oveq2d 6097 . . . . . . 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 448 . . . . . . . . 9  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  n  =  ( 2nd `  ( 1st `  o ) ) )
2019xpeq1d 4901 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( n  X.  ( 2nd `  o
) )  =  ( ( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) ) )
2120oveq2d 6097 . . . . . . 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 20 . . . . . . . . 9  |-  ( m  =  ( 1st `  ( 1st `  o ) )  ->  m  =  ( 1st `  ( 1st `  o ) ) )
2322adantr 452 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  m  =  ( 1st `  ( 1st `  o ) ) )
24 eqidd 2437 . . . . . . . 8  |-  ( ( m  =  ( 1st `  ( 1st `  o
) )  /\  n  =  ( 2nd `  ( 1st `  o ) ) )  ->  ( 2nd `  o )  =  ( 2nd `  o ) )
25 eqidd 2437 . . . . . . . . . 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 4287 . . . . . . . . 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 6097 . . . . . . . 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 6136 . . . . . . 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 6136 . . . . . 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 2480 . . . . 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 3296 . . . 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 733 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
r  =  R )
3332fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( Base `  r )  =  ( Base `  R
) )
34 mamufval.b . . . . . . 7  |-  B  =  ( Base `  R
)
3533, 34syl6eqr 2486 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( Base `  r )  =  B )
36 fveq2 5728 . . . . . . . . . 10  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 1st `  o )  =  ( 1st `  <. M ,  N ,  P >. ) )
3736fveq2d 5732 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 1st `  ( 1st `  o
) )  =  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) ) )
3837ad2antll 710 . . . . . . . 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 6361 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin )  ->  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4339, 40, 41, 42syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4443adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 1st `  ( 1st `  <. M ,  N ,  P >. ) )  =  M )
4538, 44eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 1st `  ( 1st `  o ) )  =  M )
4636fveq2d 5732 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 2nd `  ( 1st `  o
) )  =  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) ) )
4746ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  o ) )  =  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) ) )
48 ot2ndg 6362 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin  /\  P  e.  Fin )  ->  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
4939, 40, 41, 48syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
5049adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  <. M ,  N ,  P >. ) )  =  N )
5147, 50eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  ( 1st `  o ) )  =  N )
5245, 51xpeq12d 4903 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( 1st `  ( 1st `  o ) )  X.  ( 2nd `  ( 1st `  o ) ) )  =  ( M  X.  N ) )
5335, 52oveq12d 6099 . . . . 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 5728 . . . . . . . . 9  |-  ( o  =  <. M ,  N ,  P >.  ->  ( 2nd `  o )  =  ( 2nd `  <. M ,  N ,  P >. ) )
5554ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  o
)  =  ( 2nd `  <. M ,  N ,  P >. ) )
56 ot3rdg 6363 . . . . . . . . . 10  |-  ( P  e.  Fin  ->  ( 2nd `  <. M ,  N ,  P >. )  =  P )
5741, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. M ,  N ,  P >. )  =  P )
5857adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  <. M ,  N ,  P >. )  =  P )
5955, 58eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( 2nd `  o
)  =  P )
6051, 59xpeq12d 4903 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( 2nd `  ( 1st `  o ) )  X.  ( 2nd `  o
) )  =  ( N  X.  P ) )
6135, 60oveq12d 6099 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( ( Base `  r
)  ^m  ( ( 2nd `  ( 1st `  o
) )  X.  ( 2nd `  o ) ) )  =  ( B  ^m  ( N  X.  P ) ) )
6232fveq2d 5732 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( .r `  r
)  =  ( .r
`  R ) )
63 mamufval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
6462, 63syl6eqr 2486 . . . . . . . . 9  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N ,  P >. ) )  -> 
( .r `  r
)  =  .x.  )
6564oveqd 6098 . . . . . . . 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 4287 . . . . . . 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 6099 . . . . . 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 6136 . . . . 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 6136 . . . 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 2480 . . 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 2964 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
7371, 72syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
74 otex 4428 . . . 4  |-  <. M ,  N ,  P >.  e. 
_V
7574a1i 11 . . 3  |-  ( ph  -> 
<. M ,  N ,  P >.  e.  _V )
76 ovex 6106 . . . . 5  |-  ( B  ^m  ( M  X.  N ) )  e. 
_V
77 ovex 6106 . . . . 5  |-  ( B  ^m  ( N  X.  P ) )  e. 
_V
7876, 77mpt2ex 6425 . . . 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 6201 . 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 2480 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251   <.cotp 3818    e. cmpt 4266    X. cxp 4876   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348    ^m cmap 7018   Fincfn 7109   Basecbs 13469   .rcmulr 13530    gsumg cgsu 13724   maMul cmmul 27416
This theorem is referenced by:  mamuval  27421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-mamu 27418
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