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Theorem mvmulfval 18375
Description: Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
Hypotheses
Ref Expression
mvmulfval.x  |-  .X.  =  ( R maVecMul  <. M ,  N >. )
mvmulfval.b  |-  B  =  ( Base `  R
)
mvmulfval.t  |-  .x.  =  ( .r `  R )
mvmulfval.r  |-  ( ph  ->  R  e.  V )
mvmulfval.m  |-  ( ph  ->  M  e.  Fin )
mvmulfval.n  |-  ( ph  ->  N  e.  Fin )
Assertion
Ref Expression
mvmulfval  |-  ( ph  ->  .X.  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) ) )
Distinct variable groups:    i, j, x, y, ph    i, M, j, x, y    i, N, j, x, y    R, i, j, x, y    x, B, y    x,  .x. , y,
i
Allowed substitution hints:    B( i, j)    .x. ( j)    .X. ( x, y, i, j)    V( x, y, i, j)

Proof of Theorem mvmulfval
Dummy variables  m  n  o  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvmulfval.x . 2  |-  .X.  =  ( R maVecMul  <. M ,  N >. )
2 df-mvmul 18374 . . . 4  |- maVecMul  =  ( r  e.  _V , 
o  e.  _V  |->  [_ ( 1st `  o )  /  m ]_ [_ ( 2nd `  o )  /  n ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  n )  |->  ( i  e.  m  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) ) ) )
32a1i 11 . . 3  |-  ( ph  -> maVecMul 
=  ( r  e. 
_V ,  o  e. 
_V  |->  [_ ( 1st `  o
)  /  m ]_ [_ ( 2nd `  o
)  /  n ]_ ( x  e.  (
( Base `  r )  ^m  ( m  X.  n
) ) ,  y  e.  ( ( Base `  r )  ^m  n
)  |->  ( i  e.  m  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( y `  j ) ) ) ) ) ) ) )
4 fvex 5722 . . . . 5  |-  ( 1st `  o )  e.  _V
5 fvex 5722 . . . . 5  |-  ( 2nd `  o )  e.  _V
6 xpeq12 4880 . . . . . . 7  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
m  X.  n )  =  ( ( 1st `  o )  X.  ( 2nd `  o ) ) )
76oveq2d 6128 . . . . . 6  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
( Base `  r )  ^m  ( m  X.  n
) )  =  ( ( Base `  r
)  ^m  ( ( 1st `  o )  X.  ( 2nd `  o
) ) ) )
8 oveq2 6120 . . . . . . 7  |-  ( n  =  ( 2nd `  o
)  ->  ( ( Base `  r )  ^m  n )  =  ( ( Base `  r
)  ^m  ( 2nd `  o ) ) )
98adantl 466 . . . . . 6  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
( Base `  r )  ^m  n )  =  ( ( Base `  r
)  ^m  ( 2nd `  o ) ) )
10 simpl 457 . . . . . . 7  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  m  =  ( 1st `  o
) )
11 simpr 461 . . . . . . . . 9  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  n  =  ( 2nd `  o
) )
1211mpteq1d 4394 . . . . . . . 8  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
j  e.  n  |->  ( ( i x j ) ( .r `  r ) ( y `
 j ) ) )  =  ( j  e.  ( 2nd `  o
)  |->  ( ( i x j ) ( .r `  r ) ( y `  j
) ) ) )
1312oveq2d 6128 . . . . . . 7  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) )  =  ( r  gsumg  ( j  e.  ( 2nd `  o ) 
|->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) )
1410, 13mpteq12dv 4391 . . . . . 6  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
i  e.  m  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) )  =  ( i  e.  ( 1st `  o
)  |->  ( r  gsumg  ( j  e.  ( 2nd `  o
)  |->  ( ( i x j ) ( .r `  r ) ( y `  j
) ) ) ) ) )
157, 9, 14mpt2eq123dv 6169 . . . . 5  |-  ( ( m  =  ( 1st `  o )  /\  n  =  ( 2nd `  o
) )  ->  (
x  e.  ( (
Base `  r )  ^m  ( m  X.  n
) ) ,  y  e.  ( ( Base `  r )  ^m  n
)  |->  ( i  e.  m  |->  ( r  gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r `  r
) ( y `  j ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r
)  ^m  ( ( 1st `  o )  X.  ( 2nd `  o
) ) ) ,  y  e.  ( (
Base `  r )  ^m  ( 2nd `  o
) )  |->  ( i  e.  ( 1st `  o
)  |->  ( r  gsumg  ( j  e.  ( 2nd `  o
)  |->  ( ( i x j ) ( .r `  r ) ( y `  j
) ) ) ) ) ) )
164, 5, 15csbie2 3338 . . . 4  |-  [_ ( 1st `  o )  /  m ]_ [_ ( 2nd `  o )  /  n ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  n )  |->  ( i  e.  m  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) ) )  =  ( x  e.  ( ( Base `  r )  ^m  (
( 1st `  o
)  X.  ( 2nd `  o ) ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( 2nd `  o ) )  |->  ( i  e.  ( 1st `  o )  |->  ( r 
gsumg  ( j  e.  ( 2nd `  o ) 
|->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) ) )
17 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  r  =  R )
1817fveq2d 5716 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( Base `  r )  =  ( Base `  R
) )
19 mvmulfval.b . . . . . . 7  |-  B  =  ( Base `  R
)
2018, 19syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( Base `  r )  =  B )
21 fveq2 5712 . . . . . . . . 9  |-  ( o  =  <. M ,  N >.  ->  ( 1st `  o
)  =  ( 1st `  <. M ,  N >. ) )
2221ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 1st `  o )  =  ( 1st `  <. M ,  N >. )
)
23 mvmulfval.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  Fin )
24 mvmulfval.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  Fin )
25 op1stg 6610 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin )  ->  ( 1st `  <. M ,  N >. )  =  M )
2623, 24, 25syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. M ,  N >. )  =  M )
2726adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 1st `  <. M ,  N >. )  =  M )
2822, 27eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 1st `  o )  =  M )
29 fveq2 5712 . . . . . . . . 9  |-  ( o  =  <. M ,  N >.  ->  ( 2nd `  o
)  =  ( 2nd `  <. M ,  N >. ) )
3029ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 2nd `  o )  =  ( 2nd `  <. M ,  N >. )
)
31 op2ndg 6611 . . . . . . . . . 10  |-  ( ( M  e.  Fin  /\  N  e.  Fin )  ->  ( 2nd `  <. M ,  N >. )  =  N )
3223, 24, 31syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. M ,  N >. )  =  N )
3332adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 2nd `  <. M ,  N >. )  =  N )
3430, 33eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( 2nd `  o )  =  N )
3528, 34xpeq12d 4886 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
( 1st `  o
)  X.  ( 2nd `  o ) )  =  ( M  X.  N
) )
3620, 35oveq12d 6130 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
( Base `  r )  ^m  ( ( 1st `  o
)  X.  ( 2nd `  o ) ) )  =  ( B  ^m  ( M  X.  N
) ) )
3720, 34oveq12d 6130 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
( Base `  r )  ^m  ( 2nd `  o
) )  =  ( B  ^m  N ) )
38 fveq2 5712 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
3938adantr 465 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  o  =  <. M ,  N >. )  ->  ( .r `  r )  =  ( .r `  R
) )
4039adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( .r `  r )  =  ( .r `  R
) )
41 mvmulfval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
4240, 41syl6eqr 2493 . . . . . . . . 9  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  ( .r `  r )  = 
.x.  )
4342oveqd 6129 . . . . . . . 8  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
( i x j ) ( .r `  r ) ( y `
 j ) )  =  ( ( i x j )  .x.  ( y `  j
) ) )
4434, 43mpteq12dv 4391 . . . . . . 7  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
j  e.  ( 2nd `  o )  |->  ( ( i x j ) ( .r `  r
) ( y `  j ) ) )  =  ( j  e.  N  |->  ( ( i x j )  .x.  ( y `  j
) ) ) )
4517, 44oveq12d 6130 . . . . . 6  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
r  gsumg  ( j  e.  ( 2nd `  o ) 
|->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) )
4628, 45mpteq12dv 4391 . . . . 5  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
i  e.  ( 1st `  o )  |->  ( r 
gsumg  ( j  e.  ( 2nd `  o ) 
|->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) )  =  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j ) 
.x.  ( y `  j ) ) ) ) ) )
4736, 37, 46mpt2eq123dv 6169 . . . 4  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  (
x  e.  ( (
Base `  r )  ^m  ( ( 1st `  o
)  X.  ( 2nd `  o ) ) ) ,  y  e.  ( ( Base `  r
)  ^m  ( 2nd `  o ) )  |->  ( i  e.  ( 1st `  o )  |->  ( r 
gsumg  ( j  e.  ( 2nd `  o ) 
|->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) ) )  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) ) )
4816, 47syl5eq 2487 . . 3  |-  ( (
ph  /\  ( r  =  R  /\  o  =  <. M ,  N >. ) )  ->  [_ ( 1st `  o )  /  m ]_ [_ ( 2nd `  o )  /  n ]_ ( x  e.  ( ( Base `  r
)  ^m  ( m  X.  n ) ) ,  y  e.  ( (
Base `  r )  ^m  n )  |->  ( i  e.  m  |->  ( r 
gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
`  r ) ( y `  j ) ) ) ) ) )  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) ) )
49 mvmulfval.r . . . 4  |-  ( ph  ->  R  e.  V )
50 elex 3002 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
5149, 50syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
52 opex 4577 . . . 4  |-  <. M ,  N >.  e.  _V
5352a1i 11 . . 3  |-  ( ph  -> 
<. M ,  N >.  e. 
_V )
54 ovex 6137 . . . . 5  |-  ( B  ^m  ( M  X.  N ) )  e. 
_V
55 ovex 6137 . . . . 5  |-  ( B  ^m  N )  e. 
_V
5654, 55mpt2ex 6671 . . . 4  |-  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) )  e.  _V
5756a1i 11 . . 3  |-  ( ph  ->  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  N ) 
|->  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) )  e.  _V )
583, 48, 51, 53, 57ovmpt2d 6239 . 2  |-  ( ph  ->  ( R maVecMul  <. M ,  N >. )  =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) ) )
591, 58syl5eq 2487 1  |-  ( ph  ->  .X.  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993   [_csb 3309   <.cop 3904    e. cmpt 4371    X. cxp 4859   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1stc1st 6596   2ndc2nd 6597    ^m cmap 7235   Fincfn 7331   Basecbs 14195   .rcmulr 14260    gsumg cgsu 14400   maVecMul cmvmul 18373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-mvmul 18374
This theorem is referenced by:  mvmulval  18376  mavmuldm  18383  mavmul0g  18386
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