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Theorem mvmulval 18484
Description: Multiplication of a vector with a matrix. (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 )
mvmulval.x  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
mvmulval.y  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
Assertion
Ref Expression
mvmulval  |-  ( ph  ->  ( X  .X.  Y
)  =  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) ) )
Distinct variable groups:    i, j, ph    i, M, j    i, N, j    R, i, j    .x. , i    i, X, j   
i, Y, j
Allowed substitution hints:    B( i, j)    .x. ( j)    .X. ( i, j)    V( i, j)

Proof of Theorem mvmulval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvmulfval.x . . 3  |-  .X.  =  ( R maVecMul  <. M ,  N >. )
2 mvmulfval.b . . 3  |-  B  =  ( Base `  R
)
3 mvmulfval.t . . 3  |-  .x.  =  ( .r `  R )
4 mvmulfval.r . . 3  |-  ( ph  ->  R  e.  V )
5 mvmulfval.m . . 3  |-  ( ph  ->  M  e.  Fin )
6 mvmulfval.n . . 3  |-  ( ph  ->  N  e.  Fin )
71, 2, 3, 4, 5, 6mvmulfval 18483 . 2  |-  ( 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 )
) ) ) ) ) )
8 oveq 6209 . . . . . . 7  |-  ( x  =  X  ->  (
i x j )  =  ( i X j ) )
9 fveq1 5801 . . . . . . 7  |-  ( y  =  Y  ->  (
y `  j )  =  ( Y `  j ) )
108, 9oveqan12d 6222 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( i x j )  .x.  (
y `  j )
)  =  ( ( i X j ) 
.x.  ( Y `  j ) ) )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( i x j )  .x.  (
y `  j )
)  =  ( ( i X j ) 
.x.  ( Y `  j ) ) )
1211mpteq2dv 4490 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) )  =  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `
 j ) ) ) )
1312oveq2d 6219 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) )
1413mpteq2dv 4490 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
y `  j )
) ) ) )  =  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j ) 
.x.  ( Y `  j ) ) ) ) ) )
15 mvmulval.x . 2  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
16 mvmulval.y . 2  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
17 mptexg 6059 . . 3  |-  ( M  e.  Fin  ->  (
i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) )  e.  _V )
185, 17syl 16 . 2  |-  ( ph  ->  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) )  e.  _V )
197, 14, 15, 16, 18ovmpt2d 6331 1  |-  ( ph  ->  ( X  .X.  Y
)  =  ( 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 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994    |-> cmpt 4461    X. cxp 4949   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   Fincfn 7423   Basecbs 14295   .rcmulr 14361    gsumg cgsu 14501   maVecMul cmvmul 18481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-mvmul 18482
This theorem is referenced by:  mvmulfv  18485  mavmulval  18486
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