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Theorem mvmulval 18914
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 18913 . 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 6301 . . . . . . 7  |-  ( x  =  X  ->  (
i x j )  =  ( i X j ) )
9 fveq1 5871 . . . . . . 7  |-  ( y  =  Y  ->  (
y `  j )  =  ( Y `  j ) )
108, 9oveqan12d 6314 . . . . . 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 4540 . . . 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 6311 . . 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 4540 . 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 6141 . . 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 6425 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 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Fincfn 7528   Basecbs 14507   .rcmulr 14573    gsumg cgsu 14713   maVecMul cmvmul 18911
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-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-mvmul 18912
This theorem is referenced by:  mvmulfv  18915  mavmulval  18916
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