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Theorem mvmulval 19215
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 19214 . 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 6276 . . . . . . 7  |-  ( x  =  X  ->  (
i x j )  =  ( i X j ) )
9 fveq1 5847 . . . . . . 7  |-  ( y  =  Y  ->  (
y `  j )  =  ( Y `  j ) )
108, 9oveqan12d 6289 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( i x j )  .x.  (
y `  j )
)  =  ( ( i X j ) 
.x.  ( Y `  j ) ) )
1110adantl 464 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( i x j )  .x.  (
y `  j )
)  =  ( ( i X j ) 
.x.  ( Y `  j ) ) )
1211mpteq2dv 4526 . . . 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 6286 . . 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 4526 . 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 6117 . . 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 6403 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022    |-> cmpt 4497    X. cxp 4986   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Fincfn 7509   Basecbs 14719   .rcmulr 14788    gsumg cgsu 14933   maVecMul cmvmul 19212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-mvmul 19213
This theorem is referenced by:  mvmulfv  19216  mavmulval  19217
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