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Theorem mvmulfv 18468
Description: A cell/element in the vector resulting from a 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 ) )
mvmulfv.i  |-  ( ph  ->  I  e.  M )
Assertion
Ref Expression
mvmulfv  |-  ( ph  ->  ( ( X  .X.  Y ) `  I
)  =  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j )
) ) ) )
Distinct variable groups:    ph, j    j, M    j, N    R, j    j, X    j, Y    j, I
Allowed substitution hints:    B( j)    .x. ( j)    .X. ( j)    V( j)

Proof of Theorem mvmulfv
Dummy variable  i is 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 )
7 mvmulval.x . . 3  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
8 mvmulval.y . . 3  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
91, 2, 3, 4, 5, 6, 7, 8mvmulval 18467 . 2  |-  ( ph  ->  ( X  .X.  Y
)  =  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) ) )
10 oveq1 6199 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  i  =  I )  ->  (
i X j )  =  ( I X j ) )
1211oveq1d 6207 . . . 4  |-  ( (
ph  /\  i  =  I )  ->  (
( i X j )  .x.  ( Y `
 j ) )  =  ( ( I X j )  .x.  ( Y `  j ) ) )
1312mpteq2dv 4479 . . 3  |-  ( (
ph  /\  i  =  I )  ->  (
j  e.  N  |->  ( ( i X j )  .x.  ( Y `
 j ) ) )  =  ( j  e.  N  |->  ( ( I X j ) 
.x.  ( Y `  j ) ) ) )
1413oveq2d 6208 . 2  |-  ( (
ph  /\  i  =  I )  ->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j )
) ) ) )
15 mvmulfv.i . 2  |-  ( ph  ->  I  e.  M )
16 ovex 6217 . . 3  |-  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j )
) ) )  e. 
_V
1716a1i 11 . 2  |-  ( ph  ->  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j )
) ) )  e. 
_V )
189, 14, 15, 17fvmptd 5880 1  |-  ( ph  ->  ( ( X  .X.  Y ) `  I
)  =  ( 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 3070   <.cop 3983    |-> cmpt 4450    X. cxp 4938   ` cfv 5518  (class class class)co 6192    ^m cmap 7316   Fincfn 7412   Basecbs 14278   .rcmulr 14343    gsumg cgsu 14483   maVecMul cmvmul 18464
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-mvmul 18465
This theorem is referenced by:  mvmumamul1  18478
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