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Theorem mvmulfv 18806
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 18805 . 2  |-  ( ph  ->  ( X  .X.  Y
)  =  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) ) )
10 oveq1 6282 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  i  =  I )  ->  (
i X j )  =  ( I X j ) )
1211oveq1d 6290 . . . 4  |-  ( (
ph  /\  i  =  I )  ->  (
( i X j )  .x.  ( Y `
 j ) )  =  ( ( I X j )  .x.  ( Y `  j ) ) )
1312mpteq2dv 4527 . . 3  |-  ( (
ph  /\  i  =  I )  ->  (
j  e.  N  |->  ( ( i X j )  .x.  ( Y `
 j ) ) )  =  ( j  e.  N  |->  ( ( I X j ) 
.x.  ( Y `  j ) ) ) )
1413oveq2d 6291 . 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 6300 . . 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 5946 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 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026    |-> cmpt 4498    X. cxp 4990   ` cfv 5579  (class class class)co 6275    ^m cmap 7410   Fincfn 7506   Basecbs 14479   .rcmulr 14545    gsumg cgsu 14685   maVecMul cmvmul 18802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-mvmul 18803
This theorem is referenced by:  mvmumamul1  18816
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