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Theorem mvmulfv 18913
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 18912 . 2  |-  ( ph  ->  ( X  .X.  Y
)  =  ( i  e.  M  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j )
) ) ) ) )
10 oveq1 6284 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  i  =  I )  ->  (
i X j )  =  ( I X j ) )
1211oveq1d 6292 . . . 4  |-  ( (
ph  /\  i  =  I )  ->  (
( i X j )  .x.  ( Y `
 j ) )  =  ( ( I X j )  .x.  ( Y `  j ) ) )
1312mpteq2dv 4520 . . 3  |-  ( (
ph  /\  i  =  I )  ->  (
j  e.  N  |->  ( ( i X j )  .x.  ( Y `
 j ) ) )  =  ( j  e.  N  |->  ( ( I X j ) 
.x.  ( Y `  j ) ) ) )
1413oveq2d 6293 . 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 6305 . . 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 5942 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 1381    e. wcel 1802   _Vcvv 3093   <.cop 4016    |-> cmpt 4491    X. cxp 4983   ` cfv 5574  (class class class)co 6277    ^m cmap 7418   Fincfn 7514   Basecbs 14504   .rcmulr 14570    gsumg cgsu 14710   maVecMul cmvmul 18909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-mvmul 18910
This theorem is referenced by:  mvmumamul1  18923
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