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Theorem mamufv 27422
Description: A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mamufval.f  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
mamufval.b  |-  B  =  ( Base `  R
)
mamufval.t  |-  .x.  =  ( .r `  R )
mamufval.r  |-  ( ph  ->  R  e.  V )
mamufval.m  |-  ( ph  ->  M  e.  Fin )
mamufval.n  |-  ( ph  ->  N  e.  Fin )
mamufval.p  |-  ( ph  ->  P  e.  Fin )
mamuval.x  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
mamuval.y  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
mamufv.i  |-  ( ph  ->  I  e.  M )
mamufv.k  |-  ( ph  ->  K  e.  P )
Assertion
Ref Expression
mamufv  |-  ( ph  ->  ( I ( X F Y ) K )  =  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) ) )
Distinct variable groups:    j, M    j, N    P, j    R, j   
j, X    j, Y    ph, j    j, I    j, K
Allowed substitution hints:    B( j)    .x. ( j)    F( j)    V( j)

Proof of Theorem mamufv
Dummy variables  i 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamufval.f . . 3  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
2 mamufval.b . . 3  |-  B  =  ( Base `  R
)
3 mamufval.t . . 3  |-  .x.  =  ( .r `  R )
4 mamufval.r . . 3  |-  ( ph  ->  R  e.  V )
5 mamufval.m . . 3  |-  ( ph  ->  M  e.  Fin )
6 mamufval.n . . 3  |-  ( ph  ->  N  e.  Fin )
7 mamufval.p . . 3  |-  ( ph  ->  P  e.  Fin )
8 mamuval.x . . 3  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
9 mamuval.y . . 3  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9mamuval 27421 . 2  |-  ( ph  ->  ( X F Y )  =  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) ) )
11 oveq1 6088 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
12 oveq2 6089 . . . . . 6  |-  ( k  =  K  ->  (
j Y k )  =  ( j Y K ) )
1311, 12oveqan12d 6100 . . . . 5  |-  ( ( i  =  I  /\  k  =  K )  ->  ( ( i X j )  .x.  (
j Y k ) )  =  ( ( I X j ) 
.x.  ( j Y K ) ) )
1413adantl 453 . . . 4  |-  ( (
ph  /\  ( i  =  I  /\  k  =  K ) )  -> 
( ( i X j )  .x.  (
j Y k ) )  =  ( ( I X j ) 
.x.  ( j Y K ) ) )
1514mpteq2dv 4296 . . 3  |-  ( (
ph  /\  ( i  =  I  /\  k  =  K ) )  -> 
( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) )  =  ( j  e.  N  |->  ( ( I X j )  .x.  ( j Y K ) ) ) )
1615oveq2d 6097 . 2  |-  ( (
ph  /\  ( i  =  I  /\  k  =  K ) )  -> 
( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) ) )
17 mamufv.i . 2  |-  ( ph  ->  I  e.  M )
18 mamufv.k . 2  |-  ( ph  ->  K  e.  P )
19 ovex 6106 . . 3  |-  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) )  e. 
_V
2019a1i 11 . 2  |-  ( ph  ->  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) )  e. 
_V )
2110, 16, 17, 18, 20ovmpt2d 6201 1  |-  ( ph  ->  ( I ( X F Y ) K )  =  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cotp 3818    e. cmpt 4266    X. cxp 4876   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   Basecbs 13469   .rcmulr 13530    gsumg cgsu 13724   maMul cmmul 27416
This theorem is referenced by:  mamulid  27435  mamurid  27436  mamuass  27437  mamudi  27438  mamudir  27439  mamuvs1  27440  mamuvs2  27441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-mamu 27418
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