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Theorem mamufv 27445
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 27444 . 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 5865 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
12 oveq2 5866 . . . . . 6  |-  ( k  =  K  ->  (
j Y k )  =  ( j Y K ) )
1311, 12oveqan12d 5877 . . . . 5  |-  ( ( i  =  I  /\  k  =  K )  ->  ( ( i X j )  .x.  (
j Y k ) )  =  ( ( I X j ) 
.x.  ( j Y K ) ) )
1413adantl 452 . . . 4  |-  ( (
ph  /\  ( i  =  I  /\  k  =  K ) )  -> 
( ( i X j )  .x.  (
j Y k ) )  =  ( ( I X j ) 
.x.  ( j Y K ) ) )
1514mpteq2dv 4107 . . 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 5874 . 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 5883 . . 3  |-  ( R 
gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) )  e. 
_V
2019a1i 10 . 2  |-  ( ph  ->  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  (
j Y K ) ) ) )  e. 
_V )
2110, 16, 17, 18, 20ovmpt2d 5975 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cotp 3644    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   Basecbs 13148   .rcmulr 13209    gsumg cgsu 13401   maMul cmmul 27439
This theorem is referenced by:  mamulid  27458  mamurid  27459  mamuass  27460  mamudi  27461  mamudir  27462  mamuvs1  27463  mamuvs2  27464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-mamu 27441
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