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Theorem mamuval 27423
Description: 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 ) ) )
Assertion
Ref Expression
mamuval  |-  ( ph  ->  ( X F Y )  =  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) ) )
Distinct variable groups:    i, j,
k, M    i, N, j, k    P, i, j, k    R, i, j, k   
i, X, j, k   
i, Y, j, k    ph, i, j, k    .x. , i,
k
Allowed substitution hints:    B( i, j, k)    .x. ( j)    F( i,
j, k)    V( i,
j, k)

Proof of Theorem mamuval
Dummy variables  x  y 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 )
81, 2, 3, 4, 5, 6, 7mamufval 27422 . 2  |-  ( ph  ->  F  =  ( x  e.  ( B  ^m  ( M  X.  N
) ) ,  y  e.  ( B  ^m  ( N  X.  P
) )  |->  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) ) ) )
9 eqidd 2439 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  M  =  M )
10 eqidd 2439 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  P  =  P )
11 oveq 6089 . . . . . . 7  |-  ( x  =  X  ->  (
i x j )  =  ( i X j ) )
12 oveq 6089 . . . . . . 7  |-  ( y  =  Y  ->  (
j y k )  =  ( j Y k ) )
1311, 12oveqan12d 6102 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( i x j )  .x.  (
j y k ) )  =  ( ( i X j ) 
.x.  ( j Y k ) ) )
1413adantl 454 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( i x j )  .x.  (
j y k ) )  =  ( ( i X j ) 
.x.  ( j Y k ) ) )
1514mpteq2dv 4298 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) )  =  ( j  e.  N  |->  ( ( i X j )  .x.  ( j Y k ) ) ) )
1615oveq2d 6099 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) )  =  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) )
179, 10, 16mpt2eq123dv 6138 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
j y k ) ) ) ) )  =  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j ) 
.x.  ( j Y k ) ) ) ) ) )
18 mamuval.x . 2  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
19 mamuval.y . 2  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
20 mpt2exga 6426 . . 3  |-  ( ( M  e.  Fin  /\  P  e.  Fin )  ->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) )  e.  _V )
215, 7, 20syl2anc 644 . 2  |-  ( ph  ->  ( i  e.  M ,  k  e.  P  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) )  e.  _V )
228, 17, 18, 19, 21ovmpt2d 6203 1  |-  ( ph  ->  ( X F Y )  =  ( i  e.  M ,  k  e.  P  |->  ( R 
gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  (
j Y k ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cotp 3820    e. cmpt 4268    X. cxp 4878   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    ^m cmap 7020   Fincfn 7111   Basecbs 13471   .rcmulr 13532    gsumg cgsu 13726   maMul cmmul 27418
This theorem is referenced by:  mamufv  27424  mamucl  27435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-ot 3826  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-mamu 27420
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