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Theorem mamufv 27548
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 27547 . 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 5881 . . . . . 6  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
12 oveq2 5882 . . . . . 6  |-  ( k  =  K  ->  (
j Y k )  =  ( j Y K ) )
1311, 12oveqan12d 5893 . . . . 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 4123 . . 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 5890 . 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 5899 . . 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 5991 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 1632    e. wcel 1696   _Vcvv 2801   <.cotp 3657    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   Basecbs 13164   .rcmulr 13225    gsumg cgsu 13417   maMul cmmul 27542
This theorem is referenced by:  mamulid  27561  mamurid  27562  mamuass  27563  mamudi  27564  mamudir  27565  mamuvs1  27566  mamuvs2  27567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-mamu 27544
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