Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mamuval Unicode version

Theorem mamuval 27547
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 27546 . 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 2297 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  M  =  M )
10 eqidd 2297 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  P  =  P )
11 oveq 5880 . . . . . . 7  |-  ( x  =  X  ->  (
i x j )  =  ( i X j ) )
12 oveq 5880 . . . . . . 7  |-  ( y  =  Y  ->  (
j y k )  =  ( j Y k ) )
1311, 12oveqan12d 5893 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( i x j )  .x.  (
j y k ) )  =  ( ( i X j ) 
.x.  ( j Y k ) ) )
1413adantl 452 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( i x j )  .x.  (
j y k ) )  =  ( ( i X j ) 
.x.  ( j Y k ) ) )
1514mpteq2dv 4123 . . . 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 5890 . . 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 5926 . 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 6213 . . 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 642 . 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 5991 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cotp 3657    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   Fincfn 6879   Basecbs 13164   .rcmulr 13225    gsumg cgsu 13417   maMul cmmul 27542
This theorem is referenced by:  mamufv  27548  mamucl  27559
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
  Copyright terms: Public domain W3C validator