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Theorem mamures 18289
Description: Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
Hypotheses
Ref Expression
mamures.f  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
mamures.g  |-  G  =  ( R maMul  <. I ,  N ,  P >. )
mamures.b  |-  B  =  ( Base `  R
)
mamures.r  |-  ( ph  ->  R  e.  V )
mamures.m  |-  ( ph  ->  M  e.  Fin )
mamures.n  |-  ( ph  ->  N  e.  Fin )
mamures.p  |-  ( ph  ->  P  e.  Fin )
mamures.i  |-  ( ph  ->  I  C_  M )
mamures.x  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
mamures.y  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
Assertion
Ref Expression
mamures  |-  ( ph  ->  ( ( X F Y )  |`  (
I  X.  P ) )  =  ( ( X  |`  ( I  X.  N ) ) G Y ) )

Proof of Theorem mamures
Dummy variables  i 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamures.i . . . 4  |-  ( ph  ->  I  C_  M )
2 ssid 3374 . . . . 5  |-  P  C_  P
32a1i 11 . . . 4  |-  ( ph  ->  P  C_  P )
4 resmpt2 6187 . . . 4  |-  ( ( I  C_  M  /\  P  C_  P )  -> 
( ( i  e.  M ,  j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r `  R
) ( k Y j ) ) ) ) )  |`  (
I  X.  P ) )  =  ( i  e.  I ,  j  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r
`  R ) ( k Y j ) ) ) ) ) )
51, 3, 4syl2anc 661 . . 3  |-  ( ph  ->  ( ( i  e.  M ,  j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r `  R
) ( k Y j ) ) ) ) )  |`  (
I  X.  P ) )  =  ( i  e.  I ,  j  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r
`  R ) ( k Y j ) ) ) ) ) )
6 ovres 6229 . . . . . . . . 9  |-  ( ( i  e.  I  /\  k  e.  N )  ->  ( i ( X  |`  ( I  X.  N
) ) k )  =  ( i X k ) )
763ad2antl2 1151 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  I  /\  j  e.  P )  /\  k  e.  N )  ->  (
i ( X  |`  ( I  X.  N
) ) k )  =  ( i X k ) )
87eqcomd 2447 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  I  /\  j  e.  P )  /\  k  e.  N )  ->  (
i X k )  =  ( i ( X  |`  ( I  X.  N ) ) k ) )
98oveq1d 6105 . . . . . 6  |-  ( ( ( ph  /\  i  e.  I  /\  j  e.  P )  /\  k  e.  N )  ->  (
( i X k ) ( .r `  R ) ( k Y j ) )  =  ( ( i ( X  |`  (
I  X.  N ) ) k ) ( .r `  R ) ( k Y j ) ) )
109mpteq2dva 4377 . . . . 5  |-  ( (
ph  /\  i  e.  I  /\  j  e.  P
)  ->  ( k  e.  N  |->  ( ( i X k ) ( .r `  R
) ( k Y j ) ) )  =  ( k  e.  N  |->  ( ( i ( X  |`  (
I  X.  N ) ) k ) ( .r `  R ) ( k Y j ) ) ) )
1110oveq2d 6106 . . . 4  |-  ( (
ph  /\  i  e.  I  /\  j  e.  P
)  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r `  R ) ( k Y j ) ) ) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i ( X  |`  ( I  X.  N ) ) k ) ( .r `  R ) ( k Y j ) ) ) ) )
1211mpt2eq3dva 6149 . . 3  |-  ( ph  ->  ( i  e.  I ,  j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r
`  R ) ( k Y j ) ) ) ) )  =  ( i  e.  I ,  j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i ( X  |`  ( I  X.  N
) ) k ) ( .r `  R
) ( k Y j ) ) ) ) ) )
135, 12eqtrd 2474 . 2  |-  ( ph  ->  ( ( i  e.  M ,  j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r `  R
) ( k Y j ) ) ) ) )  |`  (
I  X.  P ) )  =  ( i  e.  I ,  j  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( i ( X  |`  ( I  X.  N ) ) k ) ( .r `  R ) ( k Y j ) ) ) ) ) )
14 mamures.f . . . 4  |-  F  =  ( R maMul  <. M ,  N ,  P >. )
15 mamures.b . . . 4  |-  B  =  ( Base `  R
)
16 eqid 2442 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
17 mamures.r . . . 4  |-  ( ph  ->  R  e.  V )
18 mamures.m . . . 4  |-  ( ph  ->  M  e.  Fin )
19 mamures.n . . . 4  |-  ( ph  ->  N  e.  Fin )
20 mamures.p . . . 4  |-  ( ph  ->  P  e.  Fin )
21 mamures.x . . . 4  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
22 mamures.y . . . 4  |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )
2314, 15, 16, 17, 18, 19, 20, 21, 22mamuval 18283 . . 3  |-  ( ph  ->  ( X F Y )  =  ( i  e.  M ,  j  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r
`  R ) ( k Y j ) ) ) ) ) )
2423reseq1d 5108 . 2  |-  ( ph  ->  ( ( X F Y )  |`  (
I  X.  P ) )  =  ( ( i  e.  M , 
j  e.  P  |->  ( R  gsumg  ( k  e.  N  |->  ( ( i X k ) ( .r
`  R ) ( k Y j ) ) ) ) )  |`  ( I  X.  P
) ) )
25 mamures.g . . 3  |-  G  =  ( R maMul  <. I ,  N ,  P >. )
26 ssfi 7532 . . . 4  |-  ( ( M  e.  Fin  /\  I  C_  M )  ->  I  e.  Fin )
2718, 1, 26syl2anc 661 . . 3  |-  ( ph  ->  I  e.  Fin )
28 elmapi 7233 . . . . . 6  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
2921, 28syl 16 . . . . 5  |-  ( ph  ->  X : ( M  X.  N ) --> B )
30 xpss1 4947 . . . . . 6  |-  ( I 
C_  M  ->  (
I  X.  N ) 
C_  ( M  X.  N ) )
311, 30syl 16 . . . . 5  |-  ( ph  ->  ( I  X.  N
)  C_  ( M  X.  N ) )
32 fssres 5577 . . . . 5  |-  ( ( X : ( M  X.  N ) --> B  /\  ( I  X.  N )  C_  ( M  X.  N ) )  ->  ( X  |`  ( I  X.  N
) ) : ( I  X.  N ) --> B )
3329, 31, 32syl2anc 661 . . . 4  |-  ( ph  ->  ( X  |`  (
I  X.  N ) ) : ( I  X.  N ) --> B )
34 fvex 5700 . . . . . . 7  |-  ( Base `  R )  e.  _V
3515, 34eqeltri 2512 . . . . . 6  |-  B  e. 
_V
3635a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
37 xpfi 7582 . . . . . 6  |-  ( ( I  e.  Fin  /\  N  e.  Fin )  ->  ( I  X.  N
)  e.  Fin )
3827, 19, 37syl2anc 661 . . . . 5  |-  ( ph  ->  ( I  X.  N
)  e.  Fin )
39 elmapg 7226 . . . . 5  |-  ( ( B  e.  _V  /\  ( I  X.  N
)  e.  Fin )  ->  ( ( X  |`  ( I  X.  N
) )  e.  ( B  ^m  ( I  X.  N ) )  <-> 
( X  |`  (
I  X.  N ) ) : ( I  X.  N ) --> B ) )
4036, 38, 39syl2anc 661 . . . 4  |-  ( ph  ->  ( ( X  |`  ( I  X.  N
) )  e.  ( B  ^m  ( I  X.  N ) )  <-> 
( X  |`  (
I  X.  N ) ) : ( I  X.  N ) --> B ) )
4133, 40mpbird 232 . . 3  |-  ( ph  ->  ( X  |`  (
I  X.  N ) )  e.  ( B  ^m  ( I  X.  N ) ) )
4225, 15, 16, 17, 27, 19, 20, 41, 22mamuval 18283 . 2  |-  ( ph  ->  ( ( X  |`  ( I  X.  N
) ) G Y )  =  ( i  e.  I ,  j  e.  P  |->  ( R 
gsumg  ( k  e.  N  |->  ( ( i ( X  |`  ( I  X.  N ) ) k ) ( .r `  R ) ( k Y j ) ) ) ) ) )
4313, 24, 423eqtr4d 2484 1  |-  ( ph  ->  ( ( X F Y )  |`  (
I  X.  P ) )  =  ( ( X  |`  ( I  X.  N ) ) G Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2971    C_ wss 3327   <.cotp 3884    e. cmpt 4349    X. cxp 4837    |` cres 4841   -->wf 5413   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7213   Fincfn 7309   Basecbs 14173   .rcmulr 14238    gsumg cgsu 14378   maMul cmmul 18278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-ot 3885  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-fin 7313  df-mamu 18280
This theorem is referenced by:  mdetmul  18428
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