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Theorem mamures 18687
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 3523 . . . . 5  |-  P  C_  P
32a1i 11 . . . 4  |-  ( ph  ->  P  C_  P )
4 resmpt2 6384 . . . 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 6426 . . . . . . . . 9  |-  ( ( i  e.  I  /\  k  e.  N )  ->  ( i ( X  |`  ( I  X.  N
) ) k )  =  ( i X k ) )
763ad2antl2 1159 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  I  /\  j  e.  P )  /\  k  e.  N )  ->  (
i ( X  |`  ( I  X.  N
) ) k )  =  ( i X k ) )
87eqcomd 2475 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  I  /\  j  e.  P )  /\  k  e.  N )  ->  (
i X k )  =  ( i ( X  |`  ( I  X.  N ) ) k ) )
98oveq1d 6299 . . . . . 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 4533 . . . . 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 6300 . . . 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 6345 . . 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 2508 . 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 2467 . . . 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 18683 . . 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 5272 . 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 7740 . . . 4  |-  ( ( M  e.  Fin  /\  I  C_  M )  ->  I  e.  Fin )
2718, 1, 26syl2anc 661 . . 3  |-  ( ph  ->  I  e.  Fin )
28 elmapi 7440 . . . . . 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 5111 . . . . . 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 5751 . . . . 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 5876 . . . . . . 7  |-  ( Base `  R )  e.  _V
3515, 34eqeltri 2551 . . . . . 6  |-  B  e. 
_V
3635a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
37 xpfi 7791 . . . . . 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 7433 . . . . 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 18683 . 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 2518 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   <.cotp 4035    |-> cmpt 4505    X. cxp 4997    |` cres 5001   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    ^m cmap 7420   Fincfn 7516   Basecbs 14490   .rcmulr 14556    gsumg cgsu 14696   maMul cmmul 18680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-fin 7520  df-mamu 18681
This theorem is referenced by:  mdetmul  18920
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