Theorem marepvval0 20372

Description: Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)

Hypotheses

Ref	Expression
marepvfval.a	|- A = ( N Mat R )
marepvfval.b	|- B = ( Base ` A )
marepvfval.q	|- Q = ( N matRepV R )
marepvfval.v	|- V = ( ( Base ` R ) ^m N )

Assertion

Ref	Expression
marepvval0	|- ( ( M e. B /\ C e. V ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )

Distinct variable groups:   i, N, j, k   R, i, j, k   C, i, j, k   i, M, j, k

Allowed substitution hints:   A( i, j, k)   B( i, j, k)   Q( i, j, k)   V( i, j, k)

Proof of Theorem marepvval0

Dummy variables m c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvfval.a	. . . . . 6 |- A = ( N Mat R )
2		marepvfval.b	. . . . . 6 |- B = ( Base ` A )
3	1, 2	matrcl 20218	. . . . 5 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
4	3	simpld 475	. . . 4 |- ( M e. B -> N e. Fin )
5	4	adantr 481	. . 3 |- ( ( M e. B /\ C e. V ) -> N e. Fin )
6		mptexg 6484	. . 3 |- ( N e. Fin -> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) e. _V )
7	5, 6	syl 17	. 2 |- ( ( M e. B /\ C e. V ) -> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) e. _V )
8		fveq1 6190	. . . . . . 7 |- ( c = C -> ( c ` i ) = ( C ` i ) )
9	8	adantl 482	. . . . . 6 |- ( ( m = M /\ c = C ) -> ( c ` i ) = ( C ` i ) )
10		oveq 6656	. . . . . . 7 |- ( m = M -> ( i m j ) = ( i M j ) )
11	10	adantr 481	. . . . . 6 |- ( ( m = M /\ c = C ) -> ( i m j ) = ( i M j ) )
12	9, 11	ifeq12d 4106	. . . . 5 |- ( ( m = M /\ c = C ) -> if ( j = k , ( c ` i ) , ( i m j ) ) = if ( j = k , ( C ` i ) , ( i M j ) ) )
13	12	mpt2eq3dv 6721	. . . 4 |- ( ( m = M /\ c = C ) -> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) )
14	13	mpteq2dv 4745	. . 3 |- ( ( m = M /\ c = C ) -> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )
15		marepvfval.q	. . . 4 |- Q = ( N matRepV R )
16		marepvfval.v	. . . 4 |- V = ( ( Base ` R ) ^m N )
17	1, 2, 15, 16	marepvfval 20371	. . 3 |- Q = ( m e. B , c e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) ) )
18	14, 17	ovmpt2ga 6790	. 2 |- ( ( M e. B /\ C e. V /\ ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) e. _V ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )
19	7, 18	mpd3an3 1425	1 |- ( ( M e. B /\ C e. V ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   Basecbs 15857   Mat cmat 20213   matRepV cmatrepV 20363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-mat 20214  df-marepv 20365

This theorem is referenced by:  marepvval  20373