Theorem marepvfval 20371

Description: First 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
marepvfval	|- Q = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) )

Distinct variable groups:   B, m, v   i, N, j, k, m, v   R, i, j, k, m, v   m, V, v

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

Proof of Theorem marepvfval

Dummy variables n r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvfval.q	. 2 |- Q = ( N matRepV R )
2		marepvfval.b	. . . . . 6 |- B = ( Base ` A )
3		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
4	2, 3	eqeltri 2697	. . . . 5 |- B e. _V
5		marepvfval.v	. . . . . . 7 |- V = ( ( Base ` R ) ^m N )
6		ovex 6678	. . . . . . 7 |- ( ( Base ` R ) ^m N ) e. _V
7	5, 6	eqeltri 2697	. . . . . 6 |- V e. _V
8	7	a1i 11	. . . . 5 |- ( ( N e. _V /\ R e. _V ) -> V e. _V )
9		mpt2exga 7246	. . . . 5 |- ( ( B e. _V /\ V e. _V ) -> ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) e. _V )
10	4, 8, 9	sylancr 695	. . . 4 |- ( ( N e. _V /\ R e. _V ) -> ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) e. _V )
11		oveq12 6659	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
12		marepvfval.a	. . . . . . . . 9 |- A = ( N Mat R )
13	11, 12	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = A )
14	13	fveq2d 6195	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` A ) )
15	14, 2	syl6eqr 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
16		fveq2 6191	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( Base ` r ) = ( Base ` R ) )
18		simpl 473	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> n = N )
19	17, 18	oveq12d 6668	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( ( Base ` r ) ^m n ) = ( ( Base ` R ) ^m N ) )
20	19, 5	syl6eqr 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( ( Base ` r ) ^m n ) = V )
21		eqidd 2623	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> if ( j = k , ( v ` i ) , ( i m j ) ) = if ( j = k , ( v ` i ) , ( i m j ) ) )
22	18, 18, 21	mpt2eq123dv 6717	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) )
23	18, 22	mpteq12dv 4733	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) )
24	15, 20, 23	mpt2eq123dv 6717	. . . . 5 |- ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
25		df-marepv 20365	. . . . 5 |- matRepV = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
26	24, 25	ovmpt2ga 6790	. . . 4 |- ( ( N e. _V /\ R e. _V /\ ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) e. _V ) -> ( N matRepV R ) = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
27	10, 26	mpd3an3 1425	. . 3 |- ( ( N e. _V /\ R e. _V ) -> ( N matRepV R ) = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
28	25	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( N matRepV R ) = (/) )
29		mpt20 6725	. . . . 5 |- ( m e. (/) , v e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) = (/)
30	28, 29	syl6eqr 2674	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N matRepV R ) = ( m e. (/) , v e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
31	12	fveq2i 6194	. . . . . . 7 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
32	2, 31	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat R ) )
33		matbas0pc 20215	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
34	32, 33	syl5eq 2668	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
35		mpt2eq12 6715	. . . . 5 |- ( ( B = (/) /\ V = ( ( Base ` R ) ^m N ) ) -> ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) = ( m e. (/) , v e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
36	34, 5, 35	sylancl 694	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) = ( m e. (/) , v e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
37	30, 36	eqtr4d 2659	. . 3 |- ( -. ( N e. _V /\ R e. _V ) -> ( N matRepV R ) = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
38	27, 37	pm2.61i 176	. 2 |- ( N matRepV R ) = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) )
39	1, 38	eqtri 2644	1 |- Q = ( m e. B , v e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   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:  marepvval0  20372