Theorem submafval 20385

Description: First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)

Hypotheses

Ref	Expression
submafval.a	|- A = ( N Mat R )
submafval.q	|- Q = ( N subMat R )
submafval.b	|- B = ( Base ` A )

Assertion

Ref	Expression
submafval	|- Q = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) )

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

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

Proof of Theorem submafval

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

Step	Hyp	Ref	Expression
1		submafval.q	. 2 |- Q = ( N subMat R )
2		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
3		submafval.a	. . . . . . . 8 |- A = ( N Mat R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = A )
5	4	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` A ) )
6		submafval.b	. . . . . 6 |- B = ( Base ` A )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
8		simpl 473	. . . . . 6 |- ( ( n = N /\ r = R ) -> n = N )
9		difeq1 3721	. . . . . . . 8 |- ( n = N -> ( n \ { k } ) = ( N \ { k } ) )
10	9	adantr 481	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n \ { k } ) = ( N \ { k } ) )
11		difeq1 3721	. . . . . . . 8 |- ( n = N -> ( n \ { l } ) = ( N \ { l } ) )
12	11	adantr 481	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n \ { l } ) = ( N \ { l } ) )
13		eqidd 2623	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( i m j ) = ( i m j ) )
14	10, 12, 13	mpt2eq123dv 6717	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( i e. ( n \ { k } ) , j e. ( n \ { l } ) |-> ( i m j ) ) = ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) )
15	8, 8, 14	mpt2eq123dv 6717	. . . . 5 |- ( ( n = N /\ r = R ) -> ( k e. n , l e. n |-> ( i e. ( n \ { k } ) , j e. ( n \ { l } ) |-> ( i m j ) ) ) = ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) )
16	7, 15	mpteq12dv 4733	. . . 4 |- ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. ( n \ { k } ) , j e. ( n \ { l } ) |-> ( i m j ) ) ) ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) )
17		df-subma 20383	. . . 4 |- subMat = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. ( n \ { k } ) , j e. ( n \ { l } ) |-> ( i m j ) ) ) ) )
18		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
19	6, 18	eqeltri 2697	. . . . 5 |- B e. _V
20	19	mptex 6486	. . . 4 |- ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) e. _V
21	16, 17, 20	ovmpt2a 6791	. . 3 |- ( ( N e. _V /\ R e. _V ) -> ( N subMat R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) )
22	17	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( N subMat R ) = (/) )
23		mpt0 6021	. . . . 5 |- ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) = (/)
24	22, 23	syl6eqr 2674	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N subMat R ) = ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) )
25	3	fveq2i 6194	. . . . . . 7 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
26	6, 25	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat R ) )
27		matbas0pc 20215	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
28	26, 27	syl5eq 2668	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
29	28	mpteq1d 4738	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) = ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) )
30	24, 29	eqtr4d 2659	. . 3 |- ( -. ( N e. _V /\ R e. _V ) -> ( N subMat R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) ) )
31	21, 30	pm2.61i 176	. 2 |- ( N subMat R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) )
32	1, 31	eqtri 2644	1 |- Q = ( m e. B |-> ( k e. N , l e. N |-> ( i e. ( N \ { k } ) , j e. ( N \ { l } ) |-> ( i m j ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Mat cmat 20213   subMat csubma 20382

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

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-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-slot 15861  df-base 15863  df-mat 20214  df-subma 20383

This theorem is referenced by:  submaval0  20386