Theorem minmar1fval 20452

Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)

Hypotheses

Ref	Expression
minmar1fval.a	|- A = ( N Mat R )
minmar1fval.b	|- B = ( Base ` A )
minmar1fval.q	|- Q = ( N minMatR1 R )
minmar1fval.o	|- .1. = ( 1r ` R )
minmar1fval.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
minmar1fval	|- Q = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( 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)   .1. ( i, j, k, m, l)   .0. ( i, j, k, m, l)

Proof of Theorem minmar1fval

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

Step	Hyp	Ref	Expression
1		minmar1fval.q	. 2 |- Q = ( N minMatR1 R )
2		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
3		minmar1fval.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		minmar1fval.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		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
10		minmar1fval.o	. . . . . . . . . . 11 |- .1. = ( 1r ` R )
11	9, 10	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 1r ` r ) = .1. )
12		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
13		minmar1fval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
14	12, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
15	11, 14	ifeq12d 4106	. . . . . . . . 9 |- ( r = R -> if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) = if ( j = l , .1. , .0. ) )
16	15	ifeq1d 4104	. . . . . . . 8 |- ( r = R -> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) = if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) )
17	16	adantl 482	. . . . . . 7 |- ( ( n = N /\ r = R ) -> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) = if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) )
18	8, 8, 17	mpt2eq123dv 6717	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( i e. n , j e. n |-> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) )
19	8, 8, 18	mpt2eq123dv 6717	. . . . 5 |- ( ( n = N /\ r = R ) -> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) ) ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) )
20	7, 19	mpteq12dv 4733	. . . 4 |- ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) ) ) ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) )
21		df-minmar1 20441	. . . 4 |- minMatR1 = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( k e. n , l e. n |-> ( i e. n , j e. n |-> if ( i = k , if ( j = l , ( 1r ` r ) , ( 0g ` r ) ) , ( i m j ) ) ) ) ) )
22		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
23	6, 22	eqeltri 2697	. . . . 5 |- B e. _V
24	23	mptex 6486	. . . 4 |- ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) e. _V
25	20, 21, 24	ovmpt2a 6791	. . 3 |- ( ( N e. _V /\ R e. _V ) -> ( N minMatR1 R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) )
26	21	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( N minMatR1 R ) = (/) )
27		mpt0 6021	. . . . 5 |- ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) = (/)
28	26, 27	syl6eqr 2674	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N minMatR1 R ) = ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) )
29	3	fveq2i 6194	. . . . . . 7 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
30	6, 29	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat R ) )
31		matbas0pc 20215	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
32	30, 31	syl5eq 2668	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
33	32	mpteq1d 4738	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) = ( m e. (/) |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) )
34	28, 33	eqtr4d 2659	. . 3 |- ( -. ( N e. _V /\ R e. _V ) -> ( N minMatR1 R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) ) )
35	25, 34	pm2.61i 176	. 2 |- ( N minMatR1 R ) = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i m j ) ) ) ) )
36	1, 35	eqtri 2644	1 |- Q = ( m e. B |-> ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( 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   Basecbs 15857   0gc0g 16100   1rcur 18501   Mat cmat 20213   minMatR1 cminmar1 20439

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-minmar1 20441

This theorem is referenced by:  minmar1val0  20453