Theorem scmatval 20310

Description: The set of N x N scalar matrices over (a ring) R. (Contributed by AV, 18-Dec-2019.)

Hypotheses

Ref	Expression
scmatval.k	|- K = ( Base ` R )
scmatval.a	|- A = ( N Mat R )
scmatval.b	|- B = ( Base ` A )
scmatval.1	|- .1. = ( 1r ` A )
scmatval.t	|- .x. = ( .s ` A )
scmatval.s	|- S = ( N ScMat R )

Assertion

Ref	Expression
scmatval	|- ( ( N e. Fin /\ R e. V ) -> S = { m e. B | E. c e. K m = ( c .x. .1. ) } )

Distinct variable groups:   B, m   K, c   N, c, m   R, c, m

Allowed substitution hints:   A( m, c)   B( c)   S( m, c)   .x. ( m, c)   .1. ( m, c)   K( m)   V( m, c)

Proof of Theorem scmatval

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

Step	Hyp	Ref	Expression
1		scmatval.s	. 2 |- S = ( N ScMat R )
2		df-scmat 20297	. . . 4 |- ScMat = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } )
3	2	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> ScMat = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } ) )
4		ovexd 6680	. . . . 5 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( n Mat r ) e. _V )
5		fveq2 6191	. . . . . . 7 |- ( a = ( n Mat r ) -> ( Base ` a ) = ( Base ` ( n Mat r ) ) )
6		fveq2 6191	. . . . . . . . . 10 |- ( a = ( n Mat r ) -> ( .s ` a ) = ( .s ` ( n Mat r ) ) )
7		eqidd 2623	. . . . . . . . . 10 |- ( a = ( n Mat r ) -> c = c )
8		fveq2 6191	. . . . . . . . . 10 |- ( a = ( n Mat r ) -> ( 1r ` a ) = ( 1r ` ( n Mat r ) ) )
9	6, 7, 8	oveq123d 6671	. . . . . . . . 9 |- ( a = ( n Mat r ) -> ( c ( .s ` a ) ( 1r ` a ) ) = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) )
10	9	eqeq2d 2632	. . . . . . . 8 |- ( a = ( n Mat r ) -> ( m = ( c ( .s ` a ) ( 1r ` a ) ) <-> m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) ) )
11	10	rexbidv 3052	. . . . . . 7 |- ( a = ( n Mat r ) -> ( E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) <-> E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) ) )
12	5, 11	rabeqbidv 3195	. . . . . 6 |- ( a = ( n Mat r ) -> { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } = { m e. ( Base ` ( n Mat r ) ) | E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) } )
13	12	adantl 482	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) /\ a = ( n Mat r ) ) -> { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } = { m e. ( Base ` ( n Mat r ) ) | E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) } )
14	4, 13	csbied 3560	. . . 4 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } = { m e. ( Base ` ( n Mat r ) ) | E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) } )
15		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
16	15	fveq2d 6195	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat R ) ) )
17		scmatval.b	. . . . . . . 8 |- B = ( Base ` A )
18		scmatval.a	. . . . . . . . 9 |- A = ( N Mat R )
19	18	fveq2i 6194	. . . . . . . 8 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
20	17, 19	eqtri 2644	. . . . . . 7 |- B = ( Base ` ( N Mat R ) )
21	16, 20	syl6eqr 2674	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
22		fveq2 6191	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
23		scmatval.k	. . . . . . . . 9 |- K = ( Base ` R )
24	22, 23	syl6eqr 2674	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = K )
25	24	adantl 482	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( Base ` r ) = K )
26	15	fveq2d 6195	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> ( .s ` ( n Mat r ) ) = ( .s ` ( N Mat R ) ) )
27		scmatval.t	. . . . . . . . . . 11 |- .x. = ( .s ` A )
28	18	fveq2i 6194	. . . . . . . . . . 11 |- ( .s ` A ) = ( .s ` ( N Mat R ) )
29	27, 28	eqtri 2644	. . . . . . . . . 10 |- .x. = ( .s ` ( N Mat R ) )
30	26, 29	syl6eqr 2674	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( .s ` ( n Mat r ) ) = .x. )
31		eqidd 2623	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> c = c )
32	15	fveq2d 6195	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> ( 1r ` ( n Mat r ) ) = ( 1r ` ( N Mat R ) ) )
33		scmatval.1	. . . . . . . . . . 11 |- .1. = ( 1r ` A )
34	18	fveq2i 6194	. . . . . . . . . . 11 |- ( 1r ` A ) = ( 1r ` ( N Mat R ) )
35	33, 34	eqtri 2644	. . . . . . . . . 10 |- .1. = ( 1r ` ( N Mat R ) )
36	32, 35	syl6eqr 2674	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( 1r ` ( n Mat r ) ) = .1. )
37	30, 31, 36	oveq123d 6671	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) = ( c .x. .1. ) )
38	37	eqeq2d 2632	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) <-> m = ( c .x. .1. ) ) )
39	25, 38	rexeqbidv 3153	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) <-> E. c e. K m = ( c .x. .1. ) ) )
40	21, 39	rabeqbidv 3195	. . . . 5 |- ( ( n = N /\ r = R ) -> { m e. ( Base ` ( n Mat r ) ) | E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) } = { m e. B | E. c e. K m = ( c .x. .1. ) } )
41	40	adantl 482	. . . 4 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> { m e. ( Base ` ( n Mat r ) ) | E. c e. ( Base ` r ) m = ( c ( .s ` ( n Mat r ) ) ( 1r ` ( n Mat r ) ) ) } = { m e. B | E. c e. K m = ( c .x. .1. ) } )
42	14, 41	eqtrd 2656	. . 3 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } = { m e. B | E. c e. K m = ( c .x. .1. ) } )
43		simpl 473	. . 3 |- ( ( N e. Fin /\ R e. V ) -> N e. Fin )
44		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
45	44	adantl 482	. . 3 |- ( ( N e. Fin /\ R e. V ) -> R e. _V )
46		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
47	17, 46	eqeltri 2697	. . . . 5 |- B e. _V
48	47	rabex 4813	. . . 4 |- { m e. B | E. c e. K m = ( c .x. .1. ) } e. _V
49	48	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> { m e. B | E. c e. K m = ( c .x. .1. ) } e. _V )
50	3, 42, 43, 45, 49	ovmpt2d 6788	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N ScMat R ) = { m e. B | E. c e. K m = ( c .x. .1. ) } )
51	1, 50	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> S = { m e. B | E. c e. K m = ( c .x. .1. ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   [_csb 3533   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   .scvsca 15945   1rcur 18501   Mat cmat 20213   ScMat cscmat 20295

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-scmat 20297

This theorem is referenced by:  scmatel  20311  scmatmats  20317  scmatlss  20331