Theorem dmatval 20298

Description: The set of N x N diagonal matrices over (a ring) R. (Contributed by AV, 8-Dec-2019.)

Hypotheses

Ref	Expression
dmatval.a	|- A = ( N Mat R )
dmatval.b	|- B = ( Base ` A )
dmatval.0	|- .0. = ( 0g ` R )
dmatval.d	|- D = ( N DMat R )

Assertion

Ref	Expression
dmatval	|- ( ( N e. Fin /\ R e. V ) -> D = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )

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

Allowed substitution hints:   A( i, j, m)   B( i, j)   D( i, j, m)   V( i, j, m)   .0. ( i, j, m)

Proof of Theorem dmatval

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

Step	Hyp	Ref	Expression
1		dmatval.d	. 2 |- D = ( N DMat R )
2		df-dmat 20296	. . . 4 |- DMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } )
3	2	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. V ) -> DMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } ) )
4		oveq12 6659	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
5	4	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat R ) ) )
6		dmatval.b	. . . . . . 7 |- B = ( Base ` A )
7		dmatval.a	. . . . . . . 8 |- A = ( N Mat R )
8	7	fveq2i 6194	. . . . . . 7 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
9	6, 8	eqtri 2644	. . . . . 6 |- B = ( Base ` ( N Mat R ) )
10	5, 9	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
11		simpl 473	. . . . . 6 |- ( ( n = N /\ r = R ) -> n = N )
12		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
13		dmatval.0	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
14	12, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
15	14	adantl 482	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( 0g ` r ) = .0. )
16	15	eqeq2d 2632	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( ( i m j ) = ( 0g ` r ) <-> ( i m j ) = .0. ) )
17	16	imbi2d 330	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( ( i =/= j -> ( i m j ) = ( 0g ` r ) ) <-> ( i =/= j -> ( i m j ) = .0. ) ) )
18	11, 17	raleqbidv 3152	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) <-> A. j e. N ( i =/= j -> ( i m j ) = .0. ) ) )
19	11, 18	raleqbidv 3152	. . . . 5 |- ( ( n = N /\ r = R ) -> ( A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) <-> A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) ) )
20	10, 19	rabeqbidv 3195	. . . 4 |- ( ( n = N /\ r = R ) -> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
21	20	adantl 482	. . 3 |- ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
22		simpl 473	. . 3 |- ( ( N e. Fin /\ R e. V ) -> N e. Fin )
23		elex 3212	. . . 4 |- ( R e. V -> R e. _V )
24	23	adantl 482	. . 3 |- ( ( N e. Fin /\ R e. V ) -> R e. _V )
25		fvex 6201	. . . . 5 |- ( Base ` A ) e. _V
26	6, 25	eqeltri 2697	. . . 4 |- B e. _V
27		rabexg 4812	. . . 4 |- ( B e. _V -> { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } e. _V )
28	26, 27	mp1i 13	. . 3 |- ( ( N e. Fin /\ R e. V ) -> { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } e. _V )
29	3, 21, 22, 24, 28	ovmpt2d 6788	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( N DMat R ) = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
30	1, 29	syl5eq 2668	1 |- ( ( N e. Fin /\ R e. V ) -> D = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   0gc0g 16100   Mat cmat 20213   DMat cdmat 20294

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-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-dmat 20296

This theorem is referenced by:  dmatel  20299  dmatmulcl  20306  scmatdmat  20321  dmatbas  42192