Theorem dmatel 20299

Description: A N x N diagonal matrix over (a ring) R. (Contributed by AV, 18-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
dmatel	|- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )

Distinct variable groups:   i, N, j   R, i, j   i, M, j

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

Proof of Theorem dmatel

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatval.a	. . . 4 |- A = ( N Mat R )
2		dmatval.b	. . . 4 |- B = ( Base ` A )
3		dmatval.0	. . . 4 |- .0. = ( 0g ` R )
4		dmatval.d	. . . 4 |- D = ( N DMat R )
5	1, 2, 3, 4	dmatval 20298	. . 3 |- ( ( N e. Fin /\ R e. V ) -> D = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
6	5	eleq2d 2687	. 2 |- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> M e. { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) )
7		oveq 6656	. . . . . 6 |- ( m = M -> ( i m j ) = ( i M j ) )
8	7	eqeq1d 2624	. . . . 5 |- ( m = M -> ( ( i m j ) = .0. <-> ( i M j ) = .0. ) )
9	8	imbi2d 330	. . . 4 |- ( m = M -> ( ( i =/= j -> ( i m j ) = .0. ) <-> ( i =/= j -> ( i M j ) = .0. ) ) )
10	9	2ralbidv 2989	. . 3 |- ( m = M -> ( A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) <-> A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
11	10	elrab 3363	. 2 |- ( M e. { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
12	6, 11	syl6bb 276	1 |- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   ` cfv 5888  (class class class)co 6650   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:  dmatmat  20300  dmatid  20301  dmatelnd  20302  dmatsubcl  20304  dmatscmcl  20309