Theorem dmatelnd 20302

Description: An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)

Hypotheses

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

Assertion

Ref	Expression
dmatelnd	|- ( ( ( N e. Fin /\ R e. Ring /\ X e. D ) /\ ( I e. N /\ J e. N /\ I =/= J ) ) -> ( I X J ) = .0. )

Proof of Theorem dmatelnd

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatid.a	. . . . 5 |- A = ( N Mat R )
2		dmatid.b	. . . . 5 |- B = ( Base ` A )
3		dmatid.0	. . . . 5 |- .0. = ( 0g ` R )
4		dmatid.d	. . . . 5 |- D = ( N DMat R )
5	1, 2, 3, 4	dmatel 20299	. . . 4 |- ( ( N e. Fin /\ R e. Ring ) -> ( X e. D <-> ( X e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) ) ) )
6		neeq1 2856	. . . . . . . . . . 11 |- ( i = I -> ( i =/= j <-> I =/= j ) )
7		oveq1 6657	. . . . . . . . . . . 12 |- ( i = I -> ( i X j ) = ( I X j ) )
8	7	eqeq1d 2624	. . . . . . . . . . 11 |- ( i = I -> ( ( i X j ) = .0. <-> ( I X j ) = .0. ) )
9	6, 8	imbi12d 334	. . . . . . . . . 10 |- ( i = I -> ( ( i =/= j -> ( i X j ) = .0. ) <-> ( I =/= j -> ( I X j ) = .0. ) ) )
10		neeq2 2857	. . . . . . . . . . 11 |- ( j = J -> ( I =/= j <-> I =/= J ) )
11		oveq2 6658	. . . . . . . . . . . 12 |- ( j = J -> ( I X j ) = ( I X J ) )
12	11	eqeq1d 2624	. . . . . . . . . . 11 |- ( j = J -> ( ( I X j ) = .0. <-> ( I X J ) = .0. ) )
13	10, 12	imbi12d 334	. . . . . . . . . 10 |- ( j = J -> ( ( I =/= j -> ( I X j ) = .0. ) <-> ( I =/= J -> ( I X J ) = .0. ) ) )
14	9, 13	rspc2v 3322	. . . . . . . . 9 |- ( ( I e. N /\ J e. N ) -> ( A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) -> ( I =/= J -> ( I X J ) = .0. ) ) )
15	14	com23 86	. . . . . . . 8 |- ( ( I e. N /\ J e. N ) -> ( I =/= J -> ( A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) -> ( I X J ) = .0. ) ) )
16	15	3impia 1261	. . . . . . 7 |- ( ( I e. N /\ J e. N /\ I =/= J ) -> ( A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) -> ( I X J ) = .0. ) )
17	16	com12 32	. . . . . 6 |- ( A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) -> ( ( I e. N /\ J e. N /\ I =/= J ) -> ( I X J ) = .0. ) )
18	17	2a1i 12	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> ( X e. B -> ( A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) -> ( ( I e. N /\ J e. N /\ I =/= J ) -> ( I X J ) = .0. ) ) ) )
19	18	impd 447	. . . 4 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( X e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i X j ) = .0. ) ) -> ( ( I e. N /\ J e. N /\ I =/= J ) -> ( I X J ) = .0. ) ) )
20	5, 19	sylbid 230	. . 3 |- ( ( N e. Fin /\ R e. Ring ) -> ( X e. D -> ( ( I e. N /\ J e. N /\ I =/= J ) -> ( I X J ) = .0. ) ) )
21	20	3impia 1261	. 2 |- ( ( N e. Fin /\ R e. Ring /\ X e. D ) -> ( ( I e. N /\ J e. N /\ I =/= J ) -> ( I X J ) = .0. ) )
22	21	imp 445	1 |- ( ( ( N e. Fin /\ R e. Ring /\ X e. D ) /\ ( I e. N /\ J e. N /\ I =/= J ) ) -> ( I X J ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   0gc0g 16100   Ringcrg 18547   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-ne 2795  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:  dmatmul  20303  dmatsubcl  20304