Theorem dmatcomp 18414

Description: The components of diagonal matrices. (Contributed by AV, 22-Jul-2019.)

Hypotheses

Ref	Expression
mamucl.b	|- B = ( Base ` R )
mamucl.r	|- ( ph -> R e. Ring )
mamudiag.o	|- .1. = ( 1r ` R )
mamudiag.z	|- .0. = ( 0g ` R )
mamudiag.i	|- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
mamudiag.m	|- ( ph -> M e. Fin )

Assertion

Ref	Expression
dmatcomp	|- ( ( A e. M /\ J e. M ) -> ( A I J ) = if ( A = J , .1. , .0. ) )

Distinct variable groups:   i, j, B   i, M, j   ph, i, j   .1. , i, j   .0. , i, j   A, i, j   i, J, j

Allowed substitution hints:   R( i, j)   I( i, j)

Proof of Theorem dmatcomp

Step	Hyp	Ref	Expression
1		eqeq1 2455	. . 3 |- ( i = A -> ( i = j <-> A = j ) )
2	1	ifbid 3911	. 2 |- ( i = A -> if ( i = j , .1. , .0. ) = if ( A = j , .1. , .0. ) )
3		eqeq2 2466	. . 3 |- ( j = J -> ( A = j <-> A = J ) )
4	3	ifbid 3911	. 2 |- ( j = J -> if ( A = j , .1. , .0. ) = if ( A = J , .1. , .0. ) )
5		mamudiag.i	. 2 |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
6		mamudiag.o	. . . 4 |- .1. = ( 1r ` R )
7		fvex 5801	. . . 4 |- ( 1r ` R ) e. _V
8	6, 7	eqeltri 2535	. . 3 |- .1. e. _V
9		mamudiag.z	. . . 4 |- .0. = ( 0g ` R )
10		fvex 5801	. . . 4 |- ( 0g ` R ) e. _V
11	9, 10	eqeltri 2535	. . 3 |- .0. e. _V
12	8, 11	ifex 3958	. 2 |- if ( A = J , .1. , .0. ) e. _V
13	2, 4, 5, 12	ovmpt2 6328	1 |- ( ( A e. M /\ J e. M ) -> ( A I J ) = if ( A = J , .1. , .0. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3070   ifcif 3891   ` cfv 5518  (class class class)co 6192   |-> cmpt2 6194   Fincfn 7412   Basecbs 14278   0gc0g 14482   1rcur 16710   Ringcrg 16753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197

This theorem is referenced by:  mamulid  18415  mamurid  18416