Theorem mat1comp 20246

Description: The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)

Hypotheses

Ref	Expression
mamumat1cl.b	|- B = ( Base ` R )
mamumat1cl.r	|- ( ph -> R e. Ring )
mamumat1cl.o	|- .1. = ( 1r ` R )
mamumat1cl.z	|- .0. = ( 0g ` R )
mamumat1cl.i	|- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
mamumat1cl.m	|- ( ph -> M e. Fin )

Assertion

Ref	Expression
mat1comp	|- ( ( 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   A, i, j   i, J, j   .0. , i, j   .1. , i, j

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

Proof of Theorem mat1comp

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( i = A -> ( i = j <-> A = j ) )
2	1	ifbid 4108	. 2 |- ( i = A -> if ( i = j , .1. , .0. ) = if ( A = j , .1. , .0. ) )
3		eqeq2 2633	. . 3 |- ( j = J -> ( A = j <-> A = J ) )
4	3	ifbid 4108	. 2 |- ( j = J -> if ( A = j , .1. , .0. ) = if ( A = J , .1. , .0. ) )
5		mamumat1cl.i	. 2 |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )
6		mamumat1cl.o	. . . 4 |- .1. = ( 1r ` R )
7		fvex 6201	. . . 4 |- ( 1r ` R ) e. _V
8	6, 7	eqeltri 2697	. . 3 |- .1. e. _V
9		mamumat1cl.z	. . . 4 |- .0. = ( 0g ` R )
10		fvex 6201	. . . 4 |- ( 0g ` R ) e. _V
11	9, 10	eqeltri 2697	. . 3 |- .0. e. _V
12	8, 11	ifex 4156	. 2 |- if ( A = J , .1. , .0. ) e. _V
13	2, 4, 5, 12	ovmpt2 6796	1 |- ( ( A e. M /\ J e. M ) -> ( A I J ) = if ( A = J , .1. , .0. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   0gc0g 16100   1rcur 18501   Ringcrg 18547

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

This theorem is referenced by:  mamulid  20247  mamurid  20248