Theorem mat1comp 22396

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	⊢ (𝜑 → 𝑅 ∈ Ring)
mamumat1cl.o	⊢ 1 = (1r‘𝑅)
mamumat1cl.z	⊢ 0 = (0g‘𝑅)
mamumat1cl.i	⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))
mamumat1cl.m	⊢ (𝜑 → 𝑀 ∈ Fin)

Assertion

Ref	Expression
mat1comp	⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Distinct variable groups:   𝑖,𝑗,𝐵   𝑖,𝑀,𝑗   𝜑,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 2741	. . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗))
2	1	ifbid 4505	. 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3		eqeq2 2749	. . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽))
4	3	ifbid 4505	. 2 ⊢ (𝑗 = 𝐽 → if(𝐴 = 𝑗, 1 , 0 ) = if(𝐴 = 𝐽, 1 , 0 ))
5		mamumat1cl.i	. 2 ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))
6		mamumat1cl.o	. . . 4 ⊢ 1 = (1r‘𝑅)
7	6	fvexi 6856	. . 3 ⊢ 1 ∈ V
8		mamumat1cl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	8	fvexi 6856	. . 3 ⊢ 0 ∈ V
10	7, 9	ifex 4532	. 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V
11	2, 4, 5, 10	ovmpo 7528	1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ifcif 4481  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Fincfn 8895  Basecbs 17148  0_gc0g 17371  1_rcur 20128  Ringcrg 20180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by:  mamulid  22397  mamurid  22398