Theorem mat1comp 22662

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 2764	. . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗))
2	1	ifbid 4506	. 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3		eqeq2 2772	. . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽))
4	3	ifbid 4506	. 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 6892	. . 3 ⊢ 1 ∈ V
8		mamumat1cl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	8	fvexi 6892	. . 3 ⊢ 0 ∈ V
10	7, 9	ifex 4533	. 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V
11	2, 4, 5, 10	ovmpo 7573	1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ifcif 4482  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  Fincfn 8952  Basecbs 17301  0_gc0g 17524  1_rcur 20320  Ringcrg 20372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418

This theorem is used by:  mamulid  22663  mamurid  22664