Theorem mat1comp 22761

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 4505	. 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 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 6900	. . 3 ⊢ 1 ∈ V
8		mamumat1cl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	8	fvexi 6900	. . 3 ⊢ 0 ∈ V
10	7, 9	ifex 4532	. 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V
11	2, 4, 5, 10	ovmpo 7581	1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ifcif 4481  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  Fincfn 8977  Basecbs 17393  0_gc0g 17616  1_rcur 20413  Ringcrg 20465

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426

This theorem is used by:  mamulid  22762  mamurid  22763