Theorem mat1comp 21051

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 2827	. . 3 ⊢ (𝑖 = 𝐴 → (𝑖 = 𝑗 ↔ 𝐴 = 𝑗))
2	1	ifbid 4491	. 2 ⊢ (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3		eqeq2 2835	. . 3 ⊢ (𝑗 = 𝐽 → (𝐴 = 𝑗 ↔ 𝐴 = 𝐽))
4	3	ifbid 4491	. 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 6686	. . 3 ⊢ 1 ∈ V
8		mamumat1cl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	8	fvexi 6686	. . 3 ⊢ 0 ∈ V
10	7, 9	ifex 4517	. 2 ⊢ if(𝐴 = 𝐽, 1 , 0 ) ∈ V
11	2, 4, 5, 10	ovmpo 7312	1 ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ifcif 4469  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  Fincfn 8511  Basecbs 16485  0_gc0g 16715  1_rcur 19253  Ringcrg 19299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163

This theorem is referenced by:  mamulid  21052  mamurid  21053