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Theorem mat1comp 22292
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 (𝜑𝑅 ∈ Ring)
mamumat1cl.o 1 = (1r𝑅)
mamumat1cl.z 0 = (0g𝑅)
mamumat1cl.i 𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))
mamumat1cl.m (𝜑𝑀 ∈ Fin)
Assertion
Ref Expression
mat1comp ((𝐴𝑀𝐽𝑀) → (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
StepHypRef Expression
1 eqeq1 2730 . . 3 (𝑖 = 𝐴 → (𝑖 = 𝑗𝐴 = 𝑗))
21ifbid 4546 . 2 (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3 eqeq2 2738 . . 3 (𝑗 = 𝐽 → (𝐴 = 𝑗𝐴 = 𝐽))
43ifbid 4546 . 2 (𝑗 = 𝐽 → if(𝐴 = 𝑗, 1 , 0 ) = if(𝐴 = 𝐽, 1 , 0 ))
5 mamumat1cl.i . 2 𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))
6 mamumat1cl.o . . . 4 1 = (1r𝑅)
76fvexi 6898 . . 3 1 ∈ V
8 mamumat1cl.z . . . 4 0 = (0g𝑅)
98fvexi 6898 . . 3 0 ∈ V
107, 9ifex 4573 . 2 if(𝐴 = 𝐽, 1 , 0 ) ∈ V
112, 4, 5, 10ovmpo 7563 1 ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ifcif 4523  cfv 6536  (class class class)co 7404  cmpo 7406  Fincfn 8938  Basecbs 17150  0gc0g 17391  1rcur 20083  Ringcrg 20135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  mamulid  22293  mamurid  22294
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