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Theorem mat1comp 21043
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 2825 . . 3 (𝑖 = 𝐴 → (𝑖 = 𝑗𝐴 = 𝑗))
21ifbid 4489 . 2 (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3 eqeq2 2833 . . 3 (𝑗 = 𝐽 → (𝐴 = 𝑗𝐴 = 𝐽))
43ifbid 4489 . 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 6679 . . 3 1 ∈ V
8 mamumat1cl.z . . . 4 0 = (0g𝑅)
98fvexi 6679 . . 3 0 ∈ V
107, 9ifex 4515 . 2 if(𝐴 = 𝐽, 1 , 0 ) ∈ V
112, 4, 5, 10ovmpo 7304 1 ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  ifcif 4467  cfv 6350  (class class class)co 7150  cmpo 7152  Fincfn 8503  Basecbs 16477  0gc0g 16707  1rcur 19245  Ringcrg 19291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155
This theorem is referenced by:  mamulid  21044  mamurid  21045
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