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Theorem mat1comp 22362
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 2732 . . 3 (𝑖 = 𝐴 → (𝑖 = 𝑗𝐴 = 𝑗))
21ifbid 4555 . 2 (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3 eqeq2 2740 . . 3 (𝑗 = 𝐽 → (𝐴 = 𝑗𝐴 = 𝐽))
43ifbid 4555 . 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 6916 . . 3 1 ∈ V
8 mamumat1cl.z . . . 4 0 = (0g𝑅)
98fvexi 6916 . . 3 0 ∈ V
107, 9ifex 4582 . 2 if(𝐴 = 𝐽, 1 , 0 ) ∈ V
112, 4, 5, 10ovmpo 7587 1 ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  ifcif 4532  cfv 6553  (class class class)co 7426  cmpo 7428  Fincfn 8970  Basecbs 17187  0gc0g 17428  1rcur 20128  Ringcrg 20180
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431
This theorem is referenced by:  mamulid  22363  mamurid  22364
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