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Theorem mat1comp 20463
 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 2775 . . 3 (𝑖 = 𝐴 → (𝑖 = 𝑗𝐴 = 𝑗))
21ifbid 4248 . 2 (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3 eqeq2 2782 . . 3 (𝑗 = 𝐽 → (𝐴 = 𝑗𝐴 = 𝐽))
43ifbid 4248 . 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 6345 . . 3 1 ∈ V
8 mamumat1cl.z . . . 4 0 = (0g𝑅)
98fvexi 6345 . . 3 0 ∈ V
107, 9ifex 4296 . 2 if(𝐴 = 𝐽, 1 , 0 ) ∈ V
112, 4, 5, 10ovmpt2 6947 1 ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ifcif 4226  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798  Fincfn 8113  Basecbs 16064  0gc0g 16308  1rcur 18709  Ringcrg 18755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801 This theorem is referenced by:  mamulid  20464  mamurid  20465
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