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Theorem mat1comp 20013
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 2614 . . 3 (𝑖 = 𝐴 → (𝑖 = 𝑗𝐴 = 𝑗))
21ifbid 4058 . 2 (𝑖 = 𝐴 → if(𝑖 = 𝑗, 1 , 0 ) = if(𝐴 = 𝑗, 1 , 0 ))
3 eqeq2 2621 . . 3 (𝑗 = 𝐽 → (𝐴 = 𝑗𝐴 = 𝐽))
43ifbid 4058 . 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)
7 fvex 6098 . . . 4 (1r𝑅) ∈ V
86, 7eqeltri 2684 . . 3 1 ∈ V
9 mamumat1cl.z . . . 4 0 = (0g𝑅)
10 fvex 6098 . . . 4 (0g𝑅) ∈ V
119, 10eqeltri 2684 . . 3 0 ∈ V
128, 11ifex 4106 . 2 if(𝐴 = 𝐽, 1 , 0 ) ∈ V
132, 4, 5, 12ovmpt2 6672 1 ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  ifcif 4036  cfv 5790  (class class class)co 6527  cmpt2 6529  Fincfn 7819  Basecbs 15644  0gc0g 15872  1rcur 18273  Ringcrg 18319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532
This theorem is referenced by:  mamulid  20014  mamurid  20015
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