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Theorem tendo0cbv 39657
Description: Define additive identity for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)
Hypothesis
Ref Expression
tendo0cbv.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
Assertion
Ref Expression
tendo0cbv 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Distinct variable groups:   𝐵,𝑓,𝑔   𝑇,𝑓,𝑔
Allowed substitution hints:   𝑂(𝑓,𝑔)

Proof of Theorem tendo0cbv
StepHypRef Expression
1 tendo0cbv.o . 2 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
2 eqidd 2734 . . 3 (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
32cbvmptv 5262 . 2 (𝑓𝑇 ↦ ( I ↾ 𝐵)) = (𝑔𝑇 ↦ ( I ↾ 𝐵))
41, 3eqtri 2761 1 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cmpt 5232   I cid 5574  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-mpt 5233
This theorem is referenced by:  tendo02  39658  tendo0cl  39661
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