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Theorem tendo0cbv 40788
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 2738 . . 3 (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
32cbvmptv 5255 . 2 (𝑓𝑇 ↦ ( I ↾ 𝐵)) = (𝑔𝑇 ↦ ( I ↾ 𝐵))
41, 3eqtri 2765 1 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cmpt 5225   I cid 5577  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-mpt 5226
This theorem is referenced by:  tendo02  40789  tendo0cl  40792
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