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Theorem tendo0cbv 37914
Description: Define additive identity for trace-perserving 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 2820 . . 3 (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
32cbvmptv 5160 . 2 (𝑓𝑇 ↦ ( I ↾ 𝐵)) = (𝑔𝑇 ↦ ( I ↾ 𝐵))
41, 3eqtri 2842 1 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1531  cmpt 5137   I cid 5452  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120  df-mpt 5138
This theorem is referenced by:  tendo02  37915  tendo0cl  37918
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