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Theorem tendo0cbv 37802
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 2819 . . 3 (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
32cbvmptv 5160 . 2 (𝑓𝑇 ↦ ( I ↾ 𝐵)) = (𝑔𝑇 ↦ ( I ↾ 𝐵))
41, 3eqtri 2841 1 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-mpt 5138
This theorem is referenced by:  tendo02  37803  tendo0cl  37806
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