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Theorem tendo0cbv 31645
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  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
tendo0cbv  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
Distinct variable groups:    B, f    B, g    T, f    T, g
Allowed substitution hints:    O( f, g)

Proof of Theorem tendo0cbv
StepHypRef Expression
1 tendo0cbv.o . 2  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
2 eqidd 2439 . . 3  |-  ( f  =  g  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
32cbvmptv 4302 . 2  |-  ( f  e.  T  |->  (  _I  |`  B ) )  =  ( g  e.  T  |->  (  _I  |`  B ) )
41, 3eqtri 2458 1  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. cmpt 4268    _I cid 4495    |` cres 4882
This theorem is referenced by:  tendo02  31646  tendo0cl  31649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-mpt 4270
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