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Theorem tendo0cbv 30793
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 2317 . . 3  |-  ( f  =  g  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
32cbvmptv 4148 . 2  |-  ( f  e.  T  |->  (  _I  |`  B ) )  =  ( g  e.  T  |->  (  _I  |`  B ) )
41, 3eqtri 2336 1  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. cmpt 4114    _I cid 4341    |` cres 4728
This theorem is referenced by:  tendo02  30794  tendo0cl  30797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-opab 4115  df-mpt 4116
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