Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoicbv Structured version   Unicode version

Theorem tendoicbv 31517
Description: Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
Hypothesis
Ref Expression
tendoi.i  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
Assertion
Ref Expression
tendoicbv  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Distinct variable groups:    u, s, E    f, g, s, u, T
Allowed substitution hints:    E( f, g)    I( u, f, g, s)

Proof of Theorem tendoicbv
StepHypRef Expression
1 tendoi.i . 2  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
2 fveq1 5719 . . . . . 6  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32cnveqd 5040 . . . . 5  |-  ( s  =  u  ->  `' ( s `  f
)  =  `' ( u `  f ) )
43mpteq2dv 4288 . . . 4  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( f  e.  T  |->  `' ( u `  f
) ) )
5 fveq2 5720 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
65cnveqd 5040 . . . . 5  |-  ( f  =  g  ->  `' ( u `  f
)  =  `' ( u `  g ) )
76cbvmptv 4292 . . . 4  |-  ( f  e.  T  |->  `' ( u `  f ) )  =  ( g  e.  T  |->  `' ( u `  g ) )
84, 7syl6eq 2483 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( g  e.  T  |->  `' ( u `  g
) ) )
98cbvmptv 4292 . 2  |-  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
) ) )
101, 9eqtri 2455 1  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. cmpt 4258   `'ccnv 4869   ` cfv 5446
This theorem is referenced by:  tendoi  31518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-iota 5410  df-fv 5454
  Copyright terms: Public domain W3C validator