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

Theorem tendoplcbv 30889
Description: Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
Hypothesis
Ref Expression
tendoplcbv.p  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
Assertion
Ref Expression
tendoplcbv  |-  P  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  (
v `  g )
) ) )
Distinct variable groups:    t, s, u, v, E    f, g,
s, t, u, v, T
Allowed substitution hints:    P( v, u, t, f, g, s)    E( f, g)

Proof of Theorem tendoplcbv
StepHypRef Expression
1 tendoplcbv.p . 2  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
2 fveq1 5667 . . . . 5  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32coeq1d 4974 . . . 4  |-  ( s  =  u  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( t `  f
) ) )
43mpteq2dv 4237 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  ( ( s `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( t `  f ) ) ) )
5 fveq1 5667 . . . . . 6  |-  ( t  =  v  ->  (
t `  f )  =  ( v `  f ) )
65coeq2d 4975 . . . . 5  |-  ( t  =  v  ->  (
( u `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( v `  f
) ) )
76mpteq2dv 4237 . . . 4  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) ) )
8 fveq2 5668 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
9 fveq2 5668 . . . . . 6  |-  ( f  =  g  ->  (
v `  f )  =  ( v `  g ) )
108, 9coeq12d 4977 . . . . 5  |-  ( f  =  g  ->  (
( u `  f
)  o.  ( v `
 f ) )  =  ( ( u `
 g )  o.  ( v `  g
) ) )
1110cbvmptv 4241 . . . 4  |-  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) )  =  ( g  e.  T  |->  ( ( u `
 g )  o.  ( v `  g
) ) )
127, 11syl6eq 2435 . . 3  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
134, 12cbvmpt2v 6091 . 2  |-  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  ( t `  f ) ) ) )  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
141, 13eqtri 2407 1  |-  P  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  (
v `  g )
) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. cmpt 4207    o. ccom 4822   ` cfv 5394    e. cmpt2 6022
This theorem is referenced by:  tendopl  30890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-co 4827  df-iota 5358  df-fv 5402  df-oprab 6024  df-mpt2 6025
  Copyright terms: Public domain W3C validator