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Theorem tendoplcbv 30964
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 5524 . . . . 5  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32coeq1d 4845 . . . 4  |-  ( s  =  u  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( t `  f
) ) )
43mpteq2dv 4107 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  ( ( s `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( t `  f ) ) ) )
5 fveq1 5524 . . . . . 6  |-  ( t  =  v  ->  (
t `  f )  =  ( v `  f ) )
65coeq2d 4846 . . . . 5  |-  ( t  =  v  ->  (
( u `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( v `  f
) ) )
76mpteq2dv 4107 . . . 4  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) ) )
8 fveq2 5525 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
9 fveq2 5525 . . . . . 6  |-  ( f  =  g  ->  (
v `  f )  =  ( v `  g ) )
108, 9coeq12d 4848 . . . . 5  |-  ( f  =  g  ->  (
( u `  f
)  o.  ( v `
 f ) )  =  ( ( u `
 g )  o.  ( v `  g
) ) )
1110cbvmptv 4111 . . . 4  |-  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) )  =  ( g  e.  T  |->  ( ( u `
 g )  o.  ( v `  g
) ) )
127, 11syl6eq 2331 . . 3  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
134, 12cbvmpt2v 5926 . 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 2303 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 1623    e. cmpt 4077    o. ccom 4693   ` cfv 5255    e. cmpt2 5860
This theorem is referenced by:  tendopl  30965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-co 4698  df-iota 5219  df-fv 5263  df-oprab 5862  df-mpt2 5863
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