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Theorem tendoplcbv 31586
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 5540 . . . . 5  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32coeq1d 4861 . . . 4  |-  ( s  =  u  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( t `  f
) ) )
43mpteq2dv 4123 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  ( ( s `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( t `  f ) ) ) )
5 fveq1 5540 . . . . . 6  |-  ( t  =  v  ->  (
t `  f )  =  ( v `  f ) )
65coeq2d 4862 . . . . 5  |-  ( t  =  v  ->  (
( u `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( v `  f
) ) )
76mpteq2dv 4123 . . . 4  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) ) )
8 fveq2 5541 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
9 fveq2 5541 . . . . . 6  |-  ( f  =  g  ->  (
v `  f )  =  ( v `  g ) )
108, 9coeq12d 4864 . . . . 5  |-  ( f  =  g  ->  (
( u `  f
)  o.  ( v `
 f ) )  =  ( ( u `
 g )  o.  ( v `  g
) ) )
1110cbvmptv 4127 . . . 4  |-  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) )  =  ( g  e.  T  |->  ( ( u `
 g )  o.  ( v `  g
) ) )
127, 11syl6eq 2344 . . 3  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
134, 12cbvmpt2v 5942 . 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 2316 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 1632    e. cmpt 4093    o. ccom 4709   ` cfv 5271    e. cmpt2 5876
This theorem is referenced by:  tendopl  31587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-co 4714  df-iota 5235  df-fv 5279  df-oprab 5878  df-mpt2 5879
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