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Theorem tendoplcbv 36917
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 5847 . . . . 5  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32coeq1d 5153 . . . 4  |-  ( s  =  u  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( t `  f
) ) )
43mpteq2dv 4526 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  ( ( s `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( t `  f ) ) ) )
5 fveq1 5847 . . . . . 6  |-  ( t  =  v  ->  (
t `  f )  =  ( v `  f ) )
65coeq2d 5154 . . . . 5  |-  ( t  =  v  ->  (
( u `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( v `  f
) ) )
76mpteq2dv 4526 . . . 4  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) ) )
8 fveq2 5848 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
9 fveq2 5848 . . . . . 6  |-  ( f  =  g  ->  (
v `  f )  =  ( v `  g ) )
108, 9coeq12d 5156 . . . . 5  |-  ( f  =  g  ->  (
( u `  f
)  o.  ( v `
 f ) )  =  ( ( u `
 g )  o.  ( v `  g
) ) )
1110cbvmptv 4530 . . . 4  |-  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) )  =  ( g  e.  T  |->  ( ( u `
 g )  o.  ( v `  g
) ) )
127, 11syl6eq 2511 . . 3  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
134, 12cbvmpt2v 6350 . 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 2483 1  |-  P  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  (
v `  g )
) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    |-> cmpt 4497    o. ccom 4992   ` cfv 5570    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-co 4997  df-iota 5534  df-fv 5578  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  tendopl  36918
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