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Theorem tendoplcbv 34722
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 5785 . . . . 5  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32coeq1d 5096 . . . 4  |-  ( s  =  u  ->  (
( s `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( t `  f
) ) )
43mpteq2dv 4474 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  ( ( s `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( t `  f ) ) ) )
5 fveq1 5785 . . . . . 6  |-  ( t  =  v  ->  (
t `  f )  =  ( v `  f ) )
65coeq2d 5097 . . . . 5  |-  ( t  =  v  ->  (
( u `  f
)  o.  ( t `
 f ) )  =  ( ( u `
 f )  o.  ( v `  f
) ) )
76mpteq2dv 4474 . . . 4  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) ) )
8 fveq2 5786 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
9 fveq2 5786 . . . . . 6  |-  ( f  =  g  ->  (
v `  f )  =  ( v `  g ) )
108, 9coeq12d 5099 . . . . 5  |-  ( f  =  g  ->  (
( u `  f
)  o.  ( v `
 f ) )  =  ( ( u `
 g )  o.  ( v `  g
) ) )
1110cbvmptv 4478 . . . 4  |-  ( f  e.  T  |->  ( ( u `  f )  o.  ( v `  f ) ) )  =  ( g  e.  T  |->  ( ( u `
 g )  o.  ( v `  g
) ) )
127, 11syl6eq 2507 . . 3  |-  ( t  =  v  ->  (
f  e.  T  |->  ( ( u `  f
)  o.  ( t `
 f ) ) )  =  ( g  e.  T  |->  ( ( u `  g )  o.  ( v `  g ) ) ) )
134, 12cbvmpt2v 6262 . 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 2479 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 1370    |-> cmpt 4445    o. ccom 4939   ` cfv 5513    |-> cmpt2 6189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-co 4944  df-iota 5476  df-fv 5521  df-oprab 6191  df-mpt2 6192
This theorem is referenced by:  tendopl  34723
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