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Theorem tendopl 31036
Description: Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
Hypotheses
Ref Expression
tendoplcbv.p  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
tendopl2.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
tendopl  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
Distinct variable groups:    t, s, E    f, g, s, t, T    f, W, g, s, t    U, g   
g, V
Allowed substitution hints:    P( t, f, g, s)    U( t, f, s)    E( f, g)    K( t, f, g, s)    V( t, f, s)

Proof of Theorem tendopl
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5631 . . . 4  |-  ( u  =  U  ->  (
u `  g )  =  ( U `  g ) )
21coeq1d 4948 . . 3  |-  ( u  =  U  ->  (
( u `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( v `  g
) ) )
32mpteq2dv 4209 . 2  |-  ( u  =  U  ->  (
g  e.  T  |->  ( ( u `  g
)  o.  ( v `
 g ) ) )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( v `  g ) ) ) )
4 fveq1 5631 . . . 4  |-  ( v  =  V  ->  (
v `  g )  =  ( V `  g ) )
54coeq2d 4949 . . 3  |-  ( v  =  V  ->  (
( U `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( V `  g
) ) )
65mpteq2dv 4209 . 2  |-  ( v  =  V  ->  (
g  e.  T  |->  ( ( U `  g
)  o.  ( v `
 g ) ) )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
7 tendoplcbv.p . . 3  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
87tendoplcbv 31035 . 2  |-  P  =  ( u  e.  E ,  v  e.  E  |->  ( g  e.  T  |->  ( ( u `  g )  o.  (
v `  g )
) ) )
9 tendopl2.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
10 fvex 5646 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
119, 10eqeltri 2436 . . 3  |-  T  e. 
_V
1211mptex 5866 . 2  |-  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) )  e.  _V
133, 6, 8, 12ovmpt2 6109 1  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    e. cmpt 4179    o. ccom 4796   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   LTrncltrn 30361
This theorem is referenced by:  tendopl2  31037  tendoplcl  31041  erngplus  31063  erngplus-rN  31071  dvaplusg  31269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986
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