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Theorem tendopl2 35974
Description: Value of result 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
tendopl2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
Distinct variable groups:    t, s, E    f, s, t, T   
f, W, s, t
Allowed substitution hints:    P( t, f, s)    U( t, f, s)    E( f)    F( t, f, s)    K( t, f, s)    V( t, f, s)

Proof of Theorem tendopl2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tendoplcbv.p . . . 4  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
2 tendopl2.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
31, 2tendopl 35973 . . 3  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
433adant3 1016 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
5 fveq2 5872 . . . 4  |-  ( g  =  F  ->  ( U `  g )  =  ( U `  F ) )
6 fveq2 5872 . . . 4  |-  ( g  =  F  ->  ( V `  g )  =  ( V `  F ) )
75, 6coeq12d 5173 . . 3  |-  ( g  =  F  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
87adantl 466 . 2  |-  ( ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
9 simp3 998 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  F  e.  T )
10 fvex 5882 . . . 4  |-  ( U `
 F )  e. 
_V
11 fvex 5882 . . . 4  |-  ( V `
 F )  e. 
_V
1210, 11coex 6747 . . 3  |-  ( ( U `  F )  o.  ( V `  F ) )  e. 
_V
1312a1i 11 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U `  F )  o.  ( V `  F )
)  e.  _V )
144, 8, 9, 13fvmptd 5962 1  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511    o. ccom 5009   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   LTrncltrn 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  tendoplcl2  35975  tendoplco2  35976  tendopltp  35977  tendoplcom  35979  tendoplass  35980  tendodi1  35981  tendodi2  35982  tendo0pl  35988  tendoipl  35994  tendospdi2  36220
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