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Theorem tendopl2 31575
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 31574 . . 3  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
433adant3 978 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
5 fveq2 5729 . . . 4  |-  ( g  =  F  ->  ( U `  g )  =  ( U `  F ) )
6 fveq2 5729 . . . 4  |-  ( g  =  F  ->  ( V `  g )  =  ( V `  F ) )
75, 6coeq12d 5038 . . 3  |-  ( g  =  F  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
87adantl 454 . 2  |-  ( ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
9 simp3 960 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  F  e.  T )
10 fvex 5743 . . . 4  |-  ( U `
 F )  e. 
_V
11 fvex 5743 . . . 4  |-  ( V `
 F )  e. 
_V
1210, 11coex 5414 . . 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 5811 1  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2957    e. cmpt 4267    o. ccom 4883   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   LTrncltrn 30899
This theorem is referenced by:  tendoplcl2  31576  tendoplco2  31577  tendopltp  31578  tendoplcom  31580  tendoplass  31581  tendodi1  31582  tendodi2  31583  tendo0pl  31589  tendoipl  31595  tendospdi2  31821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087
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