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Theorem tendopl2 34784
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 34783 . . 3  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
433adant3 1008 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
5 fveq2 5802 . . . 4  |-  ( g  =  F  ->  ( U `  g )  =  ( U `  F ) )
6 fveq2 5802 . . . 4  |-  ( g  =  F  ->  ( V `  g )  =  ( V `  F ) )
75, 6coeq12d 5115 . . 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 990 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  F  e.  T )
10 fvex 5812 . . . 4  |-  ( U `
 F )  e. 
_V
11 fvex 5812 . . . 4  |-  ( V `
 F )  e. 
_V
1210, 11coex 6642 . . 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 5891 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461    o. ccom 4955   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   LTrncltrn 34108
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208
This theorem is referenced by:  tendoplcl2  34785  tendoplco2  34786  tendopltp  34787  tendoplcom  34789  tendoplass  34790  tendodi1  34791  tendodi2  34792  tendo0pl  34798  tendoipl  34804  tendospdi2  35030
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