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Theorem tendopl2 30966
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 30965 . . 3  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
433adant3 975 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
5 fveq2 5525 . . . 4  |-  ( g  =  F  ->  ( U `  g )  =  ( U `  F ) )
6 fveq2 5525 . . . 4  |-  ( g  =  F  ->  ( V `  g )  =  ( V `  F ) )
75, 6coeq12d 4848 . . 3  |-  ( g  =  F  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
87adantl 452 . 2  |-  ( ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
9 simp3 957 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  F  e.  T )
10 fvex 5539 . . . 4  |-  ( U `
 F )  e. 
_V
11 fvex 5539 . . . 4  |-  ( V `
 F )  e. 
_V
1210, 11coex 5216 . . 3  |-  ( ( U `  F )  o.  ( V `  F ) )  e. 
_V
1312a1i 10 . 2  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U `  F )  o.  ( V `  F )
)  e.  _V )
144, 8, 9, 13fvmptd 5606 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 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   LTrncltrn 30290
This theorem is referenced by:  tendoplcl2  30967  tendoplco2  30968  tendopltp  30969  tendoplcom  30971  tendoplass  30972  tendodi1  30973  tendodi2  30974  tendo0pl  30980  tendoipl  30986  tendospdi2  31212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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