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Theorem tendopl 30965
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 5524 . . . 4  |-  ( u  =  U  ->  (
u `  g )  =  ( U `  g ) )
21coeq1d 4845 . . 3  |-  ( u  =  U  ->  (
( u `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( v `  g
) ) )
32mpteq2dv 4107 . 2  |-  ( u  =  U  ->  (
g  e.  T  |->  ( ( u `  g
)  o.  ( v `
 g ) ) )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( v `  g ) ) ) )
4 fveq1 5524 . . . 4  |-  ( v  =  V  ->  (
v `  g )  =  ( V `  g ) )
54coeq2d 4846 . . 3  |-  ( v  =  V  ->  (
( U `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( V `  g
) ) )
65mpteq2dv 4107 . 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 30964 . 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 5539 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
119, 10eqeltri 2353 . . 3  |-  T  e. 
_V
1211mptex 5746 . 2  |-  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) )  e.  _V
133, 6, 8, 12ovmpt2 5983 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 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:  tendopl2  30966  tendoplcl  30970  erngplus  30992  erngplus-rN  31000  dvaplusg  31198
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-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-pr 4214
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-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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