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Theorem tendopl 35789
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 5865 . . . 4  |-  ( u  =  U  ->  (
u `  g )  =  ( U `  g ) )
21coeq1d 5164 . . 3  |-  ( u  =  U  ->  (
( u `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( v `  g
) ) )
32mpteq2dv 4534 . 2  |-  ( u  =  U  ->  (
g  e.  T  |->  ( ( u `  g
)  o.  ( v `
 g ) ) )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( v `  g ) ) ) )
4 fveq1 5865 . . . 4  |-  ( v  =  V  ->  (
v `  g )  =  ( V `  g ) )
54coeq2d 5165 . . 3  |-  ( v  =  V  ->  (
( U `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( V `  g
) ) )
65mpteq2dv 4534 . 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 35788 . 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 5876 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
119, 10eqeltri 2551 . . 3  |-  T  e. 
_V
1211mptex 6132 . 2  |-  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) )  e.  _V
133, 6, 8, 12ovmpt2 6423 1  |-  ( ( U  e.  E  /\  V  e.  E )  ->  ( U P V )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505    o. ccom 5003   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   LTrncltrn 35114
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290
This theorem is referenced by:  tendopl2  35790  tendoplcl  35794  erngplus  35816  erngplus-rN  35824  dvaplusg  36022
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