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Theorem tendopl 31647
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 5730 . . . 4  |-  ( u  =  U  ->  (
u `  g )  =  ( U `  g ) )
21coeq1d 5037 . . 3  |-  ( u  =  U  ->  (
( u `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( v `  g
) ) )
32mpteq2dv 4299 . 2  |-  ( u  =  U  ->  (
g  e.  T  |->  ( ( u `  g
)  o.  ( v `
 g ) ) )  =  ( g  e.  T  |->  ( ( U `  g )  o.  ( v `  g ) ) ) )
4 fveq1 5730 . . . 4  |-  ( v  =  V  ->  (
v `  g )  =  ( V `  g ) )
54coeq2d 5038 . . 3  |-  ( v  =  V  ->  (
( U `  g
)  o.  ( v `
 g ) )  =  ( ( U `
 g )  o.  ( V `  g
) ) )
65mpteq2dv 4299 . 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 31646 . 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 5745 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
119, 10eqeltri 2508 . . 3  |-  T  e. 
_V
1211mptex 5969 . 2  |-  ( g  e.  T  |->  ( ( U `  g )  o.  ( V `  g ) ) )  e.  _V
133, 6, 8, 12ovmpt2 6212 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4269    o. ccom 4885   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   LTrncltrn 30972
This theorem is referenced by:  tendopl2  31648  tendoplcl  31652  erngplus  31674  erngplus-rN  31682  dvaplusg  31880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
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