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Theorem tendoi2 30960
Description: Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
Hypotheses
Ref Expression
tendoi.i  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
tendoi.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
tendoi2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( ( I `  S ) `  F
)  =  `' ( S `  F ) )
Distinct variable groups:    E, s    f, s, T    f, W, s
Allowed substitution hints:    S( f, s)    E( f)    F( f, s)    I( f, s)    K( f, s)

Proof of Theorem tendoi2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tendoi.i . . . 4  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
2 tendoi.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
31, 2tendoi 30959 . . 3  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
43adantr 452 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( I `  S
)  =  ( g  e.  T  |->  `' ( S `  g ) ) )
5 fveq2 5661 . . . 4  |-  ( g  =  F  ->  ( S `  g )  =  ( S `  F ) )
65cnveqd 4981 . . 3  |-  ( g  =  F  ->  `' ( S `  g )  =  `' ( S `
 F ) )
76adantl 453 . 2  |-  ( ( ( S  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  `' ( S `  g )  =  `' ( S `
 F ) )
8 simpr 448 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  F  e.  T )
9 fvex 5675 . . . 4  |-  ( S `
 F )  e. 
_V
109cnvex 5339 . . 3  |-  `' ( S `  F )  e.  _V
1110a1i 11 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  e.  _V )
124, 7, 8, 11fvmptd 5742 1  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( ( I `  S ) `  F
)  =  `' ( S `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    e. cmpt 4200   `'ccnv 4810   ` cfv 5387   LTrncltrn 30266
This theorem is referenced by:  tendoicl  30961  tendoipl  30962  dihjatcclem4  31587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395
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