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Theorem tendoi2 35591
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 35590 . . 3  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
43adantr 465 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( I `  S
)  =  ( g  e.  T  |->  `' ( S `  g ) ) )
5 fveq2 5864 . . . 4  |-  ( g  =  F  ->  ( S `  g )  =  ( S `  F ) )
65cnveqd 5176 . . 3  |-  ( g  =  F  ->  `' ( S `  g )  =  `' ( S `
 F ) )
76adantl 466 . 2  |-  ( ( ( S  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  `' ( S `  g )  =  `' ( S `
 F ) )
8 simpr 461 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  F  e.  T )
9 fvex 5874 . . . 4  |-  ( S `
 F )  e. 
_V
109cnvex 6728 . . 3  |-  `' ( S `  F )  e.  _V
1110a1i 11 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  e.  _V )
124, 7, 8, 11fvmptd 5953 1  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( ( I `  S ) `  F
)  =  `' ( S `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   `'ccnv 4998   ` cfv 5586   LTrncltrn 34897
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6574
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-pw 4012  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594
This theorem is referenced by:  tendoicl  35592  tendoipl  35593  dihjatcclem4  36218
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