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Theorem tendoi2 34758
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 34757 . . 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 5794 . . . 4  |-  ( g  =  F  ->  ( S `  g )  =  ( S `  F ) )
65cnveqd 5118 . . 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 5804 . . . 4  |-  ( S `
 F )  e. 
_V
109cnvex 6630 . . 3  |-  `' ( S `  F )  e.  _V
1110a1i 11 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  e.  _V )
124, 7, 8, 11fvmptd 5883 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 1370    e. wcel 1758   _Vcvv 3072    |-> cmpt 4453   `'ccnv 4942   ` cfv 5521   LTrncltrn 34064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529
This theorem is referenced by:  tendoicl  34759  tendoipl  34760  dihjatcclem4  35385
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