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Theorem tendoi2 30984
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 30983 . . 3  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
43adantr 451 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( I `  S
)  =  ( g  e.  T  |->  `' ( S `  g ) ) )
5 fveq2 5525 . . . 4  |-  ( g  =  F  ->  ( S `  g )  =  ( S `  F ) )
65cnveqd 4857 . . 3  |-  ( g  =  F  ->  `' ( S `  g )  =  `' ( S `
 F ) )
76adantl 452 . 2  |-  ( ( ( S  e.  E  /\  F  e.  T
)  /\  g  =  F )  ->  `' ( S `  g )  =  `' ( S `
 F ) )
8 simpr 447 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  F  e.  T )
9 fvex 5539 . . . 4  |-  ( S `
 F )  e. 
_V
109cnvex 5209 . . 3  |-  `' ( S `  F )  e.  _V
1110a1i 10 . 2  |-  ( ( S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  e.  _V )
124, 7, 8, 11fvmptd 5606 1  |-  ( ( S  e.  E  /\  F  e.  T )  ->  ( ( I `  S ) `  F
)  =  `' ( S `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   `'ccnv 4688   ` cfv 5255   LTrncltrn 30290
This theorem is referenced by:  tendoicl  30985  tendoipl  30986  dihjatcclem4  31611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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