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Theorem tendoi2 36605
Description: Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
Hypotheses
Ref Expression
tendoi.i 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
tendoi.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoi2 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
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 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
2 tendoi.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
31, 2tendoi 36604 . . 3 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
43adantr 466 . 2 ((𝑆𝐸𝐹𝑇) → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
5 fveq2 6333 . . . 4 (𝑔 = 𝐹 → (𝑆𝑔) = (𝑆𝐹))
65cnveqd 5437 . . 3 (𝑔 = 𝐹(𝑆𝑔) = (𝑆𝐹))
76adantl 467 . 2 (((𝑆𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → (𝑆𝑔) = (𝑆𝐹))
8 simpr 471 . 2 ((𝑆𝐸𝐹𝑇) → 𝐹𝑇)
9 fvex 6343 . . . 4 (𝑆𝐹) ∈ V
109cnvex 7261 . . 3 (𝑆𝐹) ∈ V
1110a1i 11 . 2 ((𝑆𝐸𝐹𝑇) → (𝑆𝐹) ∈ V)
124, 7, 8, 11fvmptd 6431 1 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4864  ccnv 5249  cfv 6032  LTrncltrn 35910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040
This theorem is referenced by:  tendoicl  36606  tendoipl  36607  dihjatcclem4  37232
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