Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoi2 Structured version   Visualization version   GIF version

Theorem tendoi2 38088
 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 38087 . . 3 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
43adantr 484 . 2 ((𝑆𝐸𝐹𝑇) → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
5 fveq2 6645 . . . 4 (𝑔 = 𝐹 → (𝑆𝑔) = (𝑆𝐹))
65cnveqd 5710 . . 3 (𝑔 = 𝐹(𝑆𝑔) = (𝑆𝐹))
76adantl 485 . 2 (((𝑆𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → (𝑆𝑔) = (𝑆𝐹))
8 simpr 488 . 2 ((𝑆𝐸𝐹𝑇) → 𝐹𝑇)
9 fvex 6658 . . . 4 (𝑆𝐹) ∈ V
109cnvex 7612 . . 3 (𝑆𝐹) ∈ V
1110a1i 11 . 2 ((𝑆𝐸𝐹𝑇) → (𝑆𝐹) ∈ V)
124, 7, 8, 11fvmptd 6752 1 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ↦ cmpt 5110  ◡ccnv 5518  ‘cfv 6324  LTrncltrn 37394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332 This theorem is referenced by:  tendoicl  38089  tendoipl  38090  dihjatcclem4  38714
 Copyright terms: Public domain W3C validator