Theorem oppgval 19378

Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)

Hypotheses

Ref	Expression
oppgval.2	⊢ + = (+g‘𝑅)
oppgval.3	⊢ 𝑂 = (oppg‘𝑅)

Assertion

Ref	Expression
oppgval	⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Proof of Theorem oppgval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppgval.3	. 2 ⊢ 𝑂 = (oppg‘𝑅)
2		id 22	. . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
3		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅))
4		oppgval.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
5	3, 4	eqtr4di 2793	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + )
6	5	tposeqd 8253	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + )
7	6	opeq2d 4885	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g‘𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
8	2, 7	oveq12d 7449	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9		df-oppg 19377	. . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩))
10		ovex 7464	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
11	8, 9, 10	fvmpt 7016	. . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12		fvprc 6899	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅)
13		reldmsets 17199	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7470	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
15	12, 14	eqtr4d 2778	. . 3 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
16	11, 15	pm2.61i 182	. 2 ⊢ (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
17	1, 16	eqtri 2763	1 ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ⟨cop 4637  ‘cfv 6563  (class class class)co 7431  tpos ctpos 8249   sSet csts 17197  ndxcnx 17227  +_gcplusg 17298  opp_gcoppg 19376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tpos 8250  df-sets 17198  df-oppg 19377

This theorem is referenced by:  oppgplusfval  19379  oppglemOLD  19382  oppgbas  19383  oppgtset  19385  oppgle  32936