Theorem oppgval 19252

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 6890	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅))
4		oppgval.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
5	3, 4	eqtr4di 2788	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + )
6	5	tposeqd 8216	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + )
7	6	opeq2d 4879	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g‘𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
8	2, 7	oveq12d 7429	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9		df-oppg 19251	. . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩))
10		ovex 7444	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
11	8, 9, 10	fvmpt 6997	. . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12		fvprc 6882	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅)
13		reldmsets 17102	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7450	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
15	12, 14	eqtr4d 2773	. . 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 2758	1 ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ∅c0 4321  ⟨cop 4633  ‘cfv 6542  (class class class)co 7411  tpos ctpos 8212   sSet csts 17100  ndxcnx 17130  +_gcplusg 17201  opp_gcoppg 19250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-tpos 8213  df-sets 17101  df-oppg 19251

This theorem is referenced by:  oppgplusfval  19253  oppglemOLD  19256  oppgbas  19257  oppgtset  19259  oppgle  32397