Theorem oppgval 18866

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 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅))
4		oppgval.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
5	3, 4	eqtr4di 2797	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + )
6	5	tposeqd 8016	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + )
7	6	opeq2d 4808	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g‘𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
8	2, 7	oveq12d 7273	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9		df-oppg 18865	. . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩))
10		ovex 7288	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
11	8, 9, 10	fvmpt 6857	. . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12		fvprc 6748	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅)
13		reldmsets 16794	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7294	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
15	12, 14	eqtr4d 2781	. . 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 2766	1 ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  tpos ctpos 8012   sSet csts 16792  ndxcnx 16822  +_gcplusg 16888  opp_gcoppg 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-tpos 8013  df-sets 16793  df-oppg 18865

This theorem is referenced by:  oppgplusfval  18867  oppglemOLD  18870  oppgbas  18871  oppgtset  18873  oppgle  31140