Theorem oppgval 18477

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 6672	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅))
4		oppgval.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
5	3, 4	syl6eqr 2876	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + )
6	5	tposeqd 7897	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + )
7	6	opeq2d 4812	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g‘𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
8	2, 7	oveq12d 7176	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9		df-oppg 18476	. . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩))
10		ovex 7191	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
11	8, 9, 10	fvmpt 6770	. . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12		fvprc 6665	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅)
13		reldmsets 16513	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7197	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
15	12, 14	eqtr4d 2861	. . 3 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
16	11, 15	pm2.61i 184	. 2 ⊢ (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
17	1, 16	eqtri 2846	1 ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  ⟨cop 4575  ‘cfv 6357  (class class class)co 7158  tpos ctpos 7893  ndxcnx 16482   sSet csts 16483  +_gcplusg 16567  opp_gcoppg 18475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-tpos 7894  df-sets 16492  df-oppg 18476

This theorem is referenced by:  oppgplusfval  18478  oppglem  18480