Theorem oppgval 19474

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 23	. . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
3		fveq2 6878	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = (+g‘𝑅))
4		oppgval.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
5	3, 4	eqtr4di 2813	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (+g‘𝑥) = + )
6	5	tposeqd 8227	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (+g‘𝑥) = tpos + )
7	6	opeq2d 4840	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g‘𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
8	2, 7	oveq12d 7431	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9		df-oppg 19473	. . . 4 ⊢ oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g‘𝑥)⟩))
10		ovex 7446	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
11	8, 9, 10	fvmpt 6986	. . 3 ⊢ (𝑅 ∈ V → (oppg‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12		fvprc 6870	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppg‘𝑅) = ∅)
13		reldmsets 17257	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7452	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
15	12, 14	eqtr4d 2798	. . 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 2783	1 ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  ⟨cop 4590  ‘cfv 6533  (class class class)co 7413  tpos ctpos 8223   sSet csts 17255  ndxcnx 17285  +_gcplusg 17342  opp_gcoppg 19472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-tpos 8224  df-sets 17256  df-oppg 19473

This theorem is used by:  oppgplusfval  19475  oppgbas  19478  oppgtset  19479  oppgle  19494