Theorem opprval 19370

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	⊢ 𝐵 = (Base‘𝑅)
opprval.2	⊢ · = (.r‘𝑅)
opprval.3	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprval	⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Proof of Theorem opprval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 ⊢ 𝑂 = (oppr‘𝑅)
2		id 22	. . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
3		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅))
4		opprval.2	. . . . . . . 8 ⊢ · = (.r‘𝑅)
5	3, 4	eqtr4di 2851	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · )
6	5	tposeqd 7878	. . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
7	6	opeq2d 4772	. . . . 5 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
8	2, 7	oveq12d 7153	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9		df-oppr 19369	. . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
10		ovex 7168	. . . 4 ⊢ (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
11	8, 9, 10	fvmpt 6745	. . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12		fvprc 6638	. . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅)
13		reldmsets 16503	. . . . 5 ⊢ Rel dom sSet
14	13	ovprc1 7174	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
15	12, 14	eqtr4d 2836	. . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
16	11, 15	pm2.61i 185	. 2 ⊢ (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
17	1, 16	eqtri 2821	1 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ⟨cop 4531  ‘cfv 6324  (class class class)co 7135  tpos ctpos 7874  ndxcnx 16472   sSet csts 16473  Basecbs 16475  ._rcmulr 16558  opp_rcoppr 19368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-tpos 7875  df-sets 16482  df-oppr 19369

This theorem is referenced by:  opprmulfval  19371  opprlem  19374