Theorem opprvalg 14081

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
opprvalg	⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))

Proof of Theorem opprvalg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 ⊢ 𝑂 = (oppr‘𝑅)
2		df-oppr 14080	. . 3 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
3		id 19	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
4		fveq2 5639	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅))
5		opprval.2	. . . . . . 7 ⊢ · = (.r‘𝑅)
6	4, 5	eqtr4di 2282	. . . . . 6 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · )
7	6	tposeqd 6413	. . . . 5 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
8	7	opeq2d 3869	. . . 4 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
9	3, 8	oveq12d 6035	. . 3 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
10		elex 2814	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11		mulrslid 13214	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
12	11	simpri 113	. . . . 5 ⊢ (.r‘ndx) ∈ ℕ
13	12	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (.r‘ndx) ∈ ℕ)
14	11	slotex 13108	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
15	5, 14	eqeltrid 2318	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
16		tposexg 6423	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
17	15, 16	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
18		setsex 13113	. . . 4 ⊢ ((𝑅 ∈ V ∧ (.r‘ndx) ∈ ℕ ∧ tpos · ∈ V) → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
19	10, 13, 17, 18	syl3anc 1273	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
20	2, 9, 10, 19	fvmptd3 5740	. 2 ⊢ (𝑅 ∈ 𝑉 → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
21	1, 20	eqtrid 2276	1 ⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ⟨cop 3672  ‘cfv 5326  (class class class)co 6017  tpos ctpos 6409  ℕcn 9142  ndxcnx 13078   sSet csts 13079  Slot cslot 13080  Basecbs 13081  ._rcmulr 13160  opp_rcoppr 14079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-sets 13088  df-mulr 13173  df-oppr 14080

This theorem is referenced by:  opprmulfvalg  14082  opprex  14085  opprsllem  14086