Theorem opprvalg 14163

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 14162	. . 3 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
3		id 19	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
4		fveq2 5648	. . . . . . 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 6457	. . . . 5 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
8	7	opeq2d 3874	. . . 4 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
9	3, 8	oveq12d 6046	. . 3 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
10		elex 2815	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11		mulrslid 13295	. . . . . 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 13189	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
15	5, 14	eqeltrid 2318	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
16		tposexg 6467	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
17	15, 16	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
18		setsex 13194	. . . 4 ⊢ ((𝑅 ∈ V ∧ (.r‘ndx) ∈ ℕ ∧ tpos · ∈ V) → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
19	10, 13, 17, 18	syl3anc 1274	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
20	2, 9, 10, 19	fvmptd3 5749	. 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 1398   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676  ‘cfv 5333  (class class class)co 6028  tpos ctpos 6453  ℕcn 9202  ndxcnx 13159   sSet csts 13160  Slot cslot 13161  Basecbs 13162  ._rcmulr 13241  opp_rcoppr 14161

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-sets 13169  df-mulr 13254  df-oppr 14162

This theorem is referenced by:  opprmulfvalg  14164  opprex  14167  opprsllem  14168