Theorem opprvalg 14048

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 14047	. . 3 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩))
3		id 19	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
4		fveq2 5629	. . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅))
5		opprval.2	. . . . . . 7 ⊢ · = (.r‘𝑅)
6	4, 5	eqtr4di 2280	. . . . . 6 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · )
7	6	tposeqd 6400	. . . . 5 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · )
8	7	opeq2d 3864	. . . 4 ⊢ (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r‘𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
9	3, 8	oveq12d 6025	. . 3 ⊢ (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r‘𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
10		elex 2811	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11		mulrslid 13181	. . . . . 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 13075	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
15	5, 14	eqeltrid 2316	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
16		tposexg 6410	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
17	15, 16	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
18		setsex 13080	. . . 4 ⊢ ((𝑅 ∈ V ∧ (.r‘ndx) ∈ ℕ ∧ tpos · ∈ V) → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
19	10, 13, 17, 18	syl3anc 1271	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
20	2, 9, 10, 19	fvmptd3 5730	. 2 ⊢ (𝑅 ∈ 𝑉 → (oppr‘𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
21	1, 20	eqtrid 2274	1 ⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669  ‘cfv 5318  (class class class)co 6007  tpos ctpos 6396  ℕcn 9121  ndxcnx 13045   sSet csts 13046  Slot cslot 13047  Basecbs 13048  ._rcmulr 13127  opp_rcoppr 14046

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-sets 13055  df-mulr 13140  df-oppr 14047

This theorem is referenced by:  opprmulfvalg  14049  opprex  14052  opprsllem  14053