Theorem ruclem13 4897

Description: Lemma for ruc 4924. A helper lemma showing the sequence builder used for our construction maps natural numbers to pairs of reals.

Hypotheses

Ref	Expression
ruclem13.0	⊢ F:ℕ–→ℝ
ruclem13.1	⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
ruclem13.2	⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}

Assertion

Ref	Expression
ruclem13	⊢ (DseqC):ℕ–→(ℝ × ℝ)

Distinct variable group(s):   x,y,z

Proof of Theorem ruclem13

Step	Hyp	Ref	Expression
1		ruclem13.2	. . . . 5 ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
2	1	ruclem9 4893	. . . 4 ⊢ D ∈ V
3		ruclem13.0	. . . . 5 ⊢ F:ℕ–→ℝ
4		ruclem13.1	. . . . 5 ⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
5	3, 4	ruclem5 4889	. . . 4 ⊢ C ∈ V
6	2, 5	seqfn 4672	. . 3 ⊢ (DseqC) Fn ℕ
7	3, 4	ruclem7 4891	. . . . 5 ⊢ (C ‘1) = ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩
8		1nn 4432	. . . . . . . . 9 ⊢ 1 ∈ ℕ
9		ffvrn 2890	. . . . . . . . 9 ⊢ ((F:ℕ–→ℝ ∧ 1 ∈ ℕ) → (F ‘1) ∈ ℝ)
10	3, 8, 9	mp2an 520	. . . . . . . 8 ⊢ (F ‘1) ∈ ℝ
11		ax1re 4064	. . . . . . . 8 ⊢ 1 ∈ ℝ
12	10, 11	readdcl 4118	. . . . . . 7 ⊢ ((F ‘1) + 1) ∈ ℝ
13		2re 4470	. . . . . . . 8 ⊢ 2 ∈ ℝ
14	10, 13	readdcl 4118	. . . . . . 7 ⊢ ((F ‘1) + 2) ∈ ℝ
15	12, 14	pm3.2i 234	. . . . . 6 ⊢ (((F ‘1) + 1) ∈ ℝ ∧ ((F ‘1) + 2) ∈ ℝ)
16		oprex 3018	. . . . . . 7 ⊢ ((F ‘1) + 2) ∈ V
17	16	opelxp 2452	. . . . . 6 ⊢ (⟨((F ‘1) + 1), ((F ‘1) + 2)⟩ ∈ (ℝ × ℝ) ↔ (((F ‘1) + 1) ∈ ℝ ∧ ((F ‘1) + 2) ∈ ℝ))
18	15, 17	mpbir 165	. . . . 5 ⊢ ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩ ∈ (ℝ × ℝ)
19	7, 18	eqeltr 1159	. . . 4 ⊢ (C ‘1) ∈ (ℝ × ℝ)
20		difss 1596	. . . . . 6 ⊢ (ℕ ∖ {1}) ⊆ ℕ
21		fssres 2764	. . . . . 6 ⊢ ((F:ℕ–→ℝ ∧ (ℕ ∖ {1}) ⊆ ℕ) → (F ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ)
22	3, 20, 21	mp2an 520	. . . . 5 ⊢ (F ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ
23	3, 4	ruclem6 4890	. . . . . 6 ⊢ (C ↾ (ℕ ∖ {1})) = (F ↾ (ℕ ∖ {1}))
24		feq1 2748	. . . . . 6 ⊢ ((C ↾ (ℕ ∖ {1})) = (F ↾ (ℕ ∖ {1})) → ((C ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ ↔ (F ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ))
25	23, 24	ax-mp 6	. . . . 5 ⊢ ((C ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ ↔ (F ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ)
26	22, 25	mpbir 165	. . . 4 ⊢ (C ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ
27	1	ruclem12 4896	. . . . . . 7 ⊢ D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}
28		opex 1893	. . . . . . . . . . . 12 ⊢ ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ V
29		opex 1893	. . . . . . . . . . . 12 ⊢ ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩ ∈ V
30	28, 29	ifex 1797	. . . . . . . . . . 11 ⊢ if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ V
31	27	oprabval4g 3053	. . . . . . . . . . 11 ⊢ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ ∧ if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ V) → (wDv) = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))
32	30, 31	mp3an3 641	. . . . . . . . . 10 ⊢ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) → (wDv) = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))
33	32	cleq2d 1112	. . . . . . . . 9 ⊢ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) → (u = (wDv) ↔ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)))
34	33	pm5.32i 489	. . . . . . . 8 ⊢ (((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = (wDv)) ↔ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)))
35	34	bioprabi 3027	. . . . . . 7 ⊢ {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = (wDv))} = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩))}
36	27, 35	eqtr4 1122	. . . . . 6 ⊢ D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = (wDv))}
37		iftrue 1780	. . . . . . . . . . . . 13 ⊢ (((1st ‘w) < v ∧ v < (2nd ‘w)) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) = ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩)
38	37	eleq1d 1155	. . . . . . . . . . . 12 ⊢ (((1st ‘w) < v ∧ v < (2nd ‘w)) → (if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ) ↔ ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ)))
39		opelxpi 2455	. . . . . . . . . . . . . . 15 ⊢ (((((2 · v) + (2nd ‘w)) / 3) ∈ ℝ ∧ ((v + (2 · (2nd ‘w))) / 3) ∈ ℝ) → ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
40		axaddrcl 4067	. . . . . . . . . . . . . . . . 17 ⊢ (((2 · v) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((2 · v) + (2nd ‘w)) ∈ ℝ)
41		axmulrcl 4069	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ v ∈ ℝ) → (2 · v) ∈ ℝ)
42	13, 41	mpan 518	. . . . . . . . . . . . . . . . 17 ⊢ (v ∈ ℝ → (2 · v) ∈ ℝ)
43	40, 42	sylan 343	. . . . . . . . . . . . . . . 16 ⊢ ((v ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((2 · v) + (2nd ‘w)) ∈ ℝ)
44		3re 4472	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
45		3pos 4480	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 3
46	44, 45	gt0ne0i 4345	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
47		redivclt 4276	. . . . . . . . . . . . . . . . . 18 ⊢ (((((2 · v) + (2nd ‘w)) ∈ ℝ ∧ 3 ∈ ℝ) ∧ 3 ≠ 0) → (((2 · v) + (2nd ‘w)) / 3) ∈ ℝ)
48	46, 47	mpan2 519	. . . . . . . . . . . . . . . . 17 ⊢ ((((2 · v) + (2nd ‘w)) ∈ ℝ ∧ 3 ∈ ℝ) → (((2 · v) + (2nd ‘w)) / 3) ∈ ℝ)
49	44, 48	mpan2 519	. . . . . . . . . . . . . . . 16 ⊢ (((2 · v) + (2nd ‘w)) ∈ ℝ → (((2 · v) + (2nd ‘w)) / 3) ∈ ℝ)
50	43, 49	syl 12	. . . . . . . . . . . . . . 15 ⊢ ((v ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → (((2 · v) + (2nd ‘w)) / 3) ∈ ℝ)
51		axaddrcl 4067	. . . . . . . . . . . . . . . . 17 ⊢ ((v ∈ ℝ ∧ (2 · (2nd ‘w)) ∈ ℝ) → (v + (2 · (2nd ‘w))) ∈ ℝ)
52		axmulrcl 4069	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → (2 · (2nd ‘w)) ∈ ℝ)
53	13, 52	mpan 518	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘w) ∈ ℝ → (2 · (2nd ‘w)) ∈ ℝ)
54	51, 53	sylan2 346	. . . . . . . . . . . . . . . 16 ⊢ ((v ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → (v + (2 · (2nd ‘w))) ∈ ℝ)
55		redivclt 4276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((v + (2 · (2nd ‘w))) ∈ ℝ ∧ 3 ∈ ℝ) ∧ 3 ≠ 0) → ((v + (2 · (2nd ‘w))) / 3) ∈ ℝ)
56	46, 55	mpan2 519	. . . . . . . . . . . . . . . . 17 ⊢ (((v + (2 · (2nd ‘w))) ∈ ℝ ∧ 3 ∈ ℝ) → ((v + (2 · (2nd ‘w))) / 3) ∈ ℝ)
57	44, 56	mpan2 519	. . . . . . . . . . . . . . . 16 ⊢ ((v + (2 · (2nd ‘w))) ∈ ℝ → ((v + (2 · (2nd ‘w))) / 3) ∈ ℝ)
58	54, 57	syl 12	. . . . . . . . . . . . . . 15 ⊢ ((v ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((v + (2 · (2nd ‘w))) / 3) ∈ ℝ)
59	39, 50, 58	sylanc 361	. . . . . . . . . . . . . 14 ⊢ ((v ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
60	59	ancoms 334	. . . . . . . . . . . . 13 ⊢ (((2nd ‘w) ∈ ℝ ∧ v ∈ ℝ) → ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
61	60	adantll 309	. . . . . . . . . . . 12 ⊢ ((((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) ∧ v ∈ ℝ) → ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
62	38, 61	syl5bir 184	. . . . . . . . . . 11 ⊢ (((1st ‘w) < v ∧ v < (2nd ‘w)) → ((((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) ∧ v ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ)))
63		iffalse 1781	. . . . . . . . . . . . . 14 ⊢ (¬ ((1st ‘w) < v ∧ v < (2nd ‘w)) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) = ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩)
64	63	eleq1d 1155	. . . . . . . . . . . . 13 ⊢ (¬ ((1st ‘w) < v ∧ v < (2nd ‘w)) → (if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ) ↔ ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ)))
65		opelxpi 2455	. . . . . . . . . . . . . 14 ⊢ (((((2 · (1st ‘w)) + (2nd ‘w)) / 3) ∈ ℝ ∧ (((1st ‘w) + (2 · (2nd ‘w))) / 3) ∈ ℝ) → ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
66		axaddrcl 4067	. . . . . . . . . . . . . . . 16 ⊢ (((2 · (1st ‘w)) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((2 · (1st ‘w)) + (2nd ‘w)) ∈ ℝ)
67		axmulrcl 4069	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ ∧ (1st ‘w) ∈ ℝ) → (2 · (1st ‘w)) ∈ ℝ)
68	13, 67	mpan 518	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘w) ∈ ℝ → (2 · (1st ‘w)) ∈ ℝ)
69	66, 68	sylan 343	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((2 · (1st ‘w)) + (2nd ‘w)) ∈ ℝ)
70		redivclt 4276	. . . . . . . . . . . . . . . . 17 ⊢ (((((2 · (1st ‘w)) + (2nd ‘w)) ∈ ℝ ∧ 3 ∈ ℝ) ∧ 3 ≠ 0) → (((2 · (1st ‘w)) + (2nd ‘w)) / 3) ∈ ℝ)
71	46, 70	mpan2 519	. . . . . . . . . . . . . . . 16 ⊢ ((((2 · (1st ‘w)) + (2nd ‘w)) ∈ ℝ ∧ 3 ∈ ℝ) → (((2 · (1st ‘w)) + (2nd ‘w)) / 3) ∈ ℝ)
72	44, 71	mpan2 519	. . . . . . . . . . . . . . 15 ⊢ (((2 · (1st ‘w)) + (2nd ‘w)) ∈ ℝ → (((2 · (1st ‘w)) + (2nd ‘w)) / 3) ∈ ℝ)
73	69, 72	syl 12	. . . . . . . . . . . . . 14 ⊢ (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → (((2 · (1st ‘w)) + (2nd ‘w)) / 3) ∈ ℝ)
74		axaddrcl 4067	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘w) ∈ ℝ ∧ (2 · (2nd ‘w)) ∈ ℝ) → ((1st ‘w) + (2 · (2nd ‘w))) ∈ ℝ)
75	74, 53	sylan2 346	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ((1st ‘w) + (2 · (2nd ‘w))) ∈ ℝ)
76		redivclt 4276	. . . . . . . . . . . . . . . . 17 ⊢ (((((1st ‘w) + (2 · (2nd ‘w))) ∈ ℝ ∧ 3 ∈ ℝ) ∧ 3 ≠ 0) → (((1st ‘w) + (2 · (2nd ‘w))) / 3) ∈ ℝ)
77	46, 76	mpan2 519	. . . . . . . . . . . . . . . 16 ⊢ ((((1st ‘w) + (2 · (2nd ‘w))) ∈ ℝ ∧ 3 ∈ ℝ) → (((1st ‘w) + (2 · (2nd ‘w))) / 3) ∈ ℝ)
78	44, 77	mpan2 519	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘w) + (2 · (2nd ‘w))) ∈ ℝ → (((1st ‘w) + (2 · (2nd ‘w))) / 3) ∈ ℝ)
79	75, 78	syl 12	. . . . . . . . . . . . . 14 ⊢ (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → (((1st ‘w) + (2 · (2nd ‘w))) / 3) ∈ ℝ)
80	65, 73, 79	sylanc 361	. . . . . . . . . . . . 13 ⊢ (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩ ∈ (ℝ × ℝ))
81	64, 80	syl5bir 184	. . . . . . . . . . . 12 ⊢ (¬ ((1st ‘w) < v ∧ v < (2nd ‘w)) → (((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ)))
82	81	adantrd 308	. . . . . . . . . . 11 ⊢ (¬ ((1st ‘w) < v ∧ v < (2nd ‘w)) → ((((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) ∧ v ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ)))
83	62, 82	pm2.61i 110	. . . . . . . . . 10 ⊢ ((((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ) ∧ v ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ))
84	83	adantll 309	. . . . . . . . 9 ⊢ (((w = ⟨(1st ‘w), (2nd ‘w)⟩ ∧ ((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ)) ∧ v ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ))
85		elxp6 3093	. . . . . . . . 9 ⊢ (w ∈ (ℝ × ℝ) ↔ (w = ⟨(1st ‘w), (2nd ‘w)⟩ ∧ ((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ)))
86	84, 85	sylanb 344	. . . . . . . 8 ⊢ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) → if(((1st ‘w) < v ∧ v < (2nd ‘w)), ⟨(((2 · v) + (2nd ‘w)) / 3), ((v + (2 · (2nd ‘w))) / 3)⟩, ⟨(((2 · (1st ‘w)) + (2nd ‘w)) / 3), (((1st ‘w) + (2 · (2nd ‘w))) / 3)⟩) ∈ (ℝ × ℝ))
87	32, 86	eqeltrd 1163	. . . . . . 7 ⊢ ((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) → (wDv) ∈ (ℝ × ℝ))
88	87	rgen2a 1264	. . . . . 6 ⊢ ∀w ∈ (ℝ × ℝ)∀v ∈ ℝ (wDv) ∈ (ℝ × ℝ)
89	36, 88	pm3.2i 234	. . . . 5 ⊢ (D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = (wDv))} ∧ ∀w ∈ (ℝ × ℝ)∀v ∈ ℝ (wDv) ∈ (ℝ × ℝ))
90		foprval 3043	. . . . 5 ⊢ (D:((ℝ × ℝ) × ℝ)–→(ℝ × ℝ) ↔ (D = {⟨⟨w, v⟩, u⟩∣((w ∈ (ℝ × ℝ) ∧ v ∈ ℝ) ∧ u = (wDv))} ∧ ∀w ∈ (ℝ × ℝ)∀v ∈ ℝ (wDv) ∈ (ℝ × ℝ)))
91	89, 90	mpbir 165	. . . 4 ⊢ D:((ℝ × ℝ) × ℝ)–→(ℝ × ℝ)
92	2, 5	seqrn 4673	. . . 4 ⊢ (((C ‘1) ∈ (ℝ × ℝ) ∧ (C ↾ (ℕ ∖ {1})):(ℕ ∖ {1})–→ℝ ∧ D:((ℝ × ℝ) × ℝ)–→(ℝ × ℝ)) → ran (DseqC) ⊆ (ℝ × ℝ))
93	19, 26, 91, 92	mp3an 642	. . 3 ⊢ ran (DseqC) ⊆ (ℝ × ℝ)
94	6, 93	pm3.2i 234	. 2 ⊢ ((DseqC) Fn ℕ ∧ ran (DseqC) ⊆ (ℝ × ℝ))
95		df-f 2434	. 2 ⊢ ((DseqC):ℕ–→(ℝ × ℝ) ↔ ((DseqC) Fn ℕ ∧ ran (DseqC) ⊆ (ℝ × ℝ)))
96	94, 95	mpbir 165	1 ⊢ (DseqC):ℕ–→(ℝ × ℝ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ≠ wne 1190  ∀wral 1201  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487  ifcif 1776  {csn 1808  ⟨cop 1810   class class class wbr 2054   × cxp 2408  ran crn 2411   ↾ cres 2412   Fn wfn 2417  –→wf 2418   ‘cfv 2422  (class class class)co 3001  {copab2 3002  1^st c1st 3085  2^nd c2nd 3086  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031   · cmulc 4032   < clt 4033   / cdiv 4091  ℕcn 4093  2c2 4454  3c3 4455  seqcseq 4660

This theorem is referenced by:  ruclem15 4899  ruclem17 4901  ruclem23 4907

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-n0 4535  df-z 4564  df-seq 4661

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