Theorem ruclem1 16287

Description: Lemma for ruc 16299 (the reals are uncountable). Substitutions for the function 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

Hypotheses

Ref	Expression
ruc.1	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
ruc.2	⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
ruclem1.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
ruclem1.4	⊢ (𝜑 → 𝐵 ∈ ℝ)
ruclem1.5	⊢ (𝜑 → 𝑀 ∈ ℝ)
ruclem1.6	⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
ruclem1.7	⊢ 𝑌 = (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀))

Assertion

Ref	Expression
ruclem1	⊢ (𝜑 → ((⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ) ∧ 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)) ∧ 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵)))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑥,𝑦   𝑚,𝐹,𝑥,𝑦   𝑚,𝑀,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)   𝑋(𝑥,𝑦,𝑚)   𝑌(𝑥,𝑦,𝑚)

Proof of Theorem ruclem1

Step	Hyp	Ref	Expression
1		ruc.2	. . . . . 6 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)))
2	1	oveqd 7428	. . . . 5 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = (⟨𝐴, 𝐵⟩(𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))𝑀))
3		ruclem1.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		ruclem1.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	3, 4	opelxpd 5701	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
6		ruclem1.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
7		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀)
8	7	breq2d 5125	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 < 𝑦 ↔ 𝑚 < 𝑀))
9		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑥 = ⟨𝐴, 𝐵⟩)
10	9	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
11	10	opeq1d 4848	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⟨(1st ‘𝑥), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩)
12	9	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
13	12	oveq2d 7427	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 + (2nd ‘𝑥)) = (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)))
14	13	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((𝑚 + (2nd ‘𝑥)) / 2) = ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
15	14, 12	opeq12d 4850	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩ = ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩)
16	8, 11, 15	ifbieq12d 4521	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩) = if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
17	16	csbeq2dv 3868	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩) = ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
18	10, 12	oveq12d 7429	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((1st ‘𝑥) + (2nd ‘𝑥)) = ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)))
19	18	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (((1st ‘𝑥) + (2nd ‘𝑥)) / 2) = (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
20	19	csbeq1d 3865	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
21	17, 20	eqtrd 2804	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩) = ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
22		eqid 2769	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩)) = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))
23		opex 5446	. . . . . . . . 9 ⊢ ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ ∈ V
24		opex 5446	. . . . . . . . 9 ⊢ ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ ∈ V
25	23, 24	ifex 4543	. . . . . . . 8 ⊢ if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
26	25	csbex 5276	. . . . . . 7 ⊢ ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
27	21, 22, 26	ovmpoa 7566	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ) ∧ 𝑀 ∈ ℝ) → (⟨𝐴, 𝐵⟩(𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))𝑀) = ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
28	5, 6, 27	syl2anc 595	. . . . 5 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩(𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)⟩))𝑀) = ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
29	2, 28	eqtrd 2804	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
30		op1stg 7998	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
31	3, 4, 30	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
32		op2ndg 7999	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
33	3, 4, 32	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
34	31, 33	oveq12d 7429	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐴 + 𝐵))
35	34	oveq1d 7426	. . . . . 6 ⊢ (𝜑 → (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((𝐴 + 𝐵) / 2))
36	35	csbeq1d 3865	. . . . 5 ⊢ (𝜑 → ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = ⦋((𝐴 + 𝐵) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
37		ovex 7444	. . . . . . 7 ⊢ ((𝐴 + 𝐵) / 2) ∈ V
38		breq1 5116	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 < 𝑀 ↔ ((𝐴 + 𝐵) / 2) < 𝑀))
39		opeq2 4843	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩)
40		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)))
41	40	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
42	41	opeq1d 4848	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ = ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩)
43	38, 39, 42	ifbieq12d 4521	. . . . . . 7 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩))
44	37, 43	csbie 3896	. . . . . 6 ⊢ ⦋((𝐴 + 𝐵) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩)
45	31	opeq1d 4848	. . . . . . 7 ⊢ (𝜑 → ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩ = ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩)
46	33	oveq2d 7427	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + 𝐵))
47	46	oveq1d 7426	. . . . . . . 8 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
48	47, 33	opeq12d 4850	. . . . . . 7 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ = ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)
49	45, 48	ifeq12d 4514	. . . . . 6 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
50	44, 49	eqtrid 2816	. . . . 5 ⊢ (𝜑 → ⦋((𝐴 + 𝐵) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
51	36, 50	eqtrd 2804	. . . 4 ⊢ (𝜑 → ⦋(((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) / 𝑚⦌if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
52	29, 51	eqtrd 2804	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
53	3, 4	readdcld 11238	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
54	53	rehalfcld 12491	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
55	3, 54	opelxpd 5701	. . . 4 ⊢ (𝜑 → ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩ ∈ (ℝ × ℝ))
56	54, 4	readdcld 11238	. . . . . 6 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + 𝐵) ∈ ℝ)
57	56	rehalfcld 12491	. . . . 5 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ ℝ)
58	57, 4	opelxpd 5701	. . . 4 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩ ∈ (ℝ × ℝ))
59	55, 58	ifcld 4539	. . 3 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) ∈ (ℝ × ℝ))
60	52, 59	eqeltrd 2869	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ))
61		ruclem1.6	. . 3 ⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
62	52	fveq2d 6886	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
63		fvif 6898	. . . . 5 ⊢ (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩), (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
64		op1stg 7998	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
65	3, 37, 64	sylancl 597	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
66		ovex 7444	. . . . . . 7 ⊢ ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V
67		op1stg 7998	. . . . . . 7 ⊢ ((((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V ∧ 𝐵 ∈ ℝ) → (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
68	66, 4, 67	sylancr 598	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
69	65, 68	ifeq12d 4514	. . . . 5 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩), (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
70	63, 69	eqtrid 2816	. . . 4 ⊢ (𝜑 → (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
71	62, 70	eqtrd 2804	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
72	61, 71	eqtrid 2816	. 2 ⊢ (𝜑 → 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
73		ruclem1.7	. . 3 ⊢ 𝑌 = (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
74	52	fveq2d 6886	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
75		fvif 6898	. . . . 5 ⊢ (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩), (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
76		op2ndg 7999	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
77	3, 37, 76	sylancl 597	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
78		op2ndg 7999	. . . . . . 7 ⊢ ((((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
79	66, 4, 78	sylancr 598	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
80	77, 79	ifeq12d 4514	. . . . 5 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩), (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
81	75, 80	eqtrid 2816	. . . 4 ⊢ (𝜑 → (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
82	74, 81	eqtrd 2804	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
83	73, 82	eqtrid 2816	. 2 ⊢ (𝜑 → 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
84	60, 72, 83	3jca 1144	1 ⊢ (𝜑 → ((⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ) ∧ 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)) ∧ 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⦋csb 3861  ifcif 4492  ⟨cop 4600   class class class wbr 5113   × cxp 5660  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  ℝcr 11099   + caddc 11103   < clt 11243   / cdiv 11871  ℕcn 12233  2c2 12295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303

This theorem is referenced by:  ruclem2  16288  ruclem3  16289  ruclem6  16291