Theorem ruclem1 15921

Description: Lemma for ruc 15933 (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 7285	. . . . 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 5626	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
6		ruclem1.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
7		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀)
8	7	breq2d 5090	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 < 𝑦 ↔ 𝑚 < 𝑀))
9		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑥 = ⟨𝐴, 𝐵⟩)
10	9	fveq2d 6772	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
11	10	opeq1d 4815	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⟨(1st ‘𝑥), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩)
12	9	fveq2d 6772	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
13	12	oveq2d 7284	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 + (2nd ‘𝑥)) = (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)))
14	13	oveq1d 7283	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((𝑚 + (2nd ‘𝑥)) / 2) = ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
15	14, 12	opeq12d 4817	. . . . . . . . . 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 4492	. . . . . . . . 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 3843	. . . . . . . 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 7286	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((1st ‘𝑥) + (2nd ‘𝑥)) = ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)))
19	18	oveq1d 7283	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (((1st ‘𝑥) + (2nd ‘𝑥)) / 2) = (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
20	19	csbeq1d 3840	. . . . . . . 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 2779	. . . . . . 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 2739	. . . . . . 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 5381	. . . . . . . . 9 ⊢ ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ ∈ V
24		opex 5381	. . . . . . . . 9 ⊢ ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ ∈ V
25	23, 24	ifex 4514	. . . . . . . 8 ⊢ if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
26	25	csbex 5238	. . . . . . 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 7419	. . . . . 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 583	. . . . 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 2779	. . . 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 7829	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
31	3, 4, 30	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
32		op2ndg 7830	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
33	3, 4, 32	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
34	31, 33	oveq12d 7286	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐴 + 𝐵))
35	34	oveq1d 7283	. . . . . 6 ⊢ (𝜑 → (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((𝐴 + 𝐵) / 2))
36	35	csbeq1d 3840	. . . . 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 7301	. . . . . . 7 ⊢ ((𝐴 + 𝐵) / 2) ∈ V
38		breq1 5081	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 < 𝑀 ↔ ((𝐴 + 𝐵) / 2) < 𝑀))
39		opeq2 4810	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩)
40		oveq1 7275	. . . . . . . . . 10 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)))
41	40	oveq1d 7283	. . . . . . . . 9 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
42	41	opeq1d 4815	. . . . . . . 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 4492	. . . . . . 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 3872	. . . . . 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 4815	. . . . . . 7 ⊢ (𝜑 → ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩ = ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩)
46	33	oveq2d 7284	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + 𝐵))
47	46	oveq1d 7283	. . . . . . . 8 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
48	47, 33	opeq12d 4817	. . . . . . 7 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ = ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)
49	45, 48	ifeq12d 4485	. . . . . 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 2791	. . . . 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 2779	. . . 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 2779	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
53	3, 4	readdcld 10988	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
54	53	rehalfcld 12203	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
55	3, 54	opelxpd 5626	. . . 4 ⊢ (𝜑 → ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩ ∈ (ℝ × ℝ))
56	54, 4	readdcld 10988	. . . . . 6 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + 𝐵) ∈ ℝ)
57	56	rehalfcld 12203	. . . . 5 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ ℝ)
58	57, 4	opelxpd 5626	. . . 4 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩ ∈ (ℝ × ℝ))
59	55, 58	ifcld 4510	. . 3 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) ∈ (ℝ × ℝ))
60	52, 59	eqeltrd 2840	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ))
61		ruclem1.6	. . 3 ⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
62	52	fveq2d 6772	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
63		fvif 6784	. . . . 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 7829	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
65	3, 37, 64	sylancl 585	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
66		ovex 7301	. . . . . . 7 ⊢ ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V
67		op1stg 7829	. . . . . . 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 586	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
69	65, 68	ifeq12d 4485	. . . . 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 2791	. . . 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 2779	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
72	61, 71	eqtrid 2791	. 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 6772	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
75		fvif 6784	. . . . 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 7830	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
77	3, 37, 76	sylancl 585	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
78		op2ndg 7830	. . . . . . 7 ⊢ ((((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
79	66, 4, 78	sylancr 586	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
80	77, 79	ifeq12d 4485	. . . . 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 2791	. . . 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 2779	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
83	73, 82	eqtrid 2791	. 2 ⊢ (𝜑 → 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
84	60, 72, 83	3jca 1126	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 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⦋csb 3836  ifcif 4464  ⟨cop 4572   class class class wbr 5078   × cxp 5586  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  1^st c1st 7815  2^nd c2nd 7816  ℝcr 10854   + caddc 10858   < clt 10993   / cdiv 11615  ℕcn 11956  2c2 12011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-po 5502  df-so 5503  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-2 12019

This theorem is referenced by:  ruclem2  15922  ruclem3  15923  ruclem6  15925