Theorem ruclem1 16267

Description: Lemma for ruc 16279 (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 7448	. . . . 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 5724	. . . . . 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 5155	. . . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
11	10	opeq1d 4879	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⟨(1st ‘𝑥), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩)
12	9	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
13	12	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 + (2nd ‘𝑥)) = (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)))
14	13	oveq1d 7446	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((𝑚 + (2nd ‘𝑥)) / 2) = ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
15	14, 12	opeq12d 4881	. . . . . . . . . 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 4554	. . . . . . . . 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 3906	. . . . . . . 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 7449	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((1st ‘𝑥) + (2nd ‘𝑥)) = ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)))
19	18	oveq1d 7446	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (((1st ‘𝑥) + (2nd ‘𝑥)) / 2) = (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
20	19	csbeq1d 3903	. . . . . . . 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 2777	. . . . . . 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 2737	. . . . . . 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 5469	. . . . . . . . 9 ⊢ ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ ∈ V
24		opex 5469	. . . . . . . . 9 ⊢ ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ ∈ V
25	23, 24	ifex 4576	. . . . . . . 8 ⊢ if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
26	25	csbex 5311	. . . . . . 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 7588	. . . . . 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 584	. . . . 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 2777	. . . 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 8026	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
31	3, 4, 30	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
32		op2ndg 8027	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
33	3, 4, 32	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
34	31, 33	oveq12d 7449	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐴 + 𝐵))
35	34	oveq1d 7446	. . . . . 6 ⊢ (𝜑 → (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((𝐴 + 𝐵) / 2))
36	35	csbeq1d 3903	. . . . 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 7464	. . . . . . 7 ⊢ ((𝐴 + 𝐵) / 2) ∈ V
38		breq1 5146	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 < 𝑀 ↔ ((𝐴 + 𝐵) / 2) < 𝑀))
39		opeq2 4874	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩)
40		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)))
41	40	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
42	41	opeq1d 4879	. . . . . . . 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 4554	. . . . . . 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 3934	. . . . . 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 4879	. . . . . . 7 ⊢ (𝜑 → ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩ = ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩)
46	33	oveq2d 7447	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + 𝐵))
47	46	oveq1d 7446	. . . . . . . 8 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
48	47, 33	opeq12d 4881	. . . . . . 7 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ = ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)
49	45, 48	ifeq12d 4547	. . . . . 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 2789	. . . . 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 2777	. . . 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 2777	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
53	3, 4	readdcld 11290	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
54	53	rehalfcld 12513	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
55	3, 54	opelxpd 5724	. . . 4 ⊢ (𝜑 → ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩ ∈ (ℝ × ℝ))
56	54, 4	readdcld 11290	. . . . . 6 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + 𝐵) ∈ ℝ)
57	56	rehalfcld 12513	. . . . 5 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ ℝ)
58	57, 4	opelxpd 5724	. . . 4 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩ ∈ (ℝ × ℝ))
59	55, 58	ifcld 4572	. . 3 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) ∈ (ℝ × ℝ))
60	52, 59	eqeltrd 2841	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ))
61		ruclem1.6	. . 3 ⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
62	52	fveq2d 6910	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
63		fvif 6922	. . . . 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 8026	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
65	3, 37, 64	sylancl 586	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
66		ovex 7464	. . . . . . 7 ⊢ ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V
67		op1stg 8026	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
69	65, 68	ifeq12d 4547	. . . . 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 2789	. . . 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 2777	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
72	61, 71	eqtrid 2789	. 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 6910	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
75		fvif 6922	. . . . 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 8027	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
77	3, 37, 76	sylancl 586	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
78		op2ndg 8027	. . . . . . 7 ⊢ ((((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
79	66, 4, 78	sylancr 587	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
80	77, 79	ifeq12d 4547	. . . . 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 2789	. . . 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 2777	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
83	73, 82	eqtrid 2789	. 2 ⊢ (𝜑 → 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
84	60, 72, 83	3jca 1129	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 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⦋csb 3899  ifcif 4525  ⟨cop 4632   class class class wbr 5143   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013  ℝcr 11154   + caddc 11158   < clt 11295   / cdiv 11920  ℕcn 12266  2c2 12321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329

This theorem is referenced by:  ruclem2  16268  ruclem3  16269  ruclem6  16271