Theorem ruclem1 16175

Description: Lemma for ruc 16187 (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 7386	. . . . 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 5670	. . . . . 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 5114	. . . . . . . . . 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 6844	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
11	10	opeq1d 4839	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ⟨(1st ‘𝑥), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩)
12	9	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
13	12	oveq2d 7385	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 + (2nd ‘𝑥)) = (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)))
14	13	oveq1d 7384	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((𝑚 + (2nd ‘𝑥)) / 2) = ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
15	14, 12	opeq12d 4841	. . . . . . . . . 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 4513	. . . . . . . . 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 3866	. . . . . . . 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 7387	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((1st ‘𝑥) + (2nd ‘𝑥)) = ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)))
19	18	oveq1d 7384	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (((1st ‘𝑥) + (2nd ‘𝑥)) / 2) = (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
20	19	csbeq1d 3863	. . . . . . . 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 2764	. . . . . . 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 2729	. . . . . . 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 5419	. . . . . . . . 9 ⊢ ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ ∈ V
24		opex 5419	. . . . . . . . 9 ⊢ ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ ∈ V
25	23, 24	ifex 4535	. . . . . . . 8 ⊢ if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
26	25	csbex 5261	. . . . . . 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 7524	. . . . . 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 2764	. . . 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 7959	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
31	3, 4, 30	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
32		op2ndg 7960	. . . . . . . . 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 7387	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐴 + 𝐵))
35	34	oveq1d 7384	. . . . . 6 ⊢ (𝜑 → (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((𝐴 + 𝐵) / 2))
36	35	csbeq1d 3863	. . . . 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 7402	. . . . . . 7 ⊢ ((𝐴 + 𝐵) / 2) ∈ V
38		breq1 5105	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 < 𝑀 ↔ ((𝐴 + 𝐵) / 2) < 𝑀))
39		opeq2 4834	. . . . . . . 8 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ = ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩)
40		oveq1 7376	. . . . . . . . . 10 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)))
41	40	oveq1d 7384	. . . . . . . . 9 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → ((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
42	41	opeq1d 4839	. . . . . . . 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 4513	. . . . . . 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 3894	. . . . . 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 4839	. . . . . . 7 ⊢ (𝜑 → ⟨(1st ‘⟨𝐴, 𝐵⟩), ((𝐴 + 𝐵) / 2)⟩ = ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩)
46	33	oveq2d 7385	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + 𝐵))
47	46	oveq1d 7384	. . . . . . . 8 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
48	47, 33	opeq12d 4841	. . . . . . 7 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ = ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)
49	45, 48	ifeq12d 4506	. . . . . 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 2776	. . . . 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 2764	. . . 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 2764	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
53	3, 4	readdcld 11179	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
54	53	rehalfcld 12405	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
55	3, 54	opelxpd 5670	. . . 4 ⊢ (𝜑 → ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩ ∈ (ℝ × ℝ))
56	54, 4	readdcld 11179	. . . . . 6 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + 𝐵) ∈ ℝ)
57	56	rehalfcld 12405	. . . . 5 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ ℝ)
58	57, 4	opelxpd 5670	. . . 4 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩ ∈ (ℝ × ℝ))
59	55, 58	ifcld 4531	. . 3 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) ∈ (ℝ × ℝ))
60	52, 59	eqeltrd 2828	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ))
61		ruclem1.6	. . 3 ⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
62	52	fveq2d 6844	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
63		fvif 6856	. . . . 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 7959	. . . . . . 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 7402	. . . . . . 7 ⊢ ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V
67		op1stg 7959	. . . . . . 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 4506	. . . . 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 2776	. . . 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 2764	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
72	61, 71	eqtrid 2776	. 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 6844	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
75		fvif 6856	. . . . 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 7960	. . . . . . 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 7960	. . . . . . 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 4506	. . . . 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 2776	. . . 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 2764	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
83	73, 82	eqtrid 2776	. 2 ⊢ (𝜑 → 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
84	60, 72, 83	3jca 1128	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 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⦋csb 3859  ifcif 4484  ⟨cop 4591   class class class wbr 5102   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  ℝcr 11043   + caddc 11047   < clt 11184   / cdiv 11811  ℕcn 12162  2c2 12217

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225

This theorem is referenced by:  ruclem2  16176  ruclem3  16177  ruclem6  16179