Theorem ruclem1 15985

Description: Lemma for ruc 15997 (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 7324	. . . . 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 5638	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
6		ruclem1.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
7		simpr 486	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀)
8	7	breq2d 5093	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 < 𝑦 ↔ 𝑚 < 𝑀))
9		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → 𝑥 = ⟨𝐴, 𝐵⟩)
10	9	fveq2d 6808	. . . . . . . . . . 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 6808	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
13	12	oveq2d 7323	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (𝑚 + (2nd ‘𝑥)) = (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)))
14	13	oveq1d 7322	. . . . . . . . . . 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 4493	. . . . . . . . 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 3844	. . . . . . . 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 7325	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → ((1st ‘𝑥) + (2nd ‘𝑥)) = ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)))
19	18	oveq1d 7322	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ 𝑦 = 𝑀) → (((1st ‘𝑥) + (2nd ‘𝑥)) / 2) = (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2))
20	19	csbeq1d 3841	. . . . . . . 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 2776	. . . . . . 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 2736	. . . . . . 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 5392	. . . . . . . . 9 ⊢ ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩ ∈ V
24		opex 5392	. . . . . . . . 9 ⊢ ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩ ∈ V
25	23, 24	ifex 4515	. . . . . . . 8 ⊢ if(𝑚 < 𝑀, ⟨(1st ‘⟨𝐴, 𝐵⟩), 𝑚⟩, ⟨((𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) / 2), (2nd ‘⟨𝐴, 𝐵⟩)⟩) ∈ V
26	25	csbex 5244	. . . . . . 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 7460	. . . . . 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 585	. . . . 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 2776	. . . 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 7875	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
31	3, 4, 30	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
32		op2ndg 7876	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
33	3, 4, 32	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
34	31, 33	oveq12d 7325	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐴 + 𝐵))
35	34	oveq1d 7322	. . . . . 6 ⊢ (𝜑 → (((1st ‘⟨𝐴, 𝐵⟩) + (2nd ‘⟨𝐴, 𝐵⟩)) / 2) = ((𝐴 + 𝐵) / 2))
36	35	csbeq1d 3841	. . . . 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 7340	. . . . . . 7 ⊢ ((𝐴 + 𝐵) / 2) ∈ V
38		breq1 5084	. . . . . . . 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 7314	. . . . . . . . . 10 ⊢ (𝑚 = ((𝐴 + 𝐵) / 2) → (𝑚 + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)))
41	40	oveq1d 7322	. . . . . . . . 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 4493	. . . . . . 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 3873	. . . . . 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 7323	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + (2nd ‘⟨𝐴, 𝐵⟩)) = (((𝐴 + 𝐵) / 2) + 𝐵))
47	46	oveq1d 7322	. . . . . . . 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 4486	. . . . . 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 2788	. . . . 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 2776	. . . 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 2776	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) = if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩))
53	3, 4	readdcld 11050	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
54	53	rehalfcld 12266	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ ℝ)
55	3, 54	opelxpd 5638	. . . 4 ⊢ (𝜑 → ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩ ∈ (ℝ × ℝ))
56	54, 4	readdcld 11050	. . . . . 6 ⊢ (𝜑 → (((𝐴 + 𝐵) / 2) + 𝐵) ∈ ℝ)
57	56	rehalfcld 12266	. . . . 5 ⊢ (𝜑 → ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ ℝ)
58	57, 4	opelxpd 5638	. . . 4 ⊢ (𝜑 → ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩ ∈ (ℝ × ℝ))
59	55, 58	ifcld 4511	. . 3 ⊢ (𝜑 → if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) ∈ (ℝ × ℝ))
60	52, 59	eqeltrd 2837	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝐷𝑀) ∈ (ℝ × ℝ))
61		ruclem1.6	. . 3 ⊢ 𝑋 = (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀))
62	52	fveq2d 6808	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (1st ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
63		fvif 6820	. . . . 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 7875	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
65	3, 37, 64	sylancl 587	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = 𝐴)
66		ovex 7340	. . . . . . 7 ⊢ ((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V
67		op1stg 7875	. . . . . . 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 588	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = ((((𝐴 + 𝐵) / 2) + 𝐵) / 2))
69	65, 68	ifeq12d 4486	. . . . 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 2788	. . . 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 2776	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)))
72	61, 71	eqtrid 2788	. 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 6808	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = (2nd ‘if(((𝐴 + 𝐵) / 2) < 𝑀, ⟨𝐴, ((𝐴 + 𝐵) / 2)⟩, ⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩)))
75		fvif 6820	. . . . 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 7876	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐴 + 𝐵) / 2) ∈ V) → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
77	3, 37, 76	sylancl 587	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐴, ((𝐴 + 𝐵) / 2)⟩) = ((𝐴 + 𝐵) / 2))
78		op2ndg 7876	. . . . . . 7 ⊢ ((((((𝐴 + 𝐵) / 2) + 𝐵) / 2) ∈ V ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
79	66, 4, 78	sylancr 588	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨((((𝐴 + 𝐵) / 2) + 𝐵) / 2), 𝐵⟩) = 𝐵)
80	77, 79	ifeq12d 4486	. . . . 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 2788	. . . 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 2776	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐴, 𝐵⟩𝐷𝑀)) = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵))
83	73, 82	eqtrid 2788	. 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 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  Vcvv 3437  ⦋csb 3837  ifcif 4465  ⟨cop 4571   class class class wbr 5081   × cxp 5598  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ∈ cmpo 7309  1^st c1st 7861  2^nd c2nd 7862  ℝcr 10916   + caddc 10920   < clt 11055   / cdiv 11678  ℕcn 12019  2c2 12074

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-po 5514  df-so 5515  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-div 11679  df-2 12082

This theorem is referenced by:  ruclem2  15986  ruclem3  15987  ruclem6  15989