addlimc - Mathbox for Glauco Siliprandi

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Theorem addlimc 46863

Description: Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
addlimc.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
addlimc.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)
addlimc.h	⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))
addlimc.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
addlimc.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
addlimc.e	⊢ (𝜑 → 𝐸 ∈ (𝐹 lim_ℂ 𝐷))
addlimc.i	⊢ (𝜑 → 𝐼 ∈ (𝐺 lim_ℂ 𝐷))

**Assertion**
Ref	Expression
addlimc	⊢ (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 lim_ℂ 𝐷))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥) 𝐸(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝐻(𝑥) 𝐼(𝑥)

Proof of Theorem addlimc Dummy variables 𝑎 𝑏 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 26225	. . . 4 ⊢ (𝐹 lim_ℂ 𝐷) ⊆ ℂ
2		addlimc.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (𝐹 lim_ℂ 𝐷))
3	1, 2	sselid 3911	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ)
4		limccl 26225	. . . 4 ⊢ (𝐺 lim_ℂ 𝐷) ⊆ ℂ
5		addlimc.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐺 lim_ℂ 𝐷))
6	4, 5	sselid 3911	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
7	3, 6	addcld 11355	. 2 ⊢ (𝜑 → (𝐸 + 𝐼) ∈ ℂ)
8		addlimc.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
9		addlimc.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	8, 9	fmptd 7138	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
11	9, 8, 2	limcmptdm 46850	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
12		limcrcl 26224	. . . . . . . . . . 11 ⊢ (𝐸 ∈ (𝐹 lim_ℂ 𝐷) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
13	2, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
14	13	simp3d 1162	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
15	10, 11, 14	ellimc3 26229	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ (𝐹 lim_ℂ 𝐷) ↔ (𝐸 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧))))
16	2, 15	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧)))
17	16	simprd 501	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧))
18		rphalfcl 13176	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 / 2) ∈ ℝ⁺)
19		breq2 5110	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 / 2) → ((abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧 ↔ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
20	19	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / 2) → (((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧) ↔ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))))
21	20	rexralbidv 3232	. . . . . . 7 ⊢ (𝑧 = (𝑦 / 2) → (∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧) ↔ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))))
22	21	rspccva 3579	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧) ∧ (𝑦 / 2) ∈ ℝ⁺) → ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
23	17, 18, 22	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
24		addlimc.c	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
25		addlimc.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)
26	24, 25	fmptd 7138	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐴⟶ℂ)
27	26, 11, 14	ellimc3 26229	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∈ (𝐺 lim_ℂ 𝐷) ↔ (𝐼 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧))))
28	5, 27	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧)))
29	28	simprd 501	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧))
30		breq2 5110	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 / 2) → ((abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧 ↔ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
31	30	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = (𝑦 / 2) → (((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧) ↔ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
32	31	rexralbidv 3232	. . . . . . 7 ⊢ (𝑧 = (𝑦 / 2) → (∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧) ↔ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
33	32	rspccva 3579	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧) ∧ (𝑦 / 2) ∈ ℝ⁺) → ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
34	29, 18, 33	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
35		reeanv 3238	. . . . 5 ⊢ (∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))) ↔ (∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
36	23, 34, 35	sylanbrc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
37		ifcl 4531	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
38	37	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
39		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑦 ∈ ℝ⁺)
40		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑣(𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)
41		nfra1 3290	. . . . . . . . . 10 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))
42		nfra1 3290	. . . . . . . . . 10 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))
43	41, 42	nfan 1934	. . . . . . . . 9 ⊢ Ⅎ𝑣(∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
44	39, 40, 43	nf3an 1936	. . . . . . . 8 ⊢ Ⅎ𝑣((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
45		simp11l 1303	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝜑)
46		simp2 1155	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑣 ∈ 𝐴)
47	45, 46	jca 521	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (𝜑 ∧ 𝑣 ∈ 𝐴))
48		rpre 13156	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
49	48	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
50	49	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → 𝑦 ∈ ℝ)
51	50	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑦 ∈ ℝ)
52		simp13l 1307	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
53		simp3l 1220	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑣 ≠ 𝐷)
54	11	sselda 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ)
55	45, 46, 54	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑣 ∈ ℂ)
56	45, 14	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝐷 ∈ ℂ)
57	55, 56	subcld 11696	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (𝑣 − 𝐷) ∈ ℂ)
58	57	abscld 15635	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘(𝑣 − 𝐷)) ∈ ℝ)
59	38	rpred 13191	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ)
60	59	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ)
61		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ ℝ⁺)
62	61	rpred 13191	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ ℝ)
63	62	3ad2ant2 1152	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → 𝑎 ∈ ℝ)
64	63	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑎 ∈ ℝ)
65		simp3r 1221	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))
66		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ ℝ⁺)
67	66	rpred 13191	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ ℝ)
68		min1 13346	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
69	62, 67, 68	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
70	69	3ad2ant2 1152	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
71	70	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
72	58, 60, 64, 65, 71	ltletrd 11497	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘(𝑣 − 𝐷)) < 𝑎)
73	53, 72	jca 521	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎))
74		rsp 3254	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) → (𝑣 ∈ 𝐴 → ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))))
75	52, 46, 73, 74	syl3c 67	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))
76	47, 51, 75	jca31 524	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
77		simp13r 1308	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
78	67	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → 𝑏 ∈ ℝ)
79	78	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → 𝑏 ∈ ℝ)
80		min2 13347	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
81	62, 67, 80	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
82	81	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
83	82	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
84	58, 60, 79, 65, 83	ltletrd 11497	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘(𝑣 − 𝐷)) < 𝑏)
85	53, 84	jca 521	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏))
86		rsp 3254	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝑣 ∈ 𝐴 → ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))))
87	77, 46, 85, 86	syl3c 67	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))
88	8, 24	addcld 11355	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ)
89		addlimc.h	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))
90	88, 89	fmptd 7138	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝐴⟶ℂ)
91	90	ffvelcdmda 7108	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) ∈ ℂ)
92	91	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐻‘𝑣) ∈ ℂ)
93		simp-4l 795	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝜑)
94	93, 7	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐸 + 𝐼) ∈ ℂ)
95	92, 94	subcld 11696	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((𝐻‘𝑣) − (𝐸 + 𝐼)) ∈ ℂ)
96	95	abscld 15635	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) ∈ ℝ)
97	10	ffvelcdmda 7108	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ ℂ)
98	97	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐹‘𝑣) ∈ ℂ)
99	93, 3	syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝐸 ∈ ℂ)
100	98, 99	subcld 11696	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((𝐹‘𝑣) − 𝐸) ∈ ℂ)
101	100	abscld 15635	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐹‘𝑣) − 𝐸)) ∈ ℝ)
102	26	ffvelcdmda 7108	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐺‘𝑣) ∈ ℂ)
103	102	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐺‘𝑣) ∈ ℂ)
104	93, 6	syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝐼 ∈ ℂ)
105	103, 104	subcld 11696	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((𝐺‘𝑣) − 𝐼) ∈ ℂ)
106	105	abscld 15635	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐺‘𝑣) − 𝐼)) ∈ ℝ)
107	101, 106	readdcld 11365	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((abs‘((𝐹‘𝑣) − 𝐸)) + (abs‘((𝐺‘𝑣) − 𝐼))) ∈ ℝ)
108		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝑦 ∈ ℝ)
109		nfv 1949	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑣 ∈ 𝐴)
110		nfmpt1 5207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))
111	89, 110	nfcxfr 2924	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐻
112		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝑣
113	111, 112	nffv 6918	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐻‘𝑣)
114		nfmpt1 5207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
115	9, 114	nfcxfr 2924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥𝐹
116	115, 112	nffv 6918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝐹‘𝑣)
117		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 +
118		nfmpt1 5207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
119	25, 118	nfcxfr 2924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥𝐺
120	119, 112	nffv 6918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝐺‘𝑣)
121	116, 117, 120	nfov 7473	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥((𝐹‘𝑣) + (𝐺‘𝑣))
122	113, 121	nfeq 2939	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣))
123	109, 122	nfim 1931	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
124		eleq1w 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
125	124	anbi2d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑣 ∈ 𝐴)))
126		fveq2 6908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝐻‘𝑥) = (𝐻‘𝑣))
127		fveq2 6908	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
128		fveq2 6908	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑣 → (𝐺‘𝑥) = (𝐺‘𝑣))
129	127, 128	oveq12d 7461	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥) + (𝐺‘𝑥)) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
130	126, 129	eqeq12d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ((𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)) ↔ (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣))))
131	125, 130	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥))) ↔ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))))
132		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
133	89	fvmpt2 7028	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐵 + 𝐶) ∈ ℂ) → (𝐻‘𝑥) = (𝐵 + 𝐶))
134	132, 88, 133	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐵 + 𝐶))
135	9	fvmpt2 7028	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (𝐹‘𝑥) = 𝐵)
136	132, 8, 135	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
137	136	eqcomd 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (𝐹‘𝑥))
138	25	fvmpt2 7028	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → (𝐺‘𝑥) = 𝐶)
139	132, 24, 138	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
140	139	eqcomd 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐺‘𝑥))
141	137, 140	oveq12d 7461	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
142	134, 141	eqtrd 2798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
143	123, 131, 142	chvarfv 2280	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
144	143	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
145	144	oveq1d 7458	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((𝐻‘𝑣) − (𝐸 + 𝐼)) = (((𝐹‘𝑣) + (𝐺‘𝑣)) − (𝐸 + 𝐼)))
146	98, 103, 99, 104	addsub4d 11743	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (((𝐹‘𝑣) + (𝐺‘𝑣)) − (𝐸 + 𝐼)) = (((𝐹‘𝑣) − 𝐸) + ((𝐺‘𝑣) − 𝐼)))
147	145, 146	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((𝐻‘𝑣) − (𝐸 + 𝐼)) = (((𝐹‘𝑣) − 𝐸) + ((𝐺‘𝑣) − 𝐼)))
148	147	fveq2d 6912	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) = (abs‘(((𝐹‘𝑣) − 𝐸) + ((𝐺‘𝑣) − 𝐼))))
149	100, 105	abstrid 15655	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘(((𝐹‘𝑣) − 𝐸) + ((𝐺‘𝑣) − 𝐼))) ≤ ((abs‘((𝐹‘𝑣) − 𝐸)) + (abs‘((𝐺‘𝑣) − 𝐼))))
150	148, 149	eqbrtrd 5130	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) ≤ ((abs‘((𝐹‘𝑣) − 𝐸)) + (abs‘((𝐺‘𝑣) − 𝐼))))
151		simplr 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))
152		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))
153	101, 106, 108, 151, 152	lt2halvesd 12621	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → ((abs‘((𝐹‘𝑣) − 𝐸)) + (abs‘((𝐺‘𝑣) − 𝐼))) < 𝑦)
154	96, 107, 108, 150, 153	lelttrd 11495	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦)
155	76, 87, 154	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) ∧ 𝑣 ∈ 𝐴 ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏))) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦)
156	155	3exp 1137	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → (𝑣 ∈ 𝐴 → ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦)))
157	44, 156	ralrimi 3264	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
158		brimralrspcev 5169	. . . . . . 7 ⊢ ((if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺ ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < if(𝑎 ≤ 𝑏, 𝑎, 𝑏)) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦)) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
159	38, 157, 158	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) ∧ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
160	159	3exp 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → ((∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))))
161	160	rexlimdvv 3222	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (∃𝑎 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦)))
162	36, 161	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
163	162	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
164	90, 11, 14	ellimc3 26229	. 2 ⊢ (𝜑 → ((𝐸 + 𝐼) ∈ (𝐻 lim_ℂ 𝐷) ↔ ((𝐸 + 𝐼) ∈ ℂ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))))
165	7, 163, 164	mpbir2and 726	1 ⊢ (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 lim_ℂ 𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 ifcif 4485 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 + caddc 11230 < clt 11370 ≤ cle 11371 − cmin 11568 / cdiv 12000 2c2 12424 ℝ⁺crp 13147 abscabs 15430 lim_ℂ climc 26212

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-map 8877 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fi 9430 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-fz 13667 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-struct 17354 df-slot 17389 df-ndx 17401 df-base 17417 df-plusg 17470 df-mulr 17471 df-starv 17472 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-rest 17622 df-topn 17623 df-topgen 17643 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cnp 23576 df-xms 24669 df-ms 24670 df-limc 26216

This theorem is used by: sublimc 46867 reclimc 46868 fourierdlem53 47374 fourierdlem60 47381 fourierdlem61 47382

W3C validator