addlimc - Mathbox for Glauco Siliprandi

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Theorem addlimc 45663

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 25910	. . . 4 ⊢ (𝐹 lim_ℂ 𝐷) ⊆ ℂ
2		addlimc.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (𝐹 lim_ℂ 𝐷))
3	1, 2	sselid 3981	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ)
4		limccl 25910	. . . 4 ⊢ (𝐺 lim_ℂ 𝐷) ⊆ ℂ
5		addlimc.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐺 lim_ℂ 𝐷))
6	4, 5	sselid 3981	. . 3 ⊢ (𝜑 → 𝐼 ∈ ℂ)
7	3, 6	addcld 11280	. 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 7134	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
11	9, 8, 2	limcmptdm 45650	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
12		limcrcl 25909	. . . . . . . . . . 11 ⊢ (𝐸 ∈ (𝐹 lim_ℂ 𝐷) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
13	2, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
14	13	simp3d 1145	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
15	10, 11, 14	ellimc3 25914	. . . . . . . 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 232	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧)))
17	16	simprd 495	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ℝ⁺ ∃𝑎 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧))
18		rphalfcl 13062	. . . . . 6 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 / 2) ∈ ℝ⁺)
19		breq2 5147	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 / 2) → ((abs‘((𝐹‘𝑣) − 𝐸)) < 𝑧 ↔ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)))
20	19	imbi2d 340	. . . . . . . 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 3223	. . . . . . 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 3621	. . . . . 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 596	. . . . 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 7134	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐴⟶ℂ)
27	26, 11, 14	ellimc3 25914	. . . . . . . 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 232	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ ℂ ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧)))
29	28	simprd 495	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ℝ⁺ ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧))
30		breq2 5147	. . . . . . . . 9 ⊢ (𝑧 = (𝑦 / 2) → ((abs‘((𝐺‘𝑣) − 𝐼)) < 𝑧 ↔ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
31	30	imbi2d 340	. . . . . . . 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 3223	. . . . . . 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 3621	. . . . . 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 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑏 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)))
35		reeanv 3229	. . . . 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 583	. . . 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 4571	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ∈ ℝ⁺)
38	37	3ad2ant2 1135	. . . . . . 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 1914	. . . . . . . . 9 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑦 ∈ ℝ⁺)
40		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑣(𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺)
41		nfra1 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑎) → (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2))
42		nfra1 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑏) → (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2))
43	41, 42	nfan 1899	. . . . . . . . 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 1901	. . . . . . . 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 1285	. . . . . . . . . . . 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 1138	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 13043	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
49	48	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
50	49	3ad2ant1 1134	. . . . . . . . . . . 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 1134	. . . . . . . . . . 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 1289	. . . . . . . . . . . 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 1202	. . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ)
55	45, 46, 54	syl2anc 584	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 11620	. . . . . . . . . . . . . . 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 15475	. . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ ℝ⁺)
62	61	rpred 13077	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑎 ∈ ℝ)
63	62	3ad2ant2 1135	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ ℝ⁺)
67	66	rpred 13077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → 𝑏 ∈ ℝ)
68		min1 13231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
69	62, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑎)
70	69	3ad2ant2 1135	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 11421	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 3247	. . . . . . . . . . . 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 66	. . . . . . . . . . 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 514	. . . . . . . . . 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 1290	. . . . . . . . . . 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 1135	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 13232	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
81	62, 67, 80	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ⁺ ∧ 𝑏 ∈ ℝ⁺) → if(𝑎 ≤ 𝑏, 𝑎, 𝑏) ≤ 𝑏)
82	81	3ad2ant2 1135	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 11421	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 3247	. . . . . . . . . . 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 66	. . . . . . . . . 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 11280	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ)
89		addlimc.h	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))
90	88, 89	fmptd 7134	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻:𝐴⟶ℂ)
91	90	ffvelcdmda 7104	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) ∈ ℂ)
92	91	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → (𝐻‘𝑣) ∈ ℂ)
93		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝜑)
94	93, 7	syl 17	. . . . . . . . . . . . 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 11620	. . . . . . . . . . . 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 15475	. . . . . . . . . . 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 7104	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ ℂ)
98	97	ad3antrrr 730	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝐸 ∈ ℂ)
100	98, 99	subcld 11620	. . . . . . . . . . . . 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 15475	. . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐺‘𝑣) ∈ ℂ)
103	102	ad3antrrr 730	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝐼 ∈ ℂ)
105	103, 104	subcld 11620	. . . . . . . . . . . . 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 15475	. . . . . . . . . . . 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 11290	. . . . . . . . . . 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 776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ (abs‘((𝐹‘𝑣) − 𝐸)) < (𝑦 / 2)) ∧ (abs‘((𝐺‘𝑣) − 𝐼)) < (𝑦 / 2)) → 𝑦 ∈ ℝ)
109		nfv 1914	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑣 ∈ 𝐴)
110		nfmpt1 5250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))
111	89, 110	nfcxfr 2903	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐻
112		nfcv 2905	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝑣
113	111, 112	nffv 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐻‘𝑣)
114		nfmpt1 5250	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
115	9, 114	nfcxfr 2903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥𝐹
116	115, 112	nffv 6916	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝐹‘𝑣)
117		nfcv 2905	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 +
118		nfmpt1 5250	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
119	25, 118	nfcxfr 2903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥𝐺
120	119, 112	nffv 6916	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝐺‘𝑣)
121	116, 117, 120	nfov 7461	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥((𝐹‘𝑣) + (𝐺‘𝑣))
122	113, 121	nfeq 2919	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣))
123	109, 122	nfim 1896	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
124		eleq1w 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
125	124	anbi2d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑣 ∈ 𝐴)))
126		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → (𝐻‘𝑥) = (𝐻‘𝑣))
127		fveq2 6906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
128		fveq2 6906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑣 → (𝐺‘𝑥) = (𝐺‘𝑣))
129	127, 128	oveq12d 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥) + (𝐺‘𝑥)) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
130	126, 129	eqeq12d 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → ((𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)) ↔ (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣))))
131	125, 130	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥))) ↔ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))))
132		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
133	89	fvmpt2 7027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐵 + 𝐶) ∈ ℂ) → (𝐻‘𝑥) = (𝐵 + 𝐶))
134	132, 88, 133	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐵 + 𝐶))
135	9	fvmpt2 7027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (𝐹‘𝑥) = 𝐵)
136	132, 8, 135	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
137	136	eqcomd 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (𝐹‘𝑥))
138	25	fvmpt2 7027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → (𝐺‘𝑥) = 𝐶)
139	132, 24, 138	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
140	139	eqcomd 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐺‘𝑥))
141	137, 140	oveq12d 7449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
142	134, 141	eqtrd 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
143	123, 131, 142	chvarfv 2240	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐻‘𝑣) = ((𝐹‘𝑣) + (𝐺‘𝑣)))
144	143	ad3antrrr 730	. . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . 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 11667	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . 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 15495	. . . . . . . . . . . 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 5165	. . . . . . . . . . 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 769	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 12514	. . . . . . . . . . 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 11419	. . . . . . . . . 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 584	. . . . . . . . 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 1120	. . . . . . . 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 3257	. . . . . . 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 5204	. . . . . . 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 584	. . . . . 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 1120	. . . . 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 3212	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐻‘𝑣) − (𝐸 + 𝐼))) < 𝑦))
163	162	ralrimiva 3146	. 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 25914	. 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 713	1 ⊢ (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 lim_ℂ 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 ℝ⁺crp 13034 abscabs 15273 lim_ℂ climc 25897

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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 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 ax-pre-sup 11233

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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 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-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cnp 23236 df-xms 24330 df-ms 24331 df-limc 25901

This theorem is referenced by: sublimc 45667 reclimc 45668 fourierdlem53 46174 fourierdlem60 46181 fourierdlem61 46182

W3C validator