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Theorem limccoap 14150

Description: Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)

**Hypotheses**
Ref	Expression
limccoap.r	⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})
limccoap.s	⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}) → 𝑆 ∈ ℂ)
limccoap.c	⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) lim_ℂ 𝑋))
limccoap.d	⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) lim_ℂ 𝐶))
limcco.1	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)

**Assertion**
Ref	Expression
limccoap	⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) lim_ℂ 𝑋))

Distinct variable groups: 𝑤,𝐴,𝑥 𝑤,𝐵,𝑥,𝑦 𝑤,𝐶,𝑥,𝑦 𝑥,𝐷,𝑦 𝑤,𝑅,𝑦 𝑥,𝑆 𝑦,𝑇 𝑤,𝑋,𝑥 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑤) 𝐴(𝑦) 𝐷(𝑤) 𝑅(𝑥) 𝑆(𝑦,𝑤) 𝑇(𝑥,𝑤) 𝑋(𝑦)

Proof of Theorem limccoap Dummy variables 𝑑 𝑒 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccoap.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) lim_ℂ 𝐶))
2		apsscn 8604	. . . . . 6 ⊢ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ⊆ ℂ
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ⊆ ℂ)
4		limcrcl 14130	. . . . . . 7 ⊢ (𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) lim_ℂ 𝐶) → ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆):dom (𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
5	1, 4	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆):dom (𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
6	5	simp3d 1011	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7		limccoap.s	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}) → 𝑆 ∈ ℂ)
8	3, 6, 7	limcmpted 14135	. . . 4 ⊢ (𝜑 → (𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) lim_ℂ 𝐶) ↔ (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒))))
9	1, 8	mpbid 147	. . 3 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)))
10	9	simpld 112	. 2 ⊢ (𝜑 → 𝐷 ∈ ℂ)
11	9	simprd 114	. . 3 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒))
12		breq2 4008	. . . . . . . . . 10 ⊢ (𝑣 = 𝑑 → ((abs‘(𝑅 − 𝐶)) < 𝑣 ↔ (abs‘(𝑅 − 𝐶)) < 𝑑))
13	12	imbi2d 230	. . . . . . . . 9 ⊢ (𝑣 = 𝑑 → (((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣) ↔ ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑)))
14	13	rexralbidv 2503	. . . . . . . 8 ⊢ (𝑣 = 𝑑 → (∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣) ↔ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑)))
15		limccoap.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) lim_ℂ 𝑋))
16		apsscn 8604	. . . . . . . . . . . . 13 ⊢ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ⊆ ℂ
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ⊆ ℂ)
18		limcrcl 14130	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) lim_ℂ 𝑋) → ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅):dom (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) ⊆ ℂ ∧ 𝑋 ∈ ℂ))
19	15, 18	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅):dom (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) ⊆ ℂ ∧ 𝑋 ∈ ℂ))
20	19	simp3d 1011	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℂ)
21		limccoap.r	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})
22	2, 21	sselid 3154	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ ℂ)
23	17, 20, 22	limcmpted 14135	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) lim_ℂ 𝑋) ↔ (𝐶 ∈ ℂ ∧ ∀𝑣 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣))))
24	15, 23	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑣 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣)))
25	24	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣))
26	25	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ∀𝑣 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑣))
27		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
28	14, 26, 27	rspcdva 2847	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑))
29	28	adantr 276	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) → ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑))
30		simp-5l 543	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝜑)
31	30, 21	sylancom 420	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})
32		breq1 4007	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑅 → (𝑤 # 𝐶 ↔ 𝑅 # 𝐶))
33	32	elrab 2894	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↔ (𝑅 ∈ 𝐵 ∧ 𝑅 # 𝐶))
34	31, 33	sylib 122	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → (𝑅 ∈ 𝐵 ∧ 𝑅 # 𝐶))
35	34	simprd 114	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 # 𝐶)
36		breq1 4007	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑅 → (𝑦 # 𝐶 ↔ 𝑅 # 𝐶))
37		fvoveq1 5898	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑅 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑅 − 𝐶)))
38	37	breq1d 4014	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑅 → ((abs‘(𝑦 − 𝐶)) < 𝑑 ↔ (abs‘(𝑅 − 𝐶)) < 𝑑))
39	36, 38	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑅 → ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) ↔ (𝑅 # 𝐶 ∧ (abs‘(𝑅 − 𝐶)) < 𝑑)))
40		limcco.1	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
41	40	fvoveq1d 5897	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑅 → (abs‘(𝑆 − 𝐷)) = (abs‘(𝑇 − 𝐷)))
42	41	breq1d 4014	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑅 → ((abs‘(𝑆 − 𝐷)) < 𝑒 ↔ (abs‘(𝑇 − 𝐷)) < 𝑒))
43	39, 42	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑅 → (((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒) ↔ ((𝑅 # 𝐶 ∧ (abs‘(𝑅 − 𝐶)) < 𝑑) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
44		simpllr 534	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒))
45	43, 44, 31	rspcdva 2847	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → ((𝑅 # 𝐶 ∧ (abs‘(𝑅 − 𝐶)) < 𝑑) → (abs‘(𝑇 − 𝐷)) < 𝑒))
46	35, 45	mpand 429	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → ((abs‘(𝑅 − 𝐶)) < 𝑑 → (abs‘(𝑇 − 𝐷)) < 𝑒))
47	46	imim2d 54	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → (((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑) → ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
48	47	ralimdva 2544	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ⁺) → (∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑) → ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
49	48	reximdva 2579	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) → (∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑅 − 𝐶)) < 𝑑) → ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
50	29, 49	mpd 13	. . . . 5 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒)) → ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒))
51	50	rexlimdva2 2597	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒) → ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
52	51	ralimdva 2544	. . 3 ⊢ (𝜑 → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦 − 𝐶)) < 𝑑) → (abs‘(𝑆 − 𝐷)) < 𝑒) → ∀𝑒 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒)))
53	11, 52	mpd 13	. 2 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒))
54	40	eleq1d 2246	. . . 4 ⊢ (𝑦 = 𝑅 → (𝑆 ∈ ℂ ↔ 𝑇 ∈ ℂ))
55	7	ralrimiva 2550	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}𝑆 ∈ ℂ)
56	55	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → ∀𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}𝑆 ∈ ℂ)
57	54, 56, 21	rspcdva 2847	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑇 ∈ ℂ)
58	17, 20, 57	limcmpted 14135	. 2 ⊢ (𝜑 → (𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) lim_ℂ 𝑋) ↔ (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥 − 𝑋)) < 𝑢) → (abs‘(𝑇 − 𝐷)) < 𝑒))))
59	10, 53, 58	mpbir2and 944	1 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) lim_ℂ 𝑋))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 {crab 2459 ⊆ wss 3130 class class class wbr 4004 ↦ cmpt 4065 dom cdm 4627 ⟶wf 5213 ‘cfv 5217 (class class class)co 5875 ℂcc 7809 < clt 7992 − cmin 8128 # cap 8538 ℝ⁺crp 9653 abscabs 11006 lim_ℂ climc 14126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-icn 7906 ax-addcl 7907 ax-mulcl 7909

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fo 5223 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-pm 6651 df-ap 8539 df-limced 14128

This theorem is referenced by: dvcoapbr 14174

W3C validator