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Theorem climcn2 11301

Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)

**Hypotheses**
Ref	Expression
climcn2.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
climcn2.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
climcn2.3a	⊢ (𝜑 → 𝐴 ∈ 𝐶)
climcn2.3b	⊢ (𝜑 → 𝐵 ∈ 𝐷)
climcn2.4	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)
climcn2.5a	⊢ (𝜑 → 𝐺 ⇝ 𝐴)
climcn2.5b	⊢ (𝜑 → 𝐻 ⇝ 𝐵)
climcn2.6	⊢ (𝜑 → 𝐾 ∈ 𝑊)
climcn2.7	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
climcn2.8a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐶)
climcn2.8b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) ∈ 𝐷)
climcn2.9	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))

**Assertion**
Ref	Expression
climcn2	⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))

Distinct variable groups: 𝑢,𝑘,𝑣,𝐶 𝐷,𝑘,𝑢,𝑣 𝑦,𝑘,𝑧,𝐻,𝑣 𝑥,𝑘,𝜑,𝑢,𝑦,𝑧,𝑣 𝐴,𝑘,𝑢,𝑣,𝑥,𝑦,𝑧 𝑘,𝐺,𝑢,𝑣,𝑦,𝑧 𝑘,𝐾,𝑥 𝑘,𝑍,𝑦,𝑧 𝐵,𝑘,𝑢,𝑣,𝑥,𝑦,𝑧 𝑘,𝐹,𝑢,𝑣,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐶(𝑥,𝑦,𝑧) 𝐷(𝑥,𝑦,𝑧) 𝐺(𝑥) 𝐻(𝑥,𝑢) 𝐾(𝑦,𝑧,𝑣,𝑢) 𝑀(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘) 𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑘) 𝑍(𝑥,𝑣,𝑢)

Proof of Theorem climcn2 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climcn2.7	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
2		climcn2.1	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3		climcn2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑀 ∈ ℤ)
5		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
6		eqidd 2178	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
7		climcn2.5a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
8	7	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐺 ⇝ 𝐴)
9	2, 4, 5, 6, 8	climi2 11280	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦)
10		simprr 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ⁺)
11		eqidd 2178	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
12		climcn2.5b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⇝ 𝐵)
13	12	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐻 ⇝ 𝐵)
14	2, 4, 10, 11, 13	climi2 11280	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)
15	2	rexanuz2 10984	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
16	9, 14, 15	sylanbrc 417	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
17	2	uztrn2 9534	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
18		climcn2.8a	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐶)
19		climcn2.8b	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) ∈ 𝐷)
20		oveq1 5876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢 − 𝐴) = ((𝐺‘𝑘) − 𝐴))
21	20	fveq2d 5515	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘(𝑢 − 𝐴)) = (abs‘((𝐺‘𝑘) − 𝐴)))
22	21	breq1d 4010	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘(𝑢 − 𝐴)) < 𝑦 ↔ (abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦))
23	22	anbi1d 465	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) ↔ ((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧)))
24		oveq1 5876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢𝐹𝑣) = ((𝐺‘𝑘)𝐹𝑣))
25	24	oveq1d 5884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)))
26	25	fveq2d 5515	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))))
27	26	breq1d 4010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
28	23, 27	imbi12d 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = (𝐺‘𝑘) → ((((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) ↔ (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)))
29		oveq1 5876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → (𝑣 − 𝐵) = ((𝐻‘𝑘) − 𝐵))
30	29	fveq2d 5515	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(𝑣 − 𝐵)) = (abs‘((𝐻‘𝑘) − 𝐵)))
31	30	breq1d 4010	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(𝑣 − 𝐵)) < 𝑧 ↔ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
32	31	anbi2d 464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐻‘𝑘) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) ↔ ((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)))
33		oveq2 5877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → ((𝐺‘𝑘)𝐹𝑣) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
34	33	oveq1d 5884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵)))
35	34	fveq2d 5515	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))))
36	35	breq1d 4010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
37	32, 36	imbi12d 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝐻‘𝑘) → ((((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) ↔ (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)))
38	28, 37	rspc2v 2854	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)))
39	18, 19, 38	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)))
40	39	imp 124	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
41	40	an32s 568	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) ∧ 𝑘 ∈ 𝑍) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
42	17, 41	sylan2 286	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
43	42	anassrs 400	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
44	43	ralimdva 2544	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
45	44	reximdva 2579	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
46	45	ex 115	. . . . . . . 8 ⊢ (𝜑 → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)))
47	46	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)))
48	16, 47	mpid 42	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
49	48	rexlimdvva 2602	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
50	49	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
51	1, 50	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)
52	51	ralrimiva 2550	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)
53		climcn2.6	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑊)
54		climcn2.9	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
55		climcn2.4	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)
56		climcn2.3a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
57		climcn2.3b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
58	55, 56, 57	caovcld 6022	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ ℂ)
59	18, 19	jca 306	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷))
60	55	ralrimivva 2559	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
61	60	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
62	24	eleq1d 2246	. . . . 5 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹𝑣) ∈ ℂ))
63	33	eleq1d 2246	. . . . 5 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
64	62, 63	rspc2v 2854	. . . 4 ⊢ (((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ → ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
65	59, 61, 64	sylc 62	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ)
66	2, 3, 53, 54, 58, 65	clim2c 11276	. 2 ⊢ (𝜑 → (𝐾 ⇝ (𝐴𝐹𝐵) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
67	52, 66	mpbird 167	1 ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 < clt 7982 − cmin 8118 ℤcz 9242 ℤ_≥cuz 9517 ℝ⁺crp 9640 abscabs 10990 ⇝ cli 11270

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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-clim 11271

This theorem is referenced by: climadd 11318 climmul 11319 climsub 11320

W3C validator