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Theorem climcn2 15541

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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑀 ∈ ℤ)
5		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
6		eqidd 2731	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
7		climcn2.5a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
8	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐺 ⇝ 𝐴)
9	2, 4, 5, 6, 8	climi2 15459	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦)
10		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ⁺)
11		eqidd 2731	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
12		climcn2.5b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⇝ 𝐵)
13	12	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐻 ⇝ 𝐵)
14	2, 4, 10, 11, 13	climi2 15459	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)
15	2	rexanuz2 15300	. . . . . . . 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 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
17	2	uztrn2 12845	. . . . . . . . . . . . 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		fvoveq1 7434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘(𝑢 − 𝐴)) = (abs‘((𝐺‘𝑘) − 𝐴)))
21	20	breq1d 5157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘(𝑢 − 𝐴)) < 𝑦 ↔ (abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦))
22	21	anbi1d 628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) ↔ ((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧)))
23		oveq1 7418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢𝐹𝑣) = ((𝐺‘𝑘)𝐹𝑣))
24	23	fvoveq1d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))))
25	24	breq1d 5157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
26	22, 25	imbi12d 343	. . . . . . . . . . . . . . . . 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)))
27		fvoveq1 7434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(𝑣 − 𝐵)) = (abs‘((𝐻‘𝑘) − 𝐵)))
28	27	breq1d 5157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(𝑣 − 𝐵)) < 𝑧 ↔ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
29	28	anbi2d 627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐻‘𝑘) → (((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) ↔ ((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)))
30		oveq2 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → ((𝐺‘𝑘)𝐹𝑣) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
31	30	fvoveq1d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))))
32	31	breq1d 5157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
33	29, 32	imbi12d 343	. . . . . . . . . . . . . . . . 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)))
34	26, 33	rspc2v 3621	. . . . . . . . . . . . . . . 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)))
35	18, 19, 34	syl2anc 582	. . . . . . . . . . . . . . 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)))
36	35	imp 405	. . . . . . . . . . . . . 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))
37	36	an32s 648	. . . . . . . . . . . . 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))
38	17, 37	sylan2 591	. . . . . . . . . . . 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))
39	38	anassrs 466	. . . . . . . . . . 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))
40	39	ralimdva 3165	. . . . . . . . . 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))
41	40	reximdva 3166	. . . . . . . . 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))
42	41	ex 411	. . . . . . . 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)))
43	42	adantr 479	. . . . . . 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)))
44	16, 43	mpid 44	. . . . . 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))
45	44	rexlimdvva 3209	. . . . 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))
46	45	adantr 479	. . . 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))
47	1, 46	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)
48	47	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥)
49		climcn2.6	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑊)
50		climcn2.9	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
51		climcn2.4	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)
52		climcn2.3a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
53		climcn2.3b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
54	51, 52, 53	caovcld 7602	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ ℂ)
55	18, 19	jca 510	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷))
56	51	ralrimivva 3198	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
57	56	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
58	23	eleq1d 2816	. . . . 5 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹𝑣) ∈ ℂ))
59	30	eleq1d 2816	. . . . 5 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
60	58, 59	rspc2v 3621	. . . 4 ⊢ (((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷) → (∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ → ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
61	55, 57, 60	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ)
62	2, 3, 49, 50, 54, 61	clim2c 15453	. 2 ⊢ (𝜑 → (𝐾 ⇝ (𝐴𝐹𝐵) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
63	48, 62	mpbird 256	1 ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 < clt 11252 − cmin 11448 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 abscabs 15185 ⇝ cli 15432

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 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-clim 15436

This theorem is referenced by: climadd 15580 climmul 15581 climsub 15582

W3C validator