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Theorem climcn2 11474

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 2197	. . . . . . . . 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 11453	. . . . . . . 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 2197	. . . . . . . . 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 11453	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)
15	2	rexanuz2 11156	. . . . . . . 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 9619	. . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢 − 𝐴) = ((𝐺‘𝑘) − 𝐴))
21	20	fveq2d 5562	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘(𝑢 − 𝐴)) = (abs‘((𝐺‘𝑘) − 𝐴)))
22	21	breq1d 4043	. . . . . . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢𝐹𝑣) = ((𝐺‘𝑘)𝐹𝑣))
25	24	oveq1d 5937	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)))
26	25	fveq2d 5562	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))))
27	26	breq1d 4043	. . . . . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → (𝑣 − 𝐵) = ((𝐻‘𝑘) − 𝐵))
30	29	fveq2d 5562	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(𝑣 − 𝐵)) = (abs‘((𝐻‘𝑘) − 𝐵)))
31	30	breq1d 4043	. . . . . . . . . . . . . . . . . . 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 5930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → ((𝐺‘𝑘)𝐹𝑣) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
34	33	oveq1d 5937	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵)))
35	34	fveq2d 5562	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))))
36	35	breq1d 4043	. . . . . . . . . . . . . . . . . 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 2881	. . . . . . . . . . . . . . . 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 2564	. . . . . . . . . 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 2599	. . . . . . . . 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 2622	. . . . 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 2570	. 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 6077	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ ℂ)
59	18, 19	jca 306	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷))
60	55	ralrimivva 2579	. . . . 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 2265	. . . . 5 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹𝑣) ∈ ℂ))
63	33	eleq1d 2265	. . . . 5 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
64	62, 63	rspc2v 2881	. . . 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 11449	. 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 1364 ∈ wcel 2167 ∀wral 2475 ∃wrex 2476 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 < clt 8061 − cmin 8197 ℤcz 9326 ℤ_≥cuz 9601 ℝ⁺crp 9728 abscabs 11162 ⇝ cli 11443

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-addcom 7979 ax-addass 7981 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-0id 7987 ax-rnegex 7988 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-inn 8991 df-n0 9250 df-z 9327 df-uz 9602 df-clim 11444

This theorem is referenced by: climadd 11491 climmul 11492 climsub 11493

W3C validator