climcn2 - Intuitionistic Logic Explorer

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Theorem climcn2 11071

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 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑀 ∈ ℤ)
5		simprl 520	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
6		eqidd 2138	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
7		climcn2.5a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
8	7	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐺 ⇝ 𝐴)
9	2, 4, 5, 6, 8	climi2 11050	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦)
10		simprr 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ⁺)
11		eqidd 2138	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
12		climcn2.5b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⇝ 𝐵)
13	12	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐻 ⇝ 𝐵)
14	2, 4, 10, 11, 13	climi2 11050	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧)
15	2	rexanuz2 10756	. . . . . . . 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 413	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
17	2	uztrn2 9336	. . . . . . . . . . . . 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 5774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢 − 𝐴) = ((𝐺‘𝑘) − 𝐴))
21	20	fveq2d 5418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘(𝑢 − 𝐴)) = (abs‘((𝐺‘𝑘) − 𝐴)))
22	21	breq1d 3934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘(𝑢 − 𝐴)) < 𝑦 ↔ (abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦))
23	22	anbi1d 460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) ↔ ((abs‘((𝐺‘𝑘) − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧)))
24		oveq1 5774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝐺‘𝑘) → (𝑢𝐹𝑣) = ((𝐺‘𝑘)𝐹𝑣))
25	24	oveq1d 5782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)))
26	25	fveq2d 5418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = (𝐺‘𝑘) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))))
27	26	breq1d 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = (𝐺‘𝑘) → ((abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))
28	23, 27	imbi12d 233	. . . . . . . . . . . . . . . . 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 5774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → (𝑣 − 𝐵) = ((𝐻‘𝑘) − 𝐵))
30	29	fveq2d 5418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(𝑣 − 𝐵)) = (abs‘((𝐻‘𝑘) − 𝐵)))
31	30	breq1d 3934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(𝑣 − 𝐵)) < 𝑧 ↔ (abs‘((𝐻‘𝑘) − 𝐵)) < 𝑧))
32	31	anbi2d 459	. . . . . . . . . . . . . . . . . 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 5775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝐻‘𝑘) → ((𝐺‘𝑘)𝐹𝑣) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))
34	33	oveq1d 5782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵)) = (((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵)))
35	34	fveq2d 5418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (𝐻‘𝑘) → (abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) = (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))))
36	35	breq1d 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐻‘𝑘) → ((abs‘(((𝐺‘𝑘)𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥 ↔ (abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
37	32, 36	imbi12d 233	. . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . 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 557	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 397	. . . . . . . . . . 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 2497	. . . . . . . . . 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 2532	. . . . . . . . 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 114	. . . . . . . 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 274	. . . . . . 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 2555	. . . . 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 274	. . . 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 2503	. 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 5917	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ ℂ)
59	18, 19	jca 304	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) ∈ 𝐶 ∧ (𝐻‘𝑘) ∈ 𝐷))
60	55	ralrimivva 2512	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
61	60	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (𝑢𝐹𝑣) ∈ ℂ)
62	24	eleq1d 2206	. . . . 5 ⊢ (𝑢 = (𝐺‘𝑘) → ((𝑢𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹𝑣) ∈ ℂ))
63	33	eleq1d 2206	. . . . 5 ⊢ (𝑣 = (𝐻‘𝑘) → (((𝐺‘𝑘)𝐹𝑣) ∈ ℂ ↔ ((𝐺‘𝑘)𝐹(𝐻‘𝑘)) ∈ ℂ))
64	62, 63	rspc2v 2797	. . . 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 11046	. 2 ⊢ (𝜑 → (𝐾 ⇝ (𝐴𝐹𝐵) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐺‘𝑘)𝐹(𝐻‘𝑘)) − (𝐴𝐹𝐵))) < 𝑥))
67	52, 66	mpbird 166	1 ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2414 ∃wrex 2415 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 < clt 7793 − cmin 7926 ℤcz 9047 ℤ_≥cuz 9319 ℝ⁺crp 9434 abscabs 10762 ⇝ cli 11040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-clim 11041

This theorem is referenced by: climadd 11088 climmul 11089 climsub 11090

W3C validator