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Theorem cncfco 24947

Description: The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
cncfco.4	⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
cncfco.5	⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→𝐶))

**Assertion**
Ref	Expression
cncfco	⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶))

Proof of Theorem cncfco Dummy variables 𝑤 𝑢 𝑥 𝑦 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfco.5	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→𝐶))
2		cncff 24933	. . . 4 ⊢ (𝐺 ∈ (𝐵–cn→𝐶) → 𝐺:𝐵⟶𝐶)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
4		cncfco.4	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))
5		cncff 24933	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7		fco 6761	. . 3 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐶)
8	3, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐶)
9	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝐺 ∈ (𝐵–cn→𝐶))
10	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝐹:𝐴⟶𝐵)
11		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ 𝐴)
12	10, 11	ffvelcdmd 7105	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → (𝐹‘𝑥) ∈ 𝐵)
13		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
14		cncfi 24934	. . . . 5 ⊢ ((𝐺 ∈ (𝐵–cn→𝐶) ∧ (𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦))
15	9, 12, 13, 14	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦))
16	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝐹 ∈ (𝐴–cn→𝐵))
17		simplrl 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝑥 ∈ 𝐴)
18		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝑢 ∈ ℝ⁺)
19		cncfi 24934	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑢 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢))
20	16, 17, 18, 19	syl3anc 1370	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢))
21	6	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵)
22		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
23	21, 22	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
24		fvoveq1 7454	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐹‘𝑤) → (abs‘(𝑣 − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))))
25	24	breq1d 5158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝐹‘𝑤) → ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 ↔ (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢))
26	25	imbrov2fvoveq 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝐹‘𝑤) → (((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) ↔ ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)))
27	26	rspcv 3618	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → (∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)))
28	23, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)))
29		fvco3 7008	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑤) = (𝐺‘(𝐹‘𝑤)))
30	21, 22, 29	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((𝐺 ∘ 𝐹)‘𝑤) = (𝐺‘(𝐹‘𝑤)))
31	17	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
32		fvco3 7008	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
33	21, 31, 32	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
34	30, 33	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥)) = ((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥))))
35	34	fveq2d 6911	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) = (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))))
36	35	breq1d 5158	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦 ↔ (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))) < 𝑦))
37	36	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦) ↔ ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘(𝐹‘𝑤)) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)))
38	28, 37	sylibrd 259	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
39	38	imp 406	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
40	39	an32s 652	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
41	40	imim2d 57	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢) → ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
42	41	anassrs 467	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ 𝐴) → (((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢) → ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
43	42	ralimdva 3165	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢) → ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
44	43	reximdva 3166	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦)) → (∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
45	44	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → (∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → (∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))))
46	20, 45	mpid 44	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → (∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
47	46	rexlimdva 3153	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → (∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ 𝐵 ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 → (abs‘((𝐺‘𝑣) − (𝐺‘(𝐹‘𝑥)))) < 𝑦) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦)))
48	15, 47	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
49	48	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
50		cncfrss 24931	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
51	4, 50	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
52		cncfrss2 24932	. . . 4 ⊢ (𝐺 ∈ (𝐵–cn→𝐶) → 𝐶 ⊆ ℂ)
53	1, 52	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ⊆ ℂ)
54		elcncf2 24930	. . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐶 ⊆ ℂ) → ((𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶) ↔ ((𝐺 ∘ 𝐹):𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))))
55	51, 53, 54	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶) ↔ ((𝐺 ∘ 𝐹):𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))))
56	8, 49, 55	mpbir2and 713	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 < clt 11293 − cmin 11490 ℝ⁺crp 13032 abscabs 15270 –cn→ccncf 24916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 df-abs 15272 df-cncf 24918

This theorem is referenced by: cncfcompt2 24948 cncfmpt1f 24954 negfcncf 24964 divcncf 25496 cniccbdd 25510 cncombf 25707 cnmbf 25708 dvlip 26047 dvlipcn 26048 itgsubstlem 26104 sincn 26503 coscn 26504 logcn 26704 lgamgulmlem2 27088 ftalem3 27133 evthiccabs 45449 mulc1cncfg 45545 expcnfg 45547 cncfcompt 45839 cncficcgt0 45844 dirkercncflem2 46060 dirkercncflem4 46062 fourierdlem18 46081 fourierdlem93 46155 fourierdlem101 46163 fourierdlem111 46173

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