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Theorem cncfco 24868

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 24854	. . . 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 24854	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7		fco 6694	. . 3 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐶)
8	3, 6, 7	syl2anc 585	. 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 7039	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → (𝐹‘𝑥) ∈ 𝐵)
13		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
14		cncfi 24855	. . . . 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 1374	. . . 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 727	. . . . . . 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 24855	. . . . . . 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 1374	. . . . . 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 731	. . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
24		fvoveq1 7391	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐹‘𝑤) → (abs‘(𝑣 − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))))
25	24	breq1d 5110	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝐹‘𝑤) → ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 ↔ (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢))
26	25	imbrov2fvoveq 7393	. . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . 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 6941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑤) = (𝐺‘(𝐹‘𝑤)))
30	21, 22, 29	syl2anc 585	. . . . . . . . . . . . . . . . . 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 6941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
33	21, 31, 32	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
34	30, 33	oveq12d 7386	. . . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . 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 653	. . . . . . . . . . 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 3150	. . . . . . . 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 3151	. . . . . . 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 3139	. . . 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 3181	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
50		cncfrss 24852	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
51	4, 50	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
52		cncfrss2 24853	. . . 4 ⊢ (𝐺 ∈ (𝐵–cn→𝐶) → 𝐶 ⊆ ℂ)
53	1, 52	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ⊆ ℂ)
54		elcncf2 24851	. . 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 585	. 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 714	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ⊆ wss 3903 class class class wbr 5100 ∘ ccom 5636 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 < clt 11178 − cmin 11376 ℝ⁺crp 12917 abscabs 15169 –cn→ccncf 24837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-cj 15034 df-re 15035 df-im 15036 df-abs 15171 df-cncf 24839

This theorem is referenced by: cncfcompt2 24869 cncfmpt1f 24875 negfcncf 24885 divcncf 25416 cniccbdd 25430 cncombf 25627 cnmbf 25628 dvlip 25966 dvlipcn 25967 itgsubstlem 26023 sincn 26422 coscn 26423 logcn 26624 lgamgulmlem2 27008 ftalem3 27053 evthiccabs 45850 mulc1cncfg 45943 expcnfg 45945 cncfcompt 46235 cncficcgt0 46240 dirkercncflem2 46456 dirkercncflem4 46458 fourierdlem18 46477 fourierdlem93 46551 fourierdlem101 46559 fourierdlem111 46569

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