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Theorem cncfco 23517

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 23503	. . . 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 23503	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7		fco 6533	. . 3 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐶)
8	3, 6, 7	syl2anc 586	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐶)
9	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝐺 ∈ (𝐵–cn→𝐶))
10	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝐹:𝐴⟶𝐵)
11		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝑥 ∈ 𝐴)
12	10, 11	ffvelrnd 6854	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → (𝐹‘𝑥) ∈ 𝐵)
13		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
14		cncfi 23504	. . . . 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 1367	. . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝐹 ∈ (𝐴–cn→𝐵))
17		simplrl 775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝑥 ∈ 𝐴)
18		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) → 𝑢 ∈ ℝ⁺)
19		cncfi 23504	. . . . . . 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 1367	. . . . . 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 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵)
22		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
23	21, 22	ffvelrnd 6854	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵)
24		fvoveq1 7181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝐹‘𝑤) → (abs‘(𝑣 − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))))
25	24	breq1d 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝐹‘𝑤) → ((abs‘(𝑣 − (𝐹‘𝑥))) < 𝑢 ↔ (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑢))
26	25	imbrov2fvoveq 7183	. . . . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . . . 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 6762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑤 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑤) = (𝐺‘(𝐹‘𝑤)))
30	21, 22, 29	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((𝐺 ∘ 𝐹)‘𝑤) = (𝐺‘(𝐹‘𝑤)))
31	17	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
32		fvco3 6762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
33	21, 31, 32	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ⁺)) ∧ 𝑢 ∈ ℝ⁺) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑤 ∈ 𝐴)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
34	30, 33	oveq12d 7176	. . . . . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . . . . 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 5078	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 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 261	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 650	. . . . . . . . . . 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 470	. . . . . . . . 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 3179	. . . . . . . 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 3276	. . . . . . 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 415	. . . . . 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 3286	. . . 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 3193	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘(((𝐺 ∘ 𝐹)‘𝑤) − ((𝐺 ∘ 𝐹)‘𝑥))) < 𝑦))
50		cncfrss 23501	. . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
51	4, 50	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
52		cncfrss2 23502	. . . 4 ⊢ (𝐺 ∈ (𝐵–cn→𝐶) → 𝐶 ⊆ ℂ)
53	1, 52	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ⊆ ℂ)
54		elcncf2 23500	. . 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 586	. 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 711	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 class class class wbr 5068 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 < clt 10677 − cmin 10872 ℝ⁺crp 12392 abscabs 14595 –cn→ccncf 23486

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-abs 14597 df-cncf 23488

This theorem is referenced by: cncfmpt1f 23523 negfcncf 23529 divcncf 24050 cniccbdd 24064 cncombf 24261 cnmbf 24262 dvlip 24592 dvlipcn 24593 itgsubstlem 24647 sincn 25034 coscn 25035 logcn 25232 lgamgulmlem2 25609 ftalem3 25654 evthiccabs 41778 mulc1cncfg 41877 expcnfg 41879 cncfcompt 42173 cncficcgt0 42178 cncfcompt2 42189 dirkercncflem2 42396 dirkercncflem4 42398 fourierdlem18 42417 fourierdlem93 42491 fourierdlem101 42499 fourierdlem111 42509

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