Theorem limccog 45659

Description: Limit of the composition of two functions. If the limit of 𝐹 at 𝐴 is 𝐵 and the limit of 𝐺 at 𝐵 is 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐴 is 𝐶. With respect to limcco 25819 and limccnp 25817, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
limccog.1	⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
limccog.2	⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐴))
limccog.3	⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))

Assertion

Ref	Expression
limccog	⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴))

Proof of Theorem limccog

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 25801	. . 3 ⊢ (𝐺 limℂ 𝐵) ⊆ ℂ
2		limccog.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))
3	1, 2	sselid 3932	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		limcrcl 25800	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝐺 limℂ 𝐵) → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	2, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
6	5	simp1d 1142	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℂ)
7	5	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐺 ⊆ ℂ)
8	5	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9		eqid 2731	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
10	6, 7, 8, 9	ellimc2 25803	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ (𝐺 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))))
11	2, 10	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
12	11	simprd 495	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
13	12	r19.21bi 3224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
14	13	imp 406	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15		simp1ll 1237	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16		simp2 1137	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17		simp3l 1202	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝐵 ∈ 𝑣)
18		limccog.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐴))
19		limcrcl 25800	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝐹 limℂ 𝐴) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
20	18, 19	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
21	20	simp1d 1142	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
22	20	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
23	20	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	21, 22, 23, 9	ellimc2 25803	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ (𝐹 limℂ 𝐴) ↔ (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))))
25	18, 24	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
26	25	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
27	26	r19.21bi 3224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
28	27	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵 ∈ 𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
29	15, 16, 17, 28	syl21anc 837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
30		imaco 6198	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
31	15	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → 𝜑)
32		simpl3r 1230	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
34		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
35		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36		imassrn 6020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37		limccog.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
38	36, 37	sstrid 3946	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
39	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
40	35, 39	ssind 4191	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41		imass2 6051	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
42	40, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
43	42	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
44		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
45	43, 44	sstrd 3945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ 𝑢)
46	31, 33, 34, 45	syl21anc 837	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ 𝑢)
47	30, 46	eqsstrid 3973	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)
48	47	ex 412	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → ((𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣 → ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
49	48	anim2d 612	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → ((𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
50	49	reximdva 3145	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → (∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
51	29, 50	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
52	51	rexlimdv3a 3137	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
53	14, 52	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
54	53	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
55	54	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56	21	ffund 6655	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
57		fdmrn 6682	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
58	56, 57	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
59	37	difss2d 4089	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ dom 𝐺)
60	58, 59	fssd 6668	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶dom 𝐺)
61		fco 6675	. . . 4 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺 ∘ 𝐹):dom 𝐹⟶ℂ)
62	6, 60, 61	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):dom 𝐹⟶ℂ)
63	62, 22, 23, 9	ellimc2 25803	. 2 ⊢ (𝜑 → (𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))))
64	3, 55, 63	mpbir2and 713	1 ⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  {csn 4576  dom cdm 5616  ran crn 5617   “ cima 5619   ∘ ccom 5620  Fun wfun 6475  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11001  TopOpenctopn 17322  ℂ_fldccnfld 21289   lim_ℂ climc 25788

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fi 9295  df-sup 9326  df-inf 9327  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-dec 12586  df-uz 12730  df-q 12844  df-rp 12888  df-xneg 13008  df-xadd 13009  df-xmul 13010  df-fz 13405  df-seq 13906  df-exp 13966  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-struct 17055  df-slot 17090  df-ndx 17102  df-base 17118  df-plusg 17171  df-mulr 17172  df-starv 17173  df-tset 17177  df-ple 17178  df-ds 17180  df-unif 17181  df-rest 17323  df-topn 17324  df-topgen 17344  df-psmet 21281  df-xmet 21282  df-met 21283  df-bl 21284  df-mopn 21285  df-cnfld 21290  df-top 22807  df-topon 22824  df-topsp 22846  df-bases 22859  df-cnp 23141  df-xms 24233  df-ms 24234  df-limc 25792

This theorem is referenced by:  dirkercncflem2  46141  fourierdlem53  46196  fourierdlem93  46236  fourierdlem111  46254