Theorem limccog 45635

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 25928 and limccnp 25926, 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 25910	. . 3 ⊢ (𝐺 limℂ 𝐵) ⊆ ℂ
2		limccog.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))
3	1, 2	sselid 3981	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		limcrcl 25909	. . . . . . . . . . . 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 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℂ)
7	5	simp2d 1144	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐺 ⊆ ℂ)
8	5	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
10	6, 7, 8, 9	ellimc2 25912	. . . . . . . . 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 3251	. . . . . 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 1138	. . . . . . . 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 25909	. . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
22	20	simp2d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
23	20	simp3d 1145	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	21, 22, 23, 9	ellimc2 25912	. . . . . . . . . . . 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 3251	. . . . . . . . 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 838	. . . . . . 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 6271	. . . . . . . . . . 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 6089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37		limccog.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
38	36, 37	sstrid 3995	. . . . . . . . . . . . . . . . 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 4241	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41		imass2 6120	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
45	43, 44	sstrd 3994	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 4022	. . . . . . . . . 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 3168	. . . . . . 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 3159	. . . . 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 3146	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56	21	ffund 6740	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
57		fdmrn 6767	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
58	56, 57	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
59	37	difss2d 4139	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ dom 𝐺)
60	58, 59	fssd 6753	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶dom 𝐺)
61		fco 6760	. . . 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 25912	. 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 1087   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  {csn 4626  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  TopOpenctopn 17466  ℂ_fldccnfld 21364   lim_ℂ climc 25897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-rest 17467  df-topn 17468  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cnp 23236  df-xms 24330  df-ms 24331  df-limc 25901

This theorem is referenced by:  dirkercncflem2  46119  fourierdlem53  46174  fourierdlem93  46214  fourierdlem111  46232