Theorem limccog 45744

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 25822 and limccnp 25820, 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 25804	. . 3 ⊢ (𝐺 limℂ 𝐵) ⊆ ℂ
2		limccog.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))
3	1, 2	sselid 3928	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		limcrcl 25803	. . . . . . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
10	6, 7, 8, 9	ellimc2 25806	. . . . . . . . 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 3225	. . . . . 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 25803	. . . . . . . . . . . . . . 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 25806	. . . . . . . . . . . 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 3225	. . . . . . . . 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 6203	. . . . . . . . . . 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 6024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37		limccog.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
38	36, 37	sstrid 3942	. . . . . . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41		imass2 6055	. . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 3969	. . . . . . . . . 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 3146	. . . . . . 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 3138	. . . . 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 3125	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56	21	ffund 6660	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
57		fdmrn 6687	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
58	56, 57	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
59	37	difss2d 4088	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ dom 𝐺)
60	58, 59	fssd 6673	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶dom 𝐺)
61		fco 6680	. . . 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 25806	. 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 2113  ∀wral 3048  ∃wrex 3057   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  {csn 4575  dom cdm 5619  ran crn 5620   “ cima 5622   ∘ ccom 5623  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  ℂcc 11011  TopOpenctopn 17327  ℂ_fldccnfld 21293   lim_ℂ climc 25791

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fi 9302  df-sup 9333  df-inf 9334  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-q 12849  df-rp 12893  df-xneg 13013  df-xadd 13014  df-xmul 13015  df-fz 13410  df-seq 13911  df-exp 13971  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-struct 17060  df-slot 17095  df-ndx 17107  df-base 17123  df-plusg 17176  df-mulr 17177  df-starv 17178  df-tset 17182  df-ple 17183  df-ds 17185  df-unif 17186  df-rest 17328  df-topn 17329  df-topgen 17349  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-cnfld 21294  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cnp 23144  df-xms 24236  df-ms 24237  df-limc 25795

This theorem is referenced by:  dirkercncflem2  46226  fourierdlem53  46281  fourierdlem93  46321  fourierdlem111  46339