Theorem limccog 46725

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 26218 and limccnp 26216, 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 26200	. . 3 ⊢ (𝐺 limℂ 𝐵) ⊆ ℂ
2		limccog.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))
3	1, 2	sselid 3906	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		limcrcl 26199	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝐺 limℂ 𝐵) → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	2, 4	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
6	5	simp1d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℂ)
7	5	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐺 ⊆ ℂ)
8	5	simp3d 1162	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9		eqid 2760	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
10	6, 7, 8, 9	ellimc2 26202	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
12	11	simprd 501	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
13	12	r19.21bi 3254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
14	13	imp 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15		simp1ll 1255	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16		simp2 1155	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17		simp3l 1220	. . . . . . . 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 26199	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝐹 limℂ 𝐴) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
20	18, 19	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
21	20	simp1d 1160	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
22	20	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
23	20	simp3d 1162	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	21, 22, 23, 9	ellimc2 26202	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
26	25	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
27	26	r19.21bi 3254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
28	27	imp 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵 ∈ 𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
29	15, 16, 17, 28	syl21anc 851	. . . . . . 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 6252	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
31	15	ad2antrr 739	. . . . . . . . . . . 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 1248	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 490	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36		imassrn 6197	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37		limccog.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
38	36, 37	sstrid 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
39	38	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
40	35, 39	ssind 4137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41		imass2 6054	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
43	42	adantlr 728	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
45	43, 44	sstrd 3918	. . . . . . . . . . . 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 851	. . . . . . . . . . 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 3946	. . . . . . . . . 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 418	. . . . . . . . 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 624	. . . . . . . 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 3175	. . . . . . 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 16	. . . . . 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 3167	. . . . 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 16	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
54	53	ex 418	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
55	54	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56	21	ffund 6714	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
57		fdmrn 6741	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
58	56, 57	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
59	37	difss2d 4075	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ dom 𝐺)
60	58, 59	fssd 6727	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶dom 𝐺)
61		fco 6734	. . . 4 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺 ∘ 𝐹):dom 𝐹⟶ℂ)
62	6, 60, 61	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):dom 𝐹⟶ℂ)
63	62, 22, 23, 9	ellimc2 26202	. 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 726	1 ⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  {csn 4584  dom cdm 5650  ran crn 5651   “ cima 5653   ∘ ccom 5654  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℂcc 11201  TopOpenctopn 17597  ℂ_fldccnfld 21683   lim_ℂ climc 26187

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-pm 8853  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-q 13079  df-rp 13124  df-xneg 13244  df-xadd 13245  df-xmul 13246  df-fz 13643  df-seq 14148  df-exp 14208  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-struct 17330  df-slot 17365  df-ndx 17377  df-base 17393  df-plusg 17446  df-mulr 17447  df-starv 17448  df-tset 17452  df-ple 17453  df-ds 17455  df-unif 17456  df-rest 17598  df-topn 17599  df-topgen 17619  df-psmet 21675  df-xmet 21676  df-met 21677  df-bl 21678  df-mopn 21679  df-cnfld 21684  df-top 23217  df-topon 23234  df-topsp 23256  df-bases 23269  df-cnp 23551  df-xms 24644  df-ms 24645  df-limc 26191

This theorem is used by:  dirkercncflem2  47207  fourierdlem53  47262  fourierdlem93  47302  fourierdlem111  47320