Theorem limccog 41443

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 24174 and limccnp 24172, 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 24156	. . 3 ⊢ (𝐺 limℂ 𝐵) ⊆ ℂ
2		limccog.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))
3	1, 2	sseldi 3887	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		limcrcl 24155	. . . . . . . . . . . 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 1135	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:dom 𝐺⟶ℂ)
7	5	simp2d 1136	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐺 ⊆ ℂ)
8	5	simp3d 1137	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9		eqid 2795	. . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
10	6, 7, 8, 9	ellimc2 24158	. . . . . . . . 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 233	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
12	11	simprd 496	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
13	12	r19.21bi 3175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
14	13	imp 407	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15		simp1ll 1229	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16		simp2 1130	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶 ∈ 𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵 ∈ 𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17		simp3l 1194	. . . . . . . 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 24155	. . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
22	20	simp2d 1136	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
23	20	simp3d 1137	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	21, 22, 23, 9	ellimc2 24158	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
26	25	simprd 496	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
27	26	r19.21bi 3175	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵 ∈ 𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
28	27	imp 407	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵 ∈ 𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
29	15, 16, 17, 28	syl21anc 834	. . . . . . 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 5979	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
31	15	ad2antrr 722	. . . . . . . . . . . 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 1222	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36		imassrn 5817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37		limccog.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
38	36, 37	syl5ss 3900	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
39	38	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
40	35, 39	ssind 4129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41		imass2 5841	. . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . 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 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
45	43, 44	sstrd 3899	. . . . . . . . . . . 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 834	. . . . . . . . . . 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	syl5eqss 3936	. . . . . . . . . 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 413	. . . . . . . . 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 611	. . . . . . . 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 3237	. . . . . . 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 3249	. . . . 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 413	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
55	54	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴 ∈ 𝑤 ∧ ((𝐺 ∘ 𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56	21	ffund 6386	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
57		fdmrn 6406	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
58	56, 57	sylib 219	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ran 𝐹)
59	37	difss2d 4032	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ dom 𝐺)
60	58, 59	fssd 6396	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶dom 𝐺)
61		fco 6399	. . . 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 24158	. 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 709	1 ⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  {csn 4472  dom cdm 5443  ran crn 5444   “ cima 5446   ∘ ccom 5447  Fun wfun 6219  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  ℂcc 10381  TopOpenctopn 16524  ℂ_fldccnfld 20227   lim_ℂ climc 24143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-sup 8752  df-inf 8753  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-fz 12743  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-plusg 16407  df-mulr 16408  df-starv 16409  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-rest 16525  df-topn 16526  df-topgen 16546  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-cnfld 20228  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-cnp 21520  df-xms 22613  df-ms 22614  df-limc 24147

This theorem is referenced by:  dirkercncflem2  41931  fourierdlem53  41986  fourierdlem93  42026  fourierdlem111  42044