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Theorem limccog 40272
 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 23777 and limccnp 23775, 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 (𝜑𝐵 ∈ (𝐹 lim 𝐴))
limccog.3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
Assertion
Ref Expression
limccog (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))

Proof of Theorem limccog
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl 23759 . . 3 (𝐺 lim 𝐵) ⊆ ℂ
2 limccog.3 . . 3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
31, 2sseldi 3707 . 2 (𝜑𝐶 ∈ ℂ)
4 limcrcl 23758 . . . . . . . . . . . 12 (𝐶 ∈ (𝐺 lim 𝐵) → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
52, 4syl 17 . . . . . . . . . . 11 (𝜑 → (𝐺:dom 𝐺⟶ℂ ∧ dom 𝐺 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
65simp1d 1134 . . . . . . . . . 10 (𝜑𝐺:dom 𝐺⟶ℂ)
75simp2d 1135 . . . . . . . . . 10 (𝜑 → dom 𝐺 ⊆ ℂ)
85simp3d 1136 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
9 eqid 2724 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106, 7, 8, 9ellimc2 23761 . . . . . . . . 9 (𝜑 → (𝐶 ∈ (𝐺 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))))
112, 10mpbid 222 . . . . . . . 8 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
1211simprd 482 . . . . . . 7 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1312r19.21bi 3034 . . . . . 6 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1413imp 444 . . . . 5 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15 simp1ll 1273 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16 simp2 1129 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17 simp3l 1220 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝐵𝑣)
18 limccog.2 . . . . . . . . . . . 12 (𝜑𝐵 ∈ (𝐹 lim 𝐴))
19 limcrcl 23758 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝐹 lim 𝐴) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐴 ∈ ℂ))
2120simp1d 1134 . . . . . . . . . . . . 13 (𝜑𝐹:dom 𝐹⟶ℂ)
2220simp2d 1135 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℂ)
2320simp3d 1136 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
2421, 22, 23, 9ellimc2 23761 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∈ (𝐹 lim 𝐴) ↔ (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))))
2518, 24mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
2625simprd 482 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2726r19.21bi 3034 . . . . . . . . 9 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2827imp 444 . . . . . . . 8 (((𝜑𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
2915, 16, 17, 28syl21anc 1438 . . . . . . 7 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
30 imaco 5753 . . . . . . . . . . 11 ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
3115ad2antrr 764 . . . . . . . . . . . 12 ((((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → 𝜑)
32 simpl3r 1265 . . . . . . . . . . . . 13 (((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
3332adantr 472 . . . . . . . . . . . 12 ((((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
34 simpr 479 . . . . . . . . . . . 12 ((((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
35 simpr 479 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36 imassrn 5587 . . . . . . . . . . . . . . . . . 18 (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
3836, 37syl5ss 3720 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
3938adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
4035, 39ssind 3945 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41 imass2 5611 . . . . . . . . . . . . . . 15 ((𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
4240, 41syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))))
4342adantlr 753 . . . . . . . . . . . . 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 809 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
4543, 44sstrd 3719 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ 𝑢)
4631, 33, 34, 45syl21anc 1438 . . . . . . . . . . 11 ((((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴})))) ⊆ 𝑢)
4730, 46syl5eqss 3755 . . . . . . . . . 10 ((((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)
4847ex 449 . . . . . . . . 9 (((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → ((𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣 → ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
4948anim2d 590 . . . . . . . 8 (((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) ∧ 𝑤 ∈ (TopOpen‘ℂfld)) → ((𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5049reximdva 3119 . . . . . . 7 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → (∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5129, 50mpd 15 . . . . . 6 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
5251rexlimdv3a 3135 . . . . 5 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5314, 52mpd 15 . . . 4 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢))
5453ex 449 . . 3 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5554ralrimiva 3068 . 2 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56 ffun 6161 . . . . . . 7 (𝐹:dom 𝐹⟶ℂ → Fun 𝐹)
5721, 56syl 17 . . . . . 6 (𝜑 → Fun 𝐹)
58 fdmrn 6177 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5957, 58sylib 208 . . . . 5 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
6037difss2d 3848 . . . . 5 (𝜑 → ran 𝐹 ⊆ dom 𝐺)
6159, 60fssd 6170 . . . 4 (𝜑𝐹:dom 𝐹⟶dom 𝐺)
62 fco 6171 . . . 4 ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺𝐹):dom 𝐹⟶ℂ)
636, 61, 62syl2anc 696 . . 3 (𝜑 → (𝐺𝐹):dom 𝐹⟶ℂ)
6463, 22, 23, 9ellimc2 23761 . 2 (𝜑 → (𝐶 ∈ ((𝐺𝐹) lim 𝐴) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))))
653, 55, 64mpbir2and 995 1 (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2103  ∀wral 3014  ∃wrex 3015   ∖ cdif 3677   ∩ cin 3679   ⊆ wss 3680  {csn 4285  dom cdm 5218  ran crn 5219   “ cima 5221   ∘ ccom 5222  Fun wfun 5995  ⟶wf 5997  ‘cfv 6001  (class class class)co 6765  ℂcc 10047  TopOpenctopn 16205  ℂfldccnfld 19869   limℂ climc 23746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fi 8433  df-sup 8464  df-inf 8465  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-q 11903  df-rp 11947  df-xneg 12060  df-xadd 12061  df-xmul 12062  df-fz 12441  df-seq 12917  df-exp 12976  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-plusg 16077  df-mulr 16078  df-starv 16079  df-tset 16083  df-ple 16084  df-ds 16087  df-unif 16088  df-rest 16206  df-topn 16207  df-topgen 16227  df-psmet 19861  df-xmet 19862  df-met 19863  df-bl 19864  df-mopn 19865  df-cnfld 19870  df-top 20822  df-topon 20839  df-topsp 20860  df-bases 20873  df-cnp 21155  df-xms 22247  df-ms 22248  df-limc 23750 This theorem is referenced by:  dirkercncflem2  40741  fourierdlem53  40796  fourierdlem93  40836  fourierdlem111  40854
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