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Theorem limccog 38687
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 𝐺𝐹 at 𝐴 is 𝐶. With respect to limcco 23463 and limccnp 23461, 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 23445 . . 3 (𝐺 lim 𝐵) ⊆ ℂ
2 limccog.3 . . 3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
31, 2sseldi 3566 . 2 (𝜑𝐶 ∈ ℂ)
4 limcrcl 23444 . . . . . . . . . . . 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 1066 . . . . . . . . . 10 (𝜑𝐺:dom 𝐺⟶ℂ)
75simp2d 1067 . . . . . . . . . 10 (𝜑 → dom 𝐺 ⊆ ℂ)
85simp3d 1068 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
9 eqid 2610 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106, 7, 8, 9ellimc2 23447 . . . . . . . . 9 (𝜑 → (𝐶 ∈ (𝐺 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))))
112, 10mpbid 221 . . . . . . . 8 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
1211simprd 478 . . . . . . 7 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1312r19.21bi 2916 . . . . . 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 1117 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16 simp2 1055 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17 simp3l 1082 . . . . . . . 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 23444 . . . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . 13 (𝜑𝐹:dom 𝐹⟶ℂ)
2220simp2d 1067 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℂ)
2320simp3d 1068 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
2421, 22, 23, 9ellimc2 23447 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∈ (𝐹 lim 𝐴) ↔ (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))))
2518, 24mpbid 221 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
2625simprd 478 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2726r19.21bi 2916 . . . . . . . . 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 1317 . . . . . . 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 5557 . . . . . . . . . . 11 ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
3115ad2antrr 758 . . . . . . . . . . . 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 1110 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36 imassrn 5396 . . . . . . . . . . . . . . . . . 18 (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
3836, 37syl5ss 3579 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
3938adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
4035, 39ssind 3799 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41 imass2 5420 . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . 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 788 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
4543, 44sstrd 3578 . . . . . . . . . . . 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 1317 . . . . . . . . . . 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 3612 . . . . . . . . . 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 587 . . . . . . . 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 3000 . . . . . . 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 3015 . . . . 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 2949 . 2 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
56 ffun 5961 . . . . . . 7 (𝐹:dom 𝐹⟶ℂ → Fun 𝐹)
5721, 56syl 17 . . . . . 6 (𝜑 → Fun 𝐹)
58 fdmrn 5977 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5957, 58sylib 207 . . . . 5 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
6037difss2d 3702 . . . . 5 (𝜑 → ran 𝐹 ⊆ dom 𝐺)
6159, 60fssd 5970 . . . 4 (𝜑𝐹:dom 𝐹⟶dom 𝐺)
62 fco 5971 . . . 4 ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺𝐹):dom 𝐹⟶ℂ)
636, 61, 62syl2anc 691 . . 3 (𝜑 → (𝐺𝐹):dom 𝐹⟶ℂ)
6463, 22, 23, 9ellimc2 23447 . 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 959 1 (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wcel 1977  wral 2896  wrex 2897  cdif 3537  cin 3539  wss 3540  {csn 4125  dom cdm 5038  ran crn 5039  cima 5041  ccom 5042  Fun wfun 5798  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  TopOpenctopn 15905  fldccnfld 19567   lim climc 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cnp 20842  df-xms 21935  df-ms 21936  df-limc 23436
This theorem is referenced by:  dirkercncflem2  38997  fourierdlem53  39052  fourierdlem93  39092  fourierdlem111  39110
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