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Theorem limccog 45241
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 25913 and limccnp 25911, 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 25895 . . 3 (𝐺 lim 𝐵) ⊆ ℂ
2 limccog.3 . . 3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
31, 2sselid 3977 . 2 (𝜑𝐶 ∈ ℂ)
4 limcrcl 25894 . . . . . . . . . . . 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 1139 . . . . . . . . . 10 (𝜑𝐺:dom 𝐺⟶ℂ)
75simp2d 1140 . . . . . . . . . 10 (𝜑 → dom 𝐺 ⊆ ℂ)
85simp3d 1141 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
9 eqid 2726 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106, 7, 8, 9ellimc2 25897 . . . . . . . . 9 (𝜑 → (𝐶 ∈ (𝐺 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))))
112, 10mpbid 231 . . . . . . . 8 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
1211simprd 494 . . . . . . 7 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1312r19.21bi 3239 . . . . . 6 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1413imp 405 . . . . 5 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15 simp1ll 1233 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16 simp2 1134 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17 simp3l 1198 . . . . . . . 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 25894 . . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . . 13 (𝜑𝐹:dom 𝐹⟶ℂ)
2220simp2d 1140 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℂ)
2320simp3d 1141 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
2421, 22, 23, 9ellimc2 25897 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∈ (𝐹 lim 𝐴) ↔ (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))))
2518, 24mpbid 231 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
2625simprd 494 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2726r19.21bi 3239 . . . . . . . . 9 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2827imp 405 . . . . . . . 8 (((𝜑𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
2915, 16, 17, 28syl21anc 836 . . . . . . 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 6262 . . . . . . . . . . 11 ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
3115ad2antrr 724 . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 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) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
34 simpr 483 . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36 imassrn 6080 . . . . . . . . . . . . . . . . . 18 (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
3836, 37sstrid 3991 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
3938adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
4035, 39ssind 4234 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41 imass2 6112 . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . 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 767 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
4543, 44sstrd 3990 . . . . . . . . . . . 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 836 . . . . . . . . . . 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, 46eqsstrid 4028 . . . . . . . . . 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 411 . . . . . . . . 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 610 . . . . . . . 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 3158 . . . . . . 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 3149 . . . . 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 411 . . 3 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5554ralrimiva 3136 . 2 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5621ffund 6732 . . . . . 6 (𝜑 → Fun 𝐹)
57 fdmrn 6760 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5856, 57sylib 217 . . . . 5 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
5937difss2d 4134 . . . . 5 (𝜑 → ran 𝐹 ⊆ dom 𝐺)
6058, 59fssd 6745 . . . 4 (𝜑𝐹:dom 𝐹⟶dom 𝐺)
61 fco 6752 . . . 4 ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺𝐹):dom 𝐹⟶ℂ)
626, 60, 61syl2anc 582 . . 3 (𝜑 → (𝐺𝐹):dom 𝐹⟶ℂ)
6362, 22, 23, 9ellimc2 25897 . 2 (𝜑 → (𝐶 ∈ ((𝐺𝐹) lim 𝐴) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))))
643, 55, 63mpbir2and 711 1 (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wcel 2099  wral 3051  wrex 3060  cdif 3944  cin 3946  wss 3947  {csn 4633  dom cdm 5682  ran crn 5683  cima 5685  ccom 5686  Fun wfun 6548  wf 6550  cfv 6554  (class class class)co 7424  cc 11156  TopOpenctopn 17436  fldccnfld 21343   lim climc 25882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fi 9454  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-q 12985  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-fz 13539  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-struct 17149  df-slot 17184  df-ndx 17196  df-base 17214  df-plusg 17279  df-mulr 17280  df-starv 17281  df-tset 17285  df-ple 17286  df-ds 17288  df-unif 17289  df-rest 17437  df-topn 17438  df-topgen 17458  df-psmet 21335  df-xmet 21336  df-met 21337  df-bl 21338  df-mopn 21339  df-cnfld 21344  df-top 22887  df-topon 22904  df-topsp 22926  df-bases 22940  df-cnp 23223  df-xms 24317  df-ms 24318  df-limc 25886
This theorem is referenced by:  dirkercncflem2  45725  fourierdlem53  45780  fourierdlem93  45820  fourierdlem111  45838
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