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Theorem limccog 43793
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 25241 and limccnp 25239, 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 25223 . . 3 (𝐺 lim 𝐵) ⊆ ℂ
2 limccog.3 . . 3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
31, 2sselid 3940 . 2 (𝜑𝐶 ∈ ℂ)
4 limcrcl 25222 . . . . . . . . . . . 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 1142 . . . . . . . . . 10 (𝜑𝐺:dom 𝐺⟶ℂ)
75simp2d 1143 . . . . . . . . . 10 (𝜑 → dom 𝐺 ⊆ ℂ)
85simp3d 1144 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
9 eqid 2736 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106, 7, 8, 9ellimc2 25225 . . . . . . . . 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 496 . . . . . . 7 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1312r19.21bi 3232 . . . . . 6 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1413imp 407 . . . . 5 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15 simp1ll 1236 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16 simp2 1137 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17 simp3l 1201 . . . . . . . 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 25222 . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . 13 (𝜑𝐹:dom 𝐹⟶ℂ)
2220simp2d 1143 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℂ)
2320simp3d 1144 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
2421, 22, 23, 9ellimc2 25225 . . . . . . . . . . . 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 496 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2726r19.21bi 3232 . . . . . . . . 9 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2827imp 407 . . . . . . . 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 6201 . . . . . . . . . . 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 1229 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 485 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36 imassrn 6022 . . . . . . . . . . . . . . . . . 18 (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
3836, 37sstrid 3953 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
3938adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
4035, 39ssind 4190 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41 imass2 6052 . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . 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 3990 . . . . . . . . . 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 413 . . . . . . . . 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 612 . . . . . . . 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 3163 . . . . . . 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 3154 . . . . 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 413 . . 3 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5554ralrimiva 3141 . 2 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5621ffund 6669 . . . . . 6 (𝜑 → Fun 𝐹)
57 fdmrn 6697 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5856, 57sylib 217 . . . . 5 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
5937difss2d 4092 . . . . 5 (𝜑 → ran 𝐹 ⊆ dom 𝐺)
6058, 59fssd 6683 . . . 4 (𝜑𝐹:dom 𝐹⟶dom 𝐺)
61 fco 6689 . . . 4 ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺𝐹):dom 𝐹⟶ℂ)
626, 60, 61syl2anc 584 . . 3 (𝜑 → (𝐺𝐹):dom 𝐹⟶ℂ)
6362, 22, 23, 9ellimc2 25225 . 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 396  w3a 1087  wcel 2106  wral 3062  wrex 3071  cdif 3905  cin 3907  wss 3908  {csn 4584  dom cdm 5631  ran crn 5632  cima 5634  ccom 5635  Fun wfun 6487  wf 6489  cfv 6493  (class class class)co 7353  cc 11045  TopOpenctopn 17295  fldccnfld 20781   lim climc 25210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-pm 8764  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fi 9343  df-sup 9374  df-inf 9375  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-div 11809  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-q 12866  df-rp 12908  df-xneg 13025  df-xadd 13026  df-xmul 13027  df-fz 13417  df-seq 13899  df-exp 13960  df-cj 14976  df-re 14977  df-im 14978  df-sqrt 15112  df-abs 15113  df-struct 17011  df-slot 17046  df-ndx 17058  df-base 17076  df-plusg 17138  df-mulr 17139  df-starv 17140  df-tset 17144  df-ple 17145  df-ds 17147  df-unif 17148  df-rest 17296  df-topn 17297  df-topgen 17317  df-psmet 20773  df-xmet 20774  df-met 20775  df-bl 20776  df-mopn 20777  df-cnfld 20782  df-top 22227  df-topon 22244  df-topsp 22266  df-bases 22280  df-cnp 22563  df-xms 23657  df-ms 23658  df-limc 25214
This theorem is referenced by:  dirkercncflem2  44277  fourierdlem53  44332  fourierdlem93  44372  fourierdlem111  44390
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