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Theorem limccog 40332
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 23871 and limccnp 23869, 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 23853 . . 3 (𝐺 lim 𝐵) ⊆ ℂ
2 limccog.3 . . 3 (𝜑𝐶 ∈ (𝐺 lim 𝐵))
31, 2sseldi 3796 . 2 (𝜑𝐶 ∈ ℂ)
4 limcrcl 23852 . . . . . . . . . . . 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 1165 . . . . . . . . . 10 (𝜑𝐺:dom 𝐺⟶ℂ)
75simp2d 1166 . . . . . . . . . 10 (𝜑 → dom 𝐺 ⊆ ℂ)
85simp3d 1167 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
9 eqid 2806 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106, 7, 8, 9ellimc2 23855 . . . . . . . . 9 (𝜑 → (𝐶 ∈ (𝐺 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))))
112, 10mpbid 223 . . . . . . . 8 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))))
1211simprd 485 . . . . . . 7 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1312r19.21bi 3120 . . . . . 6 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)))
1413imp 395 . . . . 5 (((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢))
15 simp1ll 1310 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝜑)
16 simp2 1160 . . . . . . . 8 ((((𝜑𝑢 ∈ (TopOpen‘ℂfld)) ∧ 𝐶𝑢) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ (𝐵𝑣 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)) → 𝑣 ∈ (TopOpen‘ℂfld))
17 simp3l 1251 . . . . . . . 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 23852 . . . . . . . . . . . . . . 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 1165 . . . . . . . . . . . . 13 (𝜑𝐹:dom 𝐹⟶ℂ)
2220simp2d 1166 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 ⊆ ℂ)
2320simp3d 1167 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
2421, 22, 23, 9ellimc2 23855 . . . . . . . . . . . 12 (𝜑 → (𝐵 ∈ (𝐹 lim 𝐴) ↔ (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))))
2518, 24mpbid 223 . . . . . . . . . . 11 (𝜑 → (𝐵 ∈ ℂ ∧ ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))))
2625simprd 485 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2726r19.21bi 3120 . . . . . . . . 9 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐵𝑣 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)))
2827imp 395 . . . . . . . 8 (((𝜑𝑣 ∈ (TopOpen‘ℂfld)) ∧ 𝐵𝑣) → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣))
2915, 16, 17, 28syl21anc 857 . . . . . . 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 5854 . . . . . . . . . . 11 ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) = (𝐺 “ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))))
3115ad2antrr 708 . . . . . . . . . . . 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 1296 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣)
36 imassrn 5687 . . . . . . . . . . . . . . . . . 18 (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))
3836, 37syl5ss 3809 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
3938adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (dom 𝐺 ∖ {𝐵}))
4035, 39ssind 4033 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ (𝑣 ∩ (dom 𝐺 ∖ {𝐵})))
41 imass2 5711 . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢) ∧ (𝐹 “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑣) → (𝐺 “ (𝑣 ∩ (dom 𝐺 ∖ {𝐵}))) ⊆ 𝑢)
4543, 44sstrd 3808 . . . . . . . . . . . 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 857 . . . . . . . . . . 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 3846 . . . . . . . . . 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 399 . . . . . . . . 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 601 . . . . . . . 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 3204 . . . . . . 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 3221 . . . . 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 399 . . 3 ((𝜑𝑢 ∈ (TopOpen‘ℂfld)) → (𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5554ralrimiva 3154 . 2 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝐶𝑢 → ∃𝑤 ∈ (TopOpen‘ℂfld)(𝐴𝑤 ∧ ((𝐺𝐹) “ (𝑤 ∩ (dom 𝐹 ∖ {𝐴}))) ⊆ 𝑢)))
5621ffund 6260 . . . . . 6 (𝜑 → Fun 𝐹)
57 fdmrn 6279 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
5856, 57sylib 209 . . . . 5 (𝜑𝐹:dom 𝐹⟶ran 𝐹)
5937difss2d 3939 . . . . 5 (𝜑 → ran 𝐹 ⊆ dom 𝐺)
6058, 59fssd 6270 . . . 4 (𝜑𝐹:dom 𝐹⟶dom 𝐺)
61 fco 6273 . . . 4 ((𝐺:dom 𝐺⟶ℂ ∧ 𝐹:dom 𝐹⟶dom 𝐺) → (𝐺𝐹):dom 𝐹⟶ℂ)
626, 60, 61syl2anc 575 . . 3 (𝜑 → (𝐺𝐹):dom 𝐹⟶ℂ)
6362, 22, 23, 9ellimc2 23855 . 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 695 1 (𝜑𝐶 ∈ ((𝐺𝐹) lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100  wcel 2156  wral 3096  wrex 3097  cdif 3766  cin 3768  wss 3769  {csn 4370  dom cdm 5311  ran crn 5312  cima 5314  ccom 5315  Fun wfun 6095  wf 6097  cfv 6101  (class class class)co 6874  cc 10219  TopOpenctopn 16287  fldccnfld 19954   lim climc 23840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fi 8556  df-sup 8587  df-inf 8588  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-fz 12550  df-seq 13025  df-exp 13084  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-plusg 16166  df-mulr 16167  df-starv 16168  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-rest 16288  df-topn 16289  df-topgen 16309  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-cnfld 19955  df-top 20912  df-topon 20929  df-topsp 20951  df-bases 20964  df-cnp 21246  df-xms 22338  df-ms 22339  df-limc 23844
This theorem is referenced by:  dirkercncflem2  40800  fourierdlem53  40855  fourierdlem93  40895  fourierdlem111  40913
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