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Theorem limccnp 23595
Description: If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
limccnp.f (𝜑𝐹:𝐴𝐷)
limccnp.d (𝜑𝐷 ⊆ ℂ)
limccnp.k 𝐾 = (TopOpen‘ℂfld)
limccnp.j 𝐽 = (𝐾t 𝐷)
limccnp.c (𝜑𝐶 ∈ (𝐹 lim 𝐵))
limccnp.b (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))
Assertion
Ref Expression
limccnp (𝜑 → (𝐺𝐶) ∈ ((𝐺𝐹) lim 𝐵))

Proof of Theorem limccnp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccnp.b . . . . . . . . 9 (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))
2 eqid 2621 . . . . . . . . . 10 𝐽 = 𝐽
32cnprcl 20989 . . . . . . . . 9 (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶) → 𝐶 𝐽)
41, 3syl 17 . . . . . . . 8 (𝜑𝐶 𝐽)
5 limccnp.j . . . . . . . . . 10 𝐽 = (𝐾t 𝐷)
6 limccnp.k . . . . . . . . . . . 12 𝐾 = (TopOpen‘ℂfld)
76cnfldtopon 22526 . . . . . . . . . . 11 𝐾 ∈ (TopOn‘ℂ)
8 limccnp.d . . . . . . . . . . 11 (𝜑𝐷 ⊆ ℂ)
9 resttopon 20905 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐾t 𝐷) ∈ (TopOn‘𝐷))
107, 8, 9sylancr 694 . . . . . . . . . 10 (𝜑 → (𝐾t 𝐷) ∈ (TopOn‘𝐷))
115, 10syl5eqel 2702 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝐷))
12 toponuni 20659 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝐷) → 𝐷 = 𝐽)
1311, 12syl 17 . . . . . . . 8 (𝜑𝐷 = 𝐽)
144, 13eleqtrrd 2701 . . . . . . 7 (𝜑𝐶𝐷)
1514ad2antrr 761 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶𝐷)
16 limccnp.f . . . . . . . 8 (𝜑𝐹:𝐴𝐷)
1716ad2antrr 761 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝐹:𝐴𝐷)
18 elun 3737 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
19 elsni 4172 . . . . . . . . . . . 12 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
2019orim2i 540 . . . . . . . . . . 11 ((𝑥𝐴𝑥 ∈ {𝐵}) → (𝑥𝐴𝑥 = 𝐵))
2118, 20sylbi 207 . . . . . . . . . 10 (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝑥𝐴𝑥 = 𝐵))
2221adantl 482 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥𝐴𝑥 = 𝐵))
2322orcomd 403 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 = 𝐵𝑥𝐴))
2423orcanai 951 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝐴)
2517, 24ffvelrnd 6326 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → (𝐹𝑥) ∈ 𝐷)
2615, 25ifclda 4098 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)) ∈ 𝐷)
27 eqidd 2622 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))))
287a1i 11 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘ℂ))
29 cnpf2 20994 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐷) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶)) → 𝐺:𝐷⟶ℂ)
3011, 28, 1, 29syl3anc 1323 . . . . . 6 (𝜑𝐺:𝐷⟶ℂ)
3130feqmptd 6216 . . . . 5 (𝜑𝐺 = (𝑦𝐷 ↦ (𝐺𝑦)))
32 fveq2 6158 . . . . 5 (𝑦 = if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)) → (𝐺𝑦) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))))
3326, 27, 31, 32fmptco 6362 . . . 4 (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))))
34 fvco3 6242 . . . . . . . 8 ((𝐹:𝐴𝐷𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
3517, 24, 34syl2anc 692 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
3635ifeq2da 4095 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥)) = if(𝑥 = 𝐵, (𝐺𝐶), (𝐺‘(𝐹𝑥))))
37 fvif 6171 . . . . . 6 (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) = if(𝑥 = 𝐵, (𝐺𝐶), (𝐺‘(𝐹𝑥)))
3836, 37syl6eqr 2673 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥)) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))))
3938mpteq2dva 4714 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))))
4033, 39eqtr4d 2658 . . 3 (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥))))
41 limccnp.c . . . . . . 7 (𝜑𝐶 ∈ (𝐹 lim 𝐵))
42 eqid 2621 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
43 eqid 2621 . . . . . . . 8 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))
4416, 8fssd 6024 . . . . . . . 8 (𝜑𝐹:𝐴⟶ℂ)
45 fdm 6018 . . . . . . . . . 10 (𝐹:𝐴𝐷 → dom 𝐹 = 𝐴)
4616, 45syl 17 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
47 limcrcl 23578 . . . . . . . . . . 11 (𝐶 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
4841, 47syl 17 . . . . . . . . . 10 (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
4948simp2d 1072 . . . . . . . . 9 (𝜑 → dom 𝐹 ⊆ ℂ)
5046, 49eqsstr3d 3625 . . . . . . . 8 (𝜑𝐴 ⊆ ℂ)
5148simp3d 1073 . . . . . . . 8 (𝜑𝐵 ∈ ℂ)
5242, 6, 43, 44, 50, 51ellimc 23577 . . . . . . 7 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
5341, 52mpbid 222 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
546cnfldtop 22527 . . . . . . . 8 𝐾 ∈ Top
5554a1i 11 . . . . . . 7 (𝜑𝐾 ∈ Top)
5626, 43fmptd 6351 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))):(𝐴 ∪ {𝐵})⟶𝐷)
5751snssd 4316 . . . . . . . . . . . 12 (𝜑 → {𝐵} ⊆ ℂ)
5850, 57unssd 3773 . . . . . . . . . . 11 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
59 resttopon 20905 . . . . . . . . . . 11 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
607, 58, 59sylancr 694 . . . . . . . . . 10 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
61 toponuni 20659 . . . . . . . . . 10 ((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
6260, 61syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
6362feq2d 5998 . . . . . . . 8 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))):(𝐴 ∪ {𝐵})⟶𝐷 ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))): (𝐾t (𝐴 ∪ {𝐵}))⟶𝐷))
6456, 63mpbid 222 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))): (𝐾t (𝐴 ∪ {𝐵}))⟶𝐷)
65 eqid 2621 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
667toponunii 20661 . . . . . . . 8 ℂ = 𝐾
6765, 66cnprest2 21034 . . . . . . 7 ((𝐾 ∈ Top ∧ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))): (𝐾t (𝐴 ∪ {𝐵}))⟶𝐷𝐷 ⊆ ℂ) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾t 𝐷))‘𝐵)))
6855, 64, 8, 67syl3anc 1323 . . . . . 6 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾t 𝐷))‘𝐵)))
6953, 68mpbid 222 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾t 𝐷))‘𝐵))
705oveq2i 6626 . . . . . 6 ((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾t 𝐷))
7170fveq1i 6159 . . . . 5 (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾t 𝐷))‘𝐵)
7269, 71syl6eleqr 2709 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
73 ssun2 3761 . . . . . . . 8 {𝐵} ⊆ (𝐴 ∪ {𝐵})
74 snssg 4303 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
7551, 74syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
7673, 75mpbiri 248 . . . . . . 7 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
77 iftrue 4070 . . . . . . . 8 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)) = 𝐶)
7877, 43fvmptg 6247 . . . . . . 7 ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶𝐷) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))‘𝐵) = 𝐶)
7976, 14, 78syl2anc 692 . . . . . 6 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))‘𝐵) = 𝐶)
8079fveq2d 6162 . . . . 5 (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))‘𝐵)) = ((𝐽 CnP 𝐾)‘𝐶))
811, 80eleqtrrd 2701 . . . 4 (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))‘𝐵)))
82 cnpco 21011 . . . 4 (((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))‘𝐵))) → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
8372, 81, 82syl2anc 692 . . 3 (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹𝑥)))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
8440, 83eqeltrrd 2699 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
85 eqid 2621 . . 3 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥)))
86 fco 6025 . . . 4 ((𝐺:𝐷⟶ℂ ∧ 𝐹:𝐴𝐷) → (𝐺𝐹):𝐴⟶ℂ)
8730, 16, 86syl2anc 692 . . 3 (𝜑 → (𝐺𝐹):𝐴⟶ℂ)
8842, 6, 85, 87, 50, 51ellimc 23577 . 2 (𝜑 → ((𝐺𝐶) ∈ ((𝐺𝐹) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺𝐶), ((𝐺𝐹)‘𝑥))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
8984, 88mpbird 247 1 (𝜑 → (𝐺𝐶) ∈ ((𝐺𝐹) lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cun 3558  wss 3560  ifcif 4064  {csn 4155   cuni 4409  cmpt 4683  dom cdm 5084  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  cc 9894  t crest 16021  TopOpenctopn 16022  fldccnfld 19686  Topctop 20638  TopOnctopon 20655   CnP ccnp 20969   lim climc 23566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-fz 12285  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-plusg 15894  df-mulr 15895  df-starv 15896  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-rest 16023  df-topn 16024  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-cnfld 19687  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cnp 20972  df-xms 22065  df-ms 22066  df-limc 23570
This theorem is referenced by:  limcco  23597  dvcjbr  23652  dvcnvlem  23677
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