limccnp - Metamath Proof Explorer

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Theorem limccnp 25844

Description: If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
limccnp.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐷)
limccnp.d	⊢ (𝜑 → 𝐷 ⊆ ℂ)
limccnp.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
limccnp.j	⊢ 𝐽 = (𝐾 ↾_t 𝐷)
limccnp.c	⊢ (𝜑 → 𝐶 ∈ (𝐹 lim_ℂ 𝐵))
limccnp.b	⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))

**Assertion**
Ref	Expression
limccnp	⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) lim_ℂ 𝐵))

Proof of Theorem limccnp Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccnp.j	. . . . . . 7 ⊢ 𝐽 = (𝐾 ↾_t 𝐷)
2		limccnp.k	. . . . . . . . 9 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24721	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		limccnp.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
5		resttopon 23099	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐾 ↾_t 𝐷) ∈ (TopOn‘𝐷))
6	3, 4, 5	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (𝐾 ↾_t 𝐷) ∈ (TopOn‘𝐷))
7	1, 6	eqeltrid 2838	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐷))
8	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
9		limccnp.b	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))
10		cnpf2 23188	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐷) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶)) → 𝐺:𝐷⟶ℂ)
11	7, 8, 9, 10	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐺:𝐷⟶ℂ)
12		eqid 2735	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	cnprcl 23183	. . . . . . . . 9 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶) → 𝐶 ∈ ∪ 𝐽)
14	9, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ∪ 𝐽)
15		toponuni 22852	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝐷) → 𝐷 = ∪ 𝐽)
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝐽)
17	14, 16	eleqtrrd 2837	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
18	17	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝐷)
19		limccnp.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)
20	19	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝐹:𝐴⟶𝐷)
21		elun 4128	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
22		elsni 4618	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
23	22	orim2i 910	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
24	21, 23	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
25	24	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
26	25	orcomd 871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 = 𝐵 ∨ 𝑥 ∈ 𝐴))
27	26	orcanai 1004	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
28	20, 27	ffvelcdmd 7075	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ 𝐷)
29	18, 28	ifclda 4536	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) ∈ 𝐷)
30	11, 29	cofmpt 7122	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))))
31		fvco3 6978	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐷 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
32	20, 27, 31	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
33	32	ifeq2da 4533	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥))))
34		fvif 6892	. . . . . 6 ⊢ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥)))
35	33, 34	eqtr4di 2788	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))))
36	35	mpteq2dva 5214	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))))
37	30, 36	eqtr4d 2773	. . 3 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))))
38		limccnp.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐹 lim_ℂ 𝐵))
39		eqid 2735	. . . . . . . 8 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
40		eqid 2735	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))
41	19, 4	fssd 6723	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
42	19	fdmd 6716	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43		limcrcl 25827	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
44	38, 43	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
45	44	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
46	42, 45	eqsstrrd 3994	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
47	44	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
48	39, 2, 40, 41, 46, 47	ellimc 25826	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
49	38, 48	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
50	2	cnfldtop 24722	. . . . . . . 8 ⊢ 𝐾 ∈ Top
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
52	29	fmpttd 7105	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):(𝐴 ∪ {𝐵})⟶𝐷)
53	47	snssd 4785	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
54	46, 53	unssd 4167	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
55		resttopon 23099	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
56	3, 54, 55	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
57		toponuni 22852	. . . . . . . . . 10 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
59	58	feq2d 6692	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):(𝐴 ∪ {𝐵})⟶𝐷 ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶𝐷))
60	52, 59	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶𝐷)
61		eqid 2735	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
62	3	toponunii 22854	. . . . . . . 8 ⊢ ℂ = ∪ 𝐾
63	61, 62	cnprest2 23228	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶𝐷 ∧ 𝐷 ⊆ ℂ) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)))
64	51, 60, 4, 63	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)))
65	49, 64	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵))
66	1	oveq2i 7416	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))
67	66	fveq1i 6877	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)
68	65, 67	eleqtrrdi 2845	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
69		iftrue 4506	. . . . . . 7 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) = 𝐶)
70		ssun2 4154	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
71		snssg 4759	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
72	47, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
73	70, 72	mpbiri 258	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
74	40, 69, 73, 38	fvmptd3 7009	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵) = 𝐶)
75	74	fveq2d 6880	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)) = ((𝐽 CnP 𝐾)‘𝐶))
76	9, 75	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)))
77		cnpco 23205	. . . 4 ⊢ (((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵))) → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
78	68, 76, 77	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
79	37, 78	eqeltrrd 2835	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
80		eqid 2735	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)))
81		fco 6730	. . . 4 ⊢ ((𝐺:𝐷⟶ℂ ∧ 𝐹:𝐴⟶𝐷) → (𝐺 ∘ 𝐹):𝐴⟶ℂ)
82	11, 19, 81	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶ℂ)
83	39, 2, 80, 82, 46, 47	ellimc 25826	. 2 ⊢ (𝜑 → ((𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
84	79, 83	mpbird 257	1 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∪ cun 3924 ⊆ wss 3926 ifcif 4500 {csn 4601 ∪ cuni 4883 ↦ cmpt 5201 dom cdm 5654 ∘ ccom 5658 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ↾_t crest 17434 TopOpenctopn 17435 ℂ_fldccnfld 21315 Topctop 22831 TopOnctopon 22848 CnP ccnp 23163 lim_ℂ climc 25815

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-fz 13525 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-mulr 17285 df-starv 17286 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-rest 17436 df-topn 17437 df-topgen 17457 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cnp 23166 df-xms 24259 df-ms 24260 df-limc 25819

This theorem is referenced by: limcco 25846 dvcjbr 25905 dvcnvlem 25932

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