limccnp - Metamath Proof Explorer

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Theorem limccnp 25814

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 24692	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		limccnp.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
5		resttopon 23071	. . . . . . . 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 2835	. . . . . 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 23160	. . . . . 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 2731	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	cnprcl 23155	. . . . . . . . 9 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶) → 𝐶 ∈ ∪ 𝐽)
14	9, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ∪ 𝐽)
15		toponuni 22824	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝐷) → 𝐷 = ∪ 𝐽)
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝐽)
17	14, 16	eleqtrrd 2834	. . . . . . 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 4098	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
22		elsni 4588	. . . . . . . . . . . 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 7013	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ 𝐷)
29	18, 28	ifclda 4506	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) ∈ 𝐷)
30	11, 29	cofmpt 7060	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))))
31		fvco3 6916	. . . . . . . 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 4503	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥))))
34		fvif 6833	. . . . . 6 ⊢ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥)))
35	33, 34	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))))
36	35	mpteq2dva 5179	. . . 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 2769	. . 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 2731	. . . . . . . 8 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
40		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))
41	19, 4	fssd 6663	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
42	19	fdmd 6656	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43		limcrcl 25797	. . . . . . . . . . 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 3965	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
47	44	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
48	39, 2, 40, 41, 46, 47	ellimc 25796	. . . . . . 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 24693	. . . . . . . 8 ⊢ 𝐾 ∈ Top
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
52	29	fmpttd 7043	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):(𝐴 ∪ {𝐵})⟶𝐷)
53	47	snssd 4756	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
54	46, 53	unssd 4137	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
55		resttopon 23071	. . . . . . . . . . 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 22824	. . . . . . . . . 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 6630	. . . . . . . 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 2731	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
62	3	toponunii 22826	. . . . . . . 8 ⊢ ℂ = ∪ 𝐾
63	61, 62	cnprest2 23200	. . . . . . 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 7352	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))
67	66	fveq1i 6818	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)
68	65, 67	eleqtrrdi 2842	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
69		iftrue 4476	. . . . . . 7 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) = 𝐶)
70		ssun2 4124	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
71		snssg 4731	. . . . . . . . 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 6947	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵) = 𝐶)
75	74	fveq2d 6821	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)) = ((𝐽 CnP 𝐾)‘𝐶))
76	9, 75	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)))
77		cnpco 23177	. . . 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 2832	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
80		eqid 2731	. . 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 6670	. . . 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 25796	. 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 1541 ∈ wcel 2111 ∪ cun 3895 ⊆ wss 3897 ifcif 4470 {csn 4571 ∪ cuni 4854 ↦ cmpt 5167 dom cdm 5611 ∘ ccom 5615 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ↾_t crest 17319 TopOpenctopn 17320 ℂ_fldccnfld 21286 Topctop 22803 TopOnctopon 22820 CnP ccnp 23135 lim_ℂ climc 25785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fi 9290 df-sup 9321 df-inf 9322 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-fz 13403 df-seq 13904 df-exp 13964 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-struct 17053 df-slot 17088 df-ndx 17100 df-base 17116 df-plusg 17169 df-mulr 17170 df-starv 17171 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-rest 17321 df-topn 17322 df-topgen 17342 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cnp 23138 df-xms 24230 df-ms 24231 df-limc 25789

This theorem is referenced by: limcco 25816 dvcjbr 25875 dvcnvlem 25902

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