limccnp - Metamath Proof Explorer

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Theorem limccnp 25792

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 24670	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		limccnp.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
5		resttopon 23048	. . . . . . . 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 2832	. . . . . 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 23137	. . . . . 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 2729	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	cnprcl 23132	. . . . . . . . 9 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶) → 𝐶 ∈ ∪ 𝐽)
14	9, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ∪ 𝐽)
15		toponuni 22801	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝐷) → 𝐷 = ∪ 𝐽)
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝐽)
17	14, 16	eleqtrrd 2831	. . . . . . 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 4116	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
22		elsni 4606	. . . . . . . . . . . 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 7057	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ 𝐷)
29	18, 28	ifclda 4524	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) ∈ 𝐷)
30	11, 29	cofmpt 7104	. . . 4 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))))
31		fvco3 6960	. . . . . . . 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 4521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥))))
34		fvif 6874	. . . . . 6 ⊢ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥)))
35	33, 34	eqtr4di 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))))
36	35	mpteq2dva 5200	. . . 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 2767	. . 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 2729	. . . . . . . 8 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
40		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))
41	19, 4	fssd 6705	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
42	19	fdmd 6698	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43		limcrcl 25775	. . . . . . . . . . 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 3982	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
47	44	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
48	39, 2, 40, 41, 46, 47	ellimc 25774	. . . . . . 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 24671	. . . . . . . 8 ⊢ 𝐾 ∈ Top
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
52	29	fmpttd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):(𝐴 ∪ {𝐵})⟶𝐷)
53	47	snssd 4773	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
54	46, 53	unssd 4155	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
55		resttopon 23048	. . . . . . . . . . 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 22801	. . . . . . . . . 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 6672	. . . . . . . 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 2729	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
62	3	toponunii 22803	. . . . . . . 8 ⊢ ℂ = ∪ 𝐾
63	61, 62	cnprest2 23177	. . . . . . 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 7398	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))
67	66	fveq1i 6859	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)
68	65, 67	eleqtrrdi 2839	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
69		iftrue 4494	. . . . . . 7 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) = 𝐶)
70		ssun2 4142	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
71		snssg 4747	. . . . . . . . 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 6991	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵) = 𝐶)
75	74	fveq2d 6862	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)) = ((𝐽 CnP 𝐾)‘𝐶))
76	9, 75	eleqtrrd 2831	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)))
77		cnpco 23154	. . . 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 2829	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
80		eqid 2729	. . 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 6712	. . . 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 25774	. 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 2109 ∪ cun 3912 ⊆ wss 3914 ifcif 4488 {csn 4589 ∪ cuni 4871 ↦ cmpt 5188 dom cdm 5638 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21264 Topctop 22780 TopOnctopon 22797 CnP ccnp 23112 lim_ℂ climc 25763

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-fz 13469 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cnp 23115 df-xms 24208 df-ms 24209 df-limc 25767

This theorem is referenced by: limcco 25794 dvcjbr 25853 dvcnvlem 25880

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