limccnp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > limccnp		Structured version Visualization version GIF version

Theorem limccnp 26118

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 25008	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		limccnp.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
5		resttopon 23386	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐾 ↾_t 𝐷) ∈ (TopOn‘𝐷))
6	3, 4, 5	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (𝐾 ↾_t 𝐷) ∈ (TopOn‘𝐷))
7	1, 6	eqeltrid 2864	. . . . . 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 23475	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝐷) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶)) → 𝐺:𝐷⟶ℂ)
11	7, 8, 9, 10	syl3anc 1398	. . . . 5 ⊢ (𝜑 → 𝐺:𝐷⟶ℂ)
12		eqid 2760	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	cnprcl 23470	. . . . . . . . 9 ⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶) → 𝐶 ∈ ∪ 𝐽)
14	9, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ∪ 𝐽)
15		toponuni 23139	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝐷) → 𝐷 = ∪ 𝐽)
16	7, 15	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝐽)
17	14, 16	eleqtrrd 2863	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
18	17	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝐷)
19		limccnp.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)
20	19	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝐹:𝐴⟶𝐷)
21		elun 4100	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
22		elsni 4601	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
23	22	orim2i 924	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
24	21, 23	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
25	24	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 = 𝐵))
26	25	orcomd 885	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 = 𝐵 ∨ 𝑥 ∈ 𝐴))
27	26	orcanai 1018	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
28	20, 27	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ 𝐷)
29	18, 28	ifclda 4518	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) ∈ 𝐷)
30	11, 29	cofmpt 7126	. . . 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 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
33	32	ifeq2da 4515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥))))
34		fvif 6894	. . . . . 6 ⊢ (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = if(𝑥 = 𝐵, (𝐺‘𝐶), (𝐺‘(𝐹‘𝑥)))
35	33, 34	eqtr4di 2813	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥)) = (𝐺‘if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))))
36	35	mpteq2dva 5198	. . . 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 2798	. . 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 2760	. . . . . . . 8 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
40		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))
41	19, 4	fssd 6720	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
42	19	fdmd 6713	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43		limcrcl 26101	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
44	38, 43	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
45	44	simp2d 1161	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ)
46	42, 45	eqsstrrd 3966	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
47	44	simp3d 1162	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
48	39, 2, 40, 41, 46, 47	ellimc 26100	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
50	2	cnfldtop 25009	. . . . . . . 8 ⊢ 𝐾 ∈ Top
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
52	29	fmpttd 7108	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):(𝐴 ∪ {𝐵})⟶𝐷)
53	47	snssd 4747	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
54	46, 53	unssd 4138	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
55		resttopon 23386	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
56	3, 54, 55	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
57		toponuni 23139	. . . . . . . . . 10 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
58	56, 57	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
59	58	feq2d 6686	. . . . . . . 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 235	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶𝐷)
61		eqid 2760	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
62	3	toponunii 23141	. . . . . . . 8 ⊢ ℂ = ∪ 𝐾
63	61, 62	cnprest2 23515	. . . . . . 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 1398	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵))
66	1	oveq2i 7424	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))
67	66	fveq1i 6879	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ↾_t 𝐷))‘𝐵)
68	65, 67	eleqtrrdi 2871	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
69		iftrue 4488	. . . . . . 7 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)) = 𝐶)
70		ssun2 4125	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
71		snssg 4744	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
72	47, 71	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
73	70, 72	mpbiri 261	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
74	40, 69, 73, 38	fvmptd3 7010	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵) = 𝐶)
75	74	fveq2d 6882	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)) = ((𝐽 CnP 𝐾)‘𝐶))
76	9, 75	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))‘𝐵)))
77		cnpco 23492	. . . 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 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, (𝐹‘𝑥)))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
79	37, 78	eqeltrrd 2861	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐺‘𝐶), ((𝐺 ∘ 𝐹)‘𝑥))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
80		eqid 2760	. . 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 6727	. . . 4 ⊢ ((𝐺:𝐷⟶ℂ ∧ 𝐹:𝐴⟶𝐷) → (𝐺 ∘ 𝐹):𝐴⟶ℂ)
82	11, 19, 81	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶ℂ)
83	39, 2, 80, 82, 46, 47	ellimc 26100	. 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 260	1 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) lim_ℂ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∪ cun 3897 ⊆ wss 3899 ifcif 4482 {csn 4584 ∪ cuni 4867 ↦ cmpt 5186 dom cdm 5655 ∘ ccom 5659 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ↾_t crest 17505 TopOpenctopn 17506 ℂ_fldccnfld 21585 Topctop 23118 TopOnctopon 23135 CnP ccnp 23450 lim_ℂ climc 26089

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fi 9381 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-fz 13562 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17239 df-slot 17274 df-ndx 17286 df-base 17302 df-plusg 17355 df-mulr 17356 df-starv 17357 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-rest 17507 df-topn 17508 df-topgen 17528 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cnp 23453 df-xms 24546 df-ms 24547 df-limc 26093

This theorem is used by: limcco 26120 dvcjbr 26176 dvcnvlem 26203

W3C validator