icopnfhmeo - Metamath Proof Explorer

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Theorem icopnfhmeo 24892

Description: The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

**Hypotheses**
Ref	Expression
icopnfhmeo.f	⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
icopnfhmeo.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
icopnfhmeo	⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))))

Distinct variable group: 𝑥,𝐽 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem icopnfhmeo Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		icopnfhmeo.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
2	1	icopnfcnv 24891	. . . 4 ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
3	2	simpli 483	. . 3 ⊢ 𝐹:(0[,)1)–1-1-onto→(0[,)+∞)
4		0re 11237	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
5		1xr 11294	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ^*
6		elico2 13427	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1)))
7	4, 5, 6	mp2an 692	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1))
8	7	simp1bi 1145	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)1) → 𝑥 ∈ ℝ)
9	8	ssriv 3962	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ
10	9	sseli 3954	. . . . . . 7 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 ∈ ℝ)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℝ)
12		elico2 13427	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1)))
13	4, 5, 12	mp2an 692	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1))
14	13	simp3bi 1147	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 < 1)
15	9	sseli 3954	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 ∈ ℝ)
16		1re 11235	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
17		difrp 13047	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
18	15, 16, 17	sylancl 586	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
19	14, 18	mpbid 232	. . . . . . . 8 ⊢ (𝑤 ∈ (0[,)1) → (1 − 𝑤) ∈ ℝ⁺)
20	19	rpregt0d 13057	. . . . . . 7 ⊢ (𝑤 ∈ (0[,)1) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
21	20	adantl 481	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
22	15	adantl 481	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℝ)
23		elico2 13427	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1)))
24	4, 5, 23	mp2an 692	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1))
25	24	simp3bi 1147	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 < 1)
26		difrp 13047	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
27	10, 16, 26	sylancl 586	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
28	25, 27	mpbid 232	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,)1) → (1 − 𝑧) ∈ ℝ⁺)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (1 − 𝑧) ∈ ℝ⁺)
30	29	rpregt0d 13057	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑧) ∈ ℝ ∧ 0 < (1 − 𝑧)))
31		lt2mul2div 12120	. . . . . 6 ⊢ (((𝑧 ∈ ℝ ∧ ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤))) ∧ (𝑤 ∈ ℝ ∧ ((1 − 𝑧) ∈ ℝ ∧ 0 < (1 − 𝑧)))) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
32	11, 21, 22, 30, 31	syl22anc 838	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
33	11, 22	remulcld 11265	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 𝑤) ∈ ℝ)
34	11, 22, 33	ltsub1d 11846	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
35	11	recnd 11263	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℂ)
36		1cnd 11230	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 1 ∈ ℂ)
37	22	recnd 11263	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℂ)
38	35, 36, 37	subdid 11693	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = ((𝑧 · 1) − (𝑧 · 𝑤)))
39	35	mulridd 11252	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 1) = 𝑧)
40	39	oveq1d 7420	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · 1) − (𝑧 · 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
41	38, 40	eqtrd 2770	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
42	37, 36, 35	subdid 11693	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = ((𝑤 · 1) − (𝑤 · 𝑧)))
43	37	mulridd 11252	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 1) = 𝑤)
44	37, 35	mulcomd 11256	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 𝑧) = (𝑧 · 𝑤))
45	43, 44	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑤 · 1) − (𝑤 · 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
46	42, 45	eqtrd 2770	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
47	41, 46	breq12d 5132	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
48	34, 47	bitr4d 282	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧))))
49		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
50		oveq2 7413	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
51	49, 50	oveq12d 7423	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
52		ovex 7438	. . . . . . 7 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
53	51, 1, 52	fvmpt 6986	. . . . . 6 ⊢ (𝑧 ∈ (0[,)1) → (𝐹‘𝑧) = (𝑧 / (1 − 𝑧)))
54		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
55		oveq2 7413	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
56	54, 55	oveq12d 7423	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
57		ovex 7438	. . . . . . 7 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
58	56, 1, 57	fvmpt 6986	. . . . . 6 ⊢ (𝑤 ∈ (0[,)1) → (𝐹‘𝑤) = (𝑤 / (1 − 𝑤)))
59	53, 58	breqan12d 5135	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
60	32, 48, 59	3bitr4d 311	. . . 4 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤)))
61	60	rgen2 3184	. . 3 ⊢ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))
62		df-isom 6540	. . 3 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))))
63	3, 61, 62	mpbir2an 711	. 2 ⊢ 𝐹 Isom < , < ((0[,)1), (0[,)+∞))
64		letsr 18603	. . . . . 6 ⊢ ≤ ∈ TosetRel
65	64	elexi 3482	. . . . 5 ⊢ ≤ ∈ V
66	65	inex1 5287	. . . 4 ⊢ ( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V
67	65	inex1 5287	. . . 4 ⊢ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V
68		icossxr 13449	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ^*
69		icossxr 13449	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ^*
70		leiso 14477	. . . . . . . 8 ⊢ (((0[,)1) ⊆ ℝ^* ∧ (0[,)+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))))
71	68, 69, 70	mp2an 692	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)))
72	63, 71	mpbi 230	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))
73		isores1 7327	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞)))
74	72, 73	mpbi 230	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞))
75		isores2 7326	. . . . 5 ⊢ (𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞)))
76	74, 75	mpbi 230	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞))
77		tsrps 18597	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
78	64, 77	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
79		ledm 18600	. . . . . . . 8 ⊢ ℝ^* = dom ≤
80	79	psssdm 18592	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1))
81	78, 68, 80	mp2an 692	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1)
82	81	eqcomi 2744	. . . . 5 ⊢ (0[,)1) = dom ( ≤ ∩ ((0[,)1) × (0[,)1)))
83	79	psssdm 18592	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞))
84	78, 69, 83	mp2an 692	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞)
85	84	eqcomi 2744	. . . . 5 ⊢ (0[,)+∞) = dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))
86	82, 85	ordthmeo 23740	. . . 4 ⊢ ((( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V ∧ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V ∧ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞))) → 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))))
87	66, 67, 76, 86	mp3an 1463	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
88		icopnfhmeo.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
89		eqid 2735	. . . . . . 7 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ )
90	88, 89	xrrest2 24748	. . . . . 6 ⊢ ((0[,)1) ⊆ ℝ → (𝐽 ↾_t (0[,)1)) = ((ordTop‘ ≤ ) ↾_t (0[,)1)))
91	9, 90	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)1)) = ((ordTop‘ ≤ ) ↾_t (0[,)1))
92		iccssico2 13437	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)1)) → (𝑥[,]𝑦) ⊆ (0[,)1))
93	68, 92	ordtrestixx 23160	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
94	91, 93	eqtri 2758	. . . 4 ⊢ (𝐽 ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
95		rge0ssre 13473	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
96	88, 89	xrrest2 24748	. . . . . 6 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞))
98		iccssico2 13437	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥[,]𝑦) ⊆ (0[,)+∞))
99	69, 98	ordtrestixx 23160	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
100	97, 99	eqtri 2758	. . . 4 ⊢ (𝐽 ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
101	94, 100	oveq12i 7417	. . 3 ⊢ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))) = ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
102	87, 101	eleqtrri 2833	. 2 ⊢ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞)))
103	63, 102	pm3.2i 470	1 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 ∩ cin 3925 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 –1-1-onto→wf1o 6530 ‘cfv 6531 Isom wiso 6532 (class class class)co 7405 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 ℝ⁺crp 13008 [,)cico 13364 ↾_t crest 17434 TopOpenctopn 17435 ordTopcordt 17513 PosetRelcps 18574 TosetRel ctsr 18575 ℂ_fldccnfld 21315 Homeochmeo 23691

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-iin 4970 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-isom 6540 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-2o 8481 df-er 8719 df-map 8842 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-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 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-ordt 17515 df-ps 18576 df-tsr 18577 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-cn 23165 df-hmeo 23693 df-xms 24259 df-ms 24260

This theorem is referenced by: iccpnfhmeo 24894

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