icopnfhmeo - Metamath Proof Explorer

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Theorem icopnfhmeo 23549

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 23548	. . . 4 ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
3	2	simpli 486	. . 3 ⊢ 𝐹:(0[,)1)–1-1-onto→(0[,)+∞)
4		0re 10645	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
5		1xr 10702	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ^*
6		elico2 12803	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1)))
7	4, 5, 6	mp2an 690	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1))
8	7	simp1bi 1141	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)1) → 𝑥 ∈ ℝ)
9	8	ssriv 3973	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ
10	9	sseli 3965	. . . . . . 7 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 ∈ ℝ)
11	10	adantr 483	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℝ)
12		elico2 12803	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1)))
13	4, 5, 12	mp2an 690	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1))
14	13	simp3bi 1143	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 < 1)
15	9	sseli 3965	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 ∈ ℝ)
16		1re 10643	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
17		difrp 12430	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
18	15, 16, 17	sylancl 588	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
19	14, 18	mpbid 234	. . . . . . . 8 ⊢ (𝑤 ∈ (0[,)1) → (1 − 𝑤) ∈ ℝ⁺)
20	19	rpregt0d 12440	. . . . . . 7 ⊢ (𝑤 ∈ (0[,)1) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
21	20	adantl 484	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
22	15	adantl 484	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℝ)
23		elico2 12803	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1)))
24	4, 5, 23	mp2an 690	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1))
25	24	simp3bi 1143	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 < 1)
26		difrp 12430	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
27	10, 16, 26	sylancl 588	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
28	25, 27	mpbid 234	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,)1) → (1 − 𝑧) ∈ ℝ⁺)
29	28	adantr 483	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (1 − 𝑧) ∈ ℝ⁺)
30	29	rpregt0d 12440	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑧) ∈ ℝ ∧ 0 < (1 − 𝑧)))
31		lt2mul2div 11520	. . . . . 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 836	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
33	11, 22	remulcld 10673	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 𝑤) ∈ ℝ)
34	11, 22, 33	ltsub1d 11251	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
35	11	recnd 10671	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℂ)
36		1cnd 10638	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 1 ∈ ℂ)
37	22	recnd 10671	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℂ)
38	35, 36, 37	subdid 11098	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = ((𝑧 · 1) − (𝑧 · 𝑤)))
39	35	mulid1d 10660	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 1) = 𝑧)
40	39	oveq1d 7173	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · 1) − (𝑧 · 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
41	38, 40	eqtrd 2858	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
42	37, 36, 35	subdid 11098	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = ((𝑤 · 1) − (𝑤 · 𝑧)))
43	37	mulid1d 10660	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 1) = 𝑤)
44	37, 35	mulcomd 10664	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 𝑧) = (𝑧 · 𝑤))
45	43, 44	oveq12d 7176	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑤 · 1) − (𝑤 · 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
46	42, 45	eqtrd 2858	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
47	41, 46	breq12d 5081	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
48	34, 47	bitr4d 284	. . . . 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 7166	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
51	49, 50	oveq12d 7176	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
52		ovex 7191	. . . . . . 7 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
53	51, 1, 52	fvmpt 6770	. . . . . 6 ⊢ (𝑧 ∈ (0[,)1) → (𝐹‘𝑧) = (𝑧 / (1 − 𝑧)))
54		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
55		oveq2 7166	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
56	54, 55	oveq12d 7176	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
57		ovex 7191	. . . . . . 7 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
58	56, 1, 57	fvmpt 6770	. . . . . 6 ⊢ (𝑤 ∈ (0[,)1) → (𝐹‘𝑤) = (𝑤 / (1 − 𝑤)))
59	53, 58	breqan12d 5084	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
60	32, 48, 59	3bitr4d 313	. . . 4 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤)))
61	60	rgen2 3205	. . 3 ⊢ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))
62		df-isom 6366	. . 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 709	. 2 ⊢ 𝐹 Isom < , < ((0[,)1), (0[,)+∞))
64		letsr 17839	. . . . . 6 ⊢ ≤ ∈ TosetRel
65	64	elexi 3515	. . . . 5 ⊢ ≤ ∈ V
66	65	inex1 5223	. . . 4 ⊢ ( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V
67	65	inex1 5223	. . . 4 ⊢ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V
68		icossxr 12824	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ^*
69		icossxr 12824	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ^*
70		leiso 13820	. . . . . . . 8 ⊢ (((0[,)1) ⊆ ℝ^* ∧ (0[,)+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))))
71	68, 69, 70	mp2an 690	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)))
72	63, 71	mpbi 232	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))
73		isores1 7089	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞)))
74	72, 73	mpbi 232	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞))
75		isores2 7088	. . . . 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 232	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞))
77		tsrps 17833	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
78	64, 77	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
79		ledm 17836	. . . . . . . 8 ⊢ ℝ^* = dom ≤
80	79	psssdm 17828	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1))
81	78, 68, 80	mp2an 690	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1)
82	81	eqcomi 2832	. . . . 5 ⊢ (0[,)1) = dom ( ≤ ∩ ((0[,)1) × (0[,)1)))
83	79	psssdm 17828	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞))
84	78, 69, 83	mp2an 690	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞)
85	84	eqcomi 2832	. . . . 5 ⊢ (0[,)+∞) = dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))
86	82, 85	ordthmeo 22412	. . . 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 1457	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
88		icopnfhmeo.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
89		eqid 2823	. . . . . . 7 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ )
90	88, 89	xrrest2 23418	. . . . . 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 12813	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)1)) → (𝑥[,]𝑦) ⊆ (0[,)1))
93	68, 92	ordtrestixx 21832	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
94	91, 93	eqtri 2846	. . . 4 ⊢ (𝐽 ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
95		rge0ssre 12847	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
96	88, 89	xrrest2 23418	. . . . . 6 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞))
98		iccssico2 12813	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥[,]𝑦) ⊆ (0[,)+∞))
99	69, 98	ordtrestixx 21832	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
100	97, 99	eqtri 2846	. . . 4 ⊢ (𝐽 ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
101	94, 100	oveq12i 7170	. . 3 ⊢ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))) = ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
102	87, 101	eleqtrri 2914	. 2 ⊢ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞)))
103	63, 102	pm3.2i 473	1 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ^◡ccnv 5556 dom cdm 5557 –1-1-onto→wf1o 6356 ‘cfv 6357 Isom wiso 6358 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 +∞cpnf 10674 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℝ⁺crp 12392 [,)cico 12743 ↾_t crest 16696 TopOpenctopn 16697 ordTopcordt 16774 PosetRelcps 17810 TosetRel ctsr 17811 ℂ_fldccnfld 20547 Homeochmeo 22363

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-ordt 16776 df-ps 17812 df-tsr 17813 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-hmeo 22365 df-xms 22932 df-ms 22933

This theorem is referenced by: iccpnfhmeo 23551

W3C validator