icopnfhmeo - Metamath Proof Explorer

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Theorem icopnfhmeo 24988

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 24987	. . . 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 11261	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
5		1xr 11318	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ^*
6		elico2 13448	. . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)1) → 𝑥 ∈ ℝ)
9	8	ssriv 3999	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ
10	9	sseli 3991	. . . . . . 7 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 ∈ ℝ)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℝ)
12		elico2 13448	. . . . . . . . . . 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 1146	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 < 1)
15	9	sseli 3991	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 ∈ ℝ)
16		1re 11259	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
17		difrp 13071	. . . . . . . . . 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 13081	. . . . . . 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 13448	. . . . . . . . . . 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 1146	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 < 1)
26		difrp 13071	. . . . . . . . . 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 13081	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑧) ∈ ℝ ∧ 0 < (1 − 𝑧)))
31		lt2mul2div 12144	. . . . . 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 839	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
33	11, 22	remulcld 11289	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 𝑤) ∈ ℝ)
34	11, 22, 33	ltsub1d 11870	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
35	11	recnd 11287	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℂ)
36		1cnd 11254	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 1 ∈ ℂ)
37	22	recnd 11287	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℂ)
38	35, 36, 37	subdid 11717	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = ((𝑧 · 1) − (𝑧 · 𝑤)))
39	35	mulridd 11276	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 1) = 𝑧)
40	39	oveq1d 7446	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · 1) − (𝑧 · 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
41	38, 40	eqtrd 2775	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
42	37, 36, 35	subdid 11717	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = ((𝑤 · 1) − (𝑤 · 𝑧)))
43	37	mulridd 11276	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 1) = 𝑤)
44	37, 35	mulcomd 11280	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 𝑧) = (𝑧 · 𝑤))
45	43, 44	oveq12d 7449	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑤 · 1) − (𝑤 · 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
46	42, 45	eqtrd 2775	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
47	41, 46	breq12d 5161	. . . . . 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 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
51	49, 50	oveq12d 7449	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
52		ovex 7464	. . . . . . 7 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
53	51, 1, 52	fvmpt 7016	. . . . . 6 ⊢ (𝑧 ∈ (0[,)1) → (𝐹‘𝑧) = (𝑧 / (1 − 𝑧)))
54		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
55		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
56	54, 55	oveq12d 7449	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
57		ovex 7464	. . . . . . 7 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
58	56, 1, 57	fvmpt 7016	. . . . . 6 ⊢ (𝑤 ∈ (0[,)1) → (𝐹‘𝑤) = (𝑤 / (1 − 𝑤)))
59	53, 58	breqan12d 5164	. . . . 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 3197	. . 3 ⊢ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))
62		df-isom 6572	. . 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 18651	. . . . . 6 ⊢ ≤ ∈ TosetRel
65	64	elexi 3501	. . . . 5 ⊢ ≤ ∈ V
66	65	inex1 5323	. . . 4 ⊢ ( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V
67	65	inex1 5323	. . . 4 ⊢ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V
68		icossxr 13469	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ^*
69		icossxr 13469	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ^*
70		leiso 14495	. . . . . . . 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 7354	. . . . . 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 7353	. . . . 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 18645	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
78	64, 77	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
79		ledm 18648	. . . . . . . 8 ⊢ ℝ^* = dom ≤
80	79	psssdm 18640	. . . . . . 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 18640	. . . . . . 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 23826	. . . 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 1460	. . 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 24844	. . . . . 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 13458	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)1)) → (𝑥[,]𝑦) ⊆ (0[,)1))
93	68, 92	ordtrestixx 23246	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
94	91, 93	eqtri 2763	. . . 4 ⊢ (𝐽 ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
95		rge0ssre 13493	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
96	88, 89	xrrest2 24844	. . . . . 6 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞))
98		iccssico2 13458	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥[,]𝑦) ⊆ (0[,)+∞))
99	69, 98	ordtrestixx 23246	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
100	97, 99	eqtri 2763	. . . 4 ⊢ (𝐽 ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
101	94, 100	oveq12i 7443	. . 3 ⊢ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))) = ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
102	87, 101	eleqtrri 2838	. 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 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ^◡ccnv 5688 dom cdm 5689 –1-1-onto→wf1o 6562 ‘cfv 6563 Isom wiso 6564 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℝ⁺crp 13032 [,)cico 13386 ↾_t crest 17467 TopOpenctopn 17468 ordTopcordt 17546 PosetRelcps 18622 TosetRel ctsr 18623 ℂ_fldccnfld 21382 Homeochmeo 23777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 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 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-ordt 17548 df-ps 18624 df-tsr 18625 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-hmeo 23779 df-xms 24346 df-ms 24347

This theorem is referenced by: iccpnfhmeo 24990

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