icopnfhmeo - Metamath Proof Explorer

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Theorem icopnfhmeo 24962

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 24961	. . . 4 ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
3	2	simpli 482	. . 3 ⊢ 𝐹:(0[,)1)–1-1-onto→(0[,)+∞)
4		0re 11268	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
5		1xr 11325	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ^*
6		elico2 13444	. . . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)1) → 𝑥 ∈ ℝ)
9	8	ssriv 3983	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ
10	9	sseli 3975	. . . . . . 7 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 ∈ ℝ)
11	10	adantr 479	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℝ)
12		elico2 13444	. . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 < 1)
15	9	sseli 3975	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 ∈ ℝ)
16		1re 11266	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
17		difrp 13068	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
18	15, 16, 17	sylancl 584	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
19	14, 18	mpbid 231	. . . . . . . 8 ⊢ (𝑤 ∈ (0[,)1) → (1 − 𝑤) ∈ ℝ⁺)
20	19	rpregt0d 13078	. . . . . . 7 ⊢ (𝑤 ∈ (0[,)1) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
21	20	adantl 480	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
22	15	adantl 480	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℝ)
23		elico2 13444	. . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 < 1)
26		difrp 13068	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
27	10, 16, 26	sylancl 584	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
28	25, 27	mpbid 231	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,)1) → (1 − 𝑧) ∈ ℝ⁺)
29	28	adantr 479	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (1 − 𝑧) ∈ ℝ⁺)
30	29	rpregt0d 13078	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑧) ∈ ℝ ∧ 0 < (1 − 𝑧)))
31		lt2mul2div 12146	. . . . . 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 837	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
33	11, 22	remulcld 11296	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 𝑤) ∈ ℝ)
34	11, 22, 33	ltsub1d 11875	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
35	11	recnd 11294	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℂ)
36		1cnd 11261	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 1 ∈ ℂ)
37	22	recnd 11294	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℂ)
38	35, 36, 37	subdid 11722	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = ((𝑧 · 1) − (𝑧 · 𝑤)))
39	35	mulridd 11283	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 1) = 𝑧)
40	39	oveq1d 7441	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · 1) − (𝑧 · 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
41	38, 40	eqtrd 2766	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
42	37, 36, 35	subdid 11722	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = ((𝑤 · 1) − (𝑤 · 𝑧)))
43	37	mulridd 11283	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 1) = 𝑤)
44	37, 35	mulcomd 11287	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 𝑧) = (𝑧 · 𝑤))
45	43, 44	oveq12d 7444	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑤 · 1) − (𝑤 · 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
46	42, 45	eqtrd 2766	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
47	41, 46	breq12d 5168	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
48	34, 47	bitr4d 281	. . . . 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 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
51	49, 50	oveq12d 7444	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
52		ovex 7459	. . . . . . 7 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
53	51, 1, 52	fvmpt 7011	. . . . . 6 ⊢ (𝑧 ∈ (0[,)1) → (𝐹‘𝑧) = (𝑧 / (1 − 𝑧)))
54		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
55		oveq2 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
56	54, 55	oveq12d 7444	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
57		ovex 7459	. . . . . . 7 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
58	56, 1, 57	fvmpt 7011	. . . . . 6 ⊢ (𝑤 ∈ (0[,)1) → (𝐹‘𝑤) = (𝑤 / (1 − 𝑤)))
59	53, 58	breqan12d 5171	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
60	32, 48, 59	3bitr4d 310	. . . 4 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤)))
61	60	rgen2 3188	. . 3 ⊢ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))
62		df-isom 6565	. . 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 18620	. . . . . 6 ⊢ ≤ ∈ TosetRel
65	64	elexi 3484	. . . . 5 ⊢ ≤ ∈ V
66	65	inex1 5324	. . . 4 ⊢ ( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V
67	65	inex1 5324	. . . 4 ⊢ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V
68		icossxr 13465	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ^*
69		icossxr 13465	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ^*
70		leiso 14480	. . . . . . . 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 229	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))
73		isores1 7348	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞)))
74	72, 73	mpbi 229	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞))
75		isores2 7347	. . . . 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 229	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞))
77		tsrps 18614	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
78	64, 77	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
79		ledm 18617	. . . . . . . 8 ⊢ ℝ^* = dom ≤
80	79	psssdm 18609	. . . . . . 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 2735	. . . . 5 ⊢ (0[,)1) = dom ( ≤ ∩ ((0[,)1) × (0[,)1)))
83	79	psssdm 18609	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞))
84	78, 69, 83	mp2an 690	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞)
85	84	eqcomi 2735	. . . . 5 ⊢ (0[,)+∞) = dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))
86	82, 85	ordthmeo 23800	. . . 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 1458	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
88		icopnfhmeo.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
89		eqid 2726	. . . . . . 7 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ )
90	88, 89	xrrest2 24818	. . . . . 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 13454	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)1)) → (𝑥[,]𝑦) ⊆ (0[,)1))
93	68, 92	ordtrestixx 23220	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
94	91, 93	eqtri 2754	. . . 4 ⊢ (𝐽 ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
95		rge0ssre 13489	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
96	88, 89	xrrest2 24818	. . . . . 6 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞))
98		iccssico2 13454	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥[,]𝑦) ⊆ (0[,)+∞))
99	69, 98	ordtrestixx 23220	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
100	97, 99	eqtri 2754	. . . 4 ⊢ (𝐽 ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
101	94, 100	oveq12i 7438	. . 3 ⊢ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))) = ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
102	87, 101	eleqtrri 2825	. 2 ⊢ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞)))
103	63, 102	pm3.2i 469	1 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5155 ↦ cmpt 5238 × cxp 5682 ^◡ccnv 5683 dom cdm 5684 –1-1-onto→wf1o 6555 ‘cfv 6556 Isom wiso 6557 (class class class)co 7426 ℝcr 11159 0cc0 11160 1c1 11161 + caddc 11163 · cmul 11165 +∞cpnf 11297 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301 − cmin 11496 / cdiv 11923 ℝ⁺crp 13030 [,)cico 13382 ↾_t crest 17437 TopOpenctopn 17438 ordTopcordt 17516 PosetRelcps 18591 TosetRel ctsr 18592 ℂ_fldccnfld 21345 Homeochmeo 23751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fi 9456 df-sup 9487 df-inf 9488 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-q 12987 df-rp 13031 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13384 df-ioc 13385 df-ico 13386 df-icc 13387 df-fz 13541 df-seq 14024 df-exp 14084 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-struct 17151 df-slot 17186 df-ndx 17198 df-base 17216 df-plusg 17281 df-mulr 17282 df-starv 17283 df-tset 17287 df-ple 17288 df-ds 17290 df-unif 17291 df-rest 17439 df-topn 17440 df-topgen 17460 df-ordt 17518 df-ps 18593 df-tsr 18594 df-psmet 21337 df-xmet 21338 df-met 21339 df-bl 21340 df-mopn 21341 df-cnfld 21346 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cn 23225 df-hmeo 23753 df-xms 24320 df-ms 24321

This theorem is referenced by: iccpnfhmeo 24964

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