icopnfhmeo - Metamath Proof Explorer

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

Theorem icopnfhmeo 25172

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 25171	. . . 4 ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
3	2	simpli 489	. . 3 ⊢ 𝐹:(0[,)1)–1-1-onto→(0[,)+∞)
4		0re 11237	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
5		1xr 11295	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ^*
6		elico2 13465	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1)))
7	4, 5, 6	mp2an 705	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,)1) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1))
8	7	simp1bi 1163	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)1) → 𝑥 ∈ ℝ)
9	8	ssriv 3938	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ
10	9	sseli 3930	. . . . . . 7 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 ∈ ℝ)
11	10	adantr 486	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℝ)
12		elico2 13465	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1)))
13	4, 5, 12	mp2an 705	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1))
14	13	simp3bi 1165	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 < 1)
15	9	sseli 3930	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → 𝑤 ∈ ℝ)
16		1re 11235	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
17		difrp 13084	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
18	15, 16, 17	sylancl 598	. . . . . . . . 9 ⊢ (𝑤 ∈ (0[,)1) → (𝑤 < 1 ↔ (1 − 𝑤) ∈ ℝ⁺))
19	14, 18	mpbid 235	. . . . . . . 8 ⊢ (𝑤 ∈ (0[,)1) → (1 − 𝑤) ∈ ℝ⁺)
20	19	rpregt0d 13094	. . . . . . 7 ⊢ (𝑤 ∈ (0[,)1) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
21	20	adantl 487	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((1 − 𝑤) ∈ ℝ ∧ 0 < (1 − 𝑤)))
22	15	adantl 487	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℝ)
23		elico2 13465	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1)))
24	4, 5, 23	mp2an 705	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1))
25	24	simp3bi 1165	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → 𝑧 < 1)
26		difrp 13084	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
27	10, 16, 26	sylancl 598	. . . . . . . . 9 ⊢ (𝑧 ∈ (0[,)1) → (𝑧 < 1 ↔ (1 − 𝑧) ∈ ℝ⁺))
28	25, 27	mpbid 235	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,)1) → (1 − 𝑧) ∈ ℝ⁺)
29	28	adantr 486	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (1 − 𝑧) ∈ ℝ⁺)
30	29	rpregt0d 13094	. . . . . 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 852	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
33	11, 22	remulcld 11266	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 𝑤) ∈ ℝ)
34	11, 22, 33	ltsub1d 11850	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
35	11	recnd 11264	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑧 ∈ ℂ)
36		1cnd 11229	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 1 ∈ ℂ)
37	22	recnd 11264	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → 𝑤 ∈ ℂ)
38	35, 36, 37	subdid 11697	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = ((𝑧 · 1) − (𝑧 · 𝑤)))
39	35	mulridd 11253	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · 1) = 𝑧)
40	39	oveq1d 7431	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · 1) − (𝑧 · 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
41	38, 40	eqtrd 2797	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 · (1 − 𝑤)) = (𝑧 − (𝑧 · 𝑤)))
42	37, 36, 35	subdid 11697	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = ((𝑤 · 1) − (𝑤 · 𝑧)))
43	37	mulridd 11253	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 1) = 𝑤)
44	37, 35	mulcomd 11257	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · 𝑧) = (𝑧 · 𝑤))
45	43, 44	oveq12d 7434	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑤 · 1) − (𝑤 · 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
46	42, 45	eqtrd 2797	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑤 · (1 − 𝑧)) = (𝑤 − (𝑧 · 𝑤)))
47	41, 46	breq12d 5120	. . . . . 6 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧)) ↔ (𝑧 − (𝑧 · 𝑤)) < (𝑤 − (𝑧 · 𝑤))))
48	34, 47	bitr4d 285	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝑧 · (1 − 𝑤)) < (𝑤 · (1 − 𝑧))))
49		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
50		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
51	49, 50	oveq12d 7434	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
52		ovex 7449	. . . . . . 7 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
53	51, 1, 52	fvmpt 6990	. . . . . 6 ⊢ (𝑧 ∈ (0[,)1) → (𝐹‘𝑧) = (𝑧 / (1 − 𝑧)))
54		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
55		oveq2 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
56	54, 55	oveq12d 7434	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
57		ovex 7449	. . . . . . 7 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
58	56, 1, 57	fvmpt 6990	. . . . . 6 ⊢ (𝑤 ∈ (0[,)1) → (𝐹‘𝑤) = (𝑤 / (1 − 𝑤)))
59	53, 58	breqan12d 5123	. . . . 5 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤))))
60	32, 48, 59	3bitr4d 314	. . . 4 ⊢ ((𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤)))
61	60	rgen2 3204	. . 3 ⊢ ∀𝑧 ∈ (0[,)1)∀𝑤 ∈ (0[,)1)(𝑧 < 𝑤 ↔ (𝐹‘𝑧) < (𝐹‘𝑤))
62		df-isom 6546	. . 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 724	. 2 ⊢ 𝐹 Isom < , < ((0[,)1), (0[,)+∞))
64		letsr 18685	. . . . . 6 ⊢ ≤ ∈ TosetRel
65	64	elexi 3475	. . . . 5 ⊢ ≤ ∈ V
66	65	inex1 5284	. . . 4 ⊢ ( ≤ ∩ ((0[,)1) × (0[,)1))) ∈ V
67	65	inex1 5284	. . . 4 ⊢ ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) ∈ V
68		icossxr 13487	. . . . . . . 8 ⊢ (0[,)1) ⊆ ℝ^*
69		icossxr 13487	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ^*
70		leiso 14526	. . . . . . . 8 ⊢ (((0[,)1) ⊆ ℝ^* ∧ (0[,)+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))))
71	68, 69, 70	mp2an 705	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)))
72	63, 71	mpbi 233	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞))
73		isores1 7338	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,)1), (0[,)+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞)))
74	72, 73	mpbi 233	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ≤ ((0[,)1), (0[,)+∞))
75		isores2 7337	. . . . 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 233	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,)1) × (0[,)1))), ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))((0[,)1), (0[,)+∞))
77		tsrps 18679	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
78	64, 77	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
79		ledm 18682	. . . . . . . 8 ⊢ ℝ^* = dom ≤
80	79	psssdm 18674	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1))
81	78, 68, 80	mp2an 705	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)1) × (0[,)1))) = (0[,)1)
82	81	eqcomi 2771	. . . . 5 ⊢ (0[,)1) = dom ( ≤ ∩ ((0[,)1) × (0[,)1)))
83	79	psssdm 18674	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,)+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞))
84	78, 69, 83	mp2an 705	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞))) = (0[,)+∞)
85	84	eqcomi 2771	. . . . 5 ⊢ (0[,)+∞) = dom ( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))
86	82, 85	ordthmeo 24029	. . . 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 1490	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
88		icopnfhmeo.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
89		eqid 2762	. . . . . . 7 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ )
90	88, 89	xrrest2 25036	. . . . . 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 13475	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)1) ∧ 𝑦 ∈ (0[,)1)) → (𝑥[,]𝑦) ⊆ (0[,)1))
93	68, 92	ordtrestixx 23448	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
94	91, 93	eqtri 2785	. . . 4 ⊢ (𝐽 ↾_t (0[,)1)) = (ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))
95		rge0ssre 13511	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
96	88, 89	xrrest2 25036	. . . . . 6 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞)))
97	95, 96	ax-mp 5	. . . . 5 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((ordTop‘ ≤ ) ↾_t (0[,)+∞))
98		iccssico2 13475	. . . . . 6 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥[,]𝑦) ⊆ (0[,)+∞))
99	69, 98	ordtrestixx 23448	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
100	97, 99	eqtri 2785	. . . 4 ⊢ (𝐽 ↾_t (0[,)+∞)) = (ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞))))
101	94, 100	oveq12i 7428	. . 3 ⊢ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))) = ((ordTop‘( ≤ ∩ ((0[,)1) × (0[,)1))))Homeo(ordTop‘( ≤ ∩ ((0[,)+∞) × (0[,)+∞)))))
102	87, 101	eleqtrri 2861	. 2 ⊢ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞)))
103	63, 102	pm3.2i 476	1 ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾_t (0[,)1))Homeo(𝐽 ↾_t (0[,)+∞))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 Vcvv 3453 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5107 ↦ cmpt 5190 × cxp 5657 ^◡ccnv 5658 dom cdm 5659 –1-1-onto→wf1o 6536 ‘cfv 6537 Isom wiso 6538 (class class class)co 7416 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11468 / cdiv 11898 ℝ⁺crp 13044 [,)cico 13402 ↾_t crest 17509 TopOpenctopn 17510 ordTopcordt 17589 PosetRelcps 18656 TosetRel ctsr 18657 ℂ_fldccnfld 21586 Homeochmeo 23980

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9384 df-sup 9415 df-inf 9416 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-seq 14068 df-exp 14128 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-struct 17243 df-slot 17278 df-ndx 17290 df-base 17306 df-plusg 17359 df-mulr 17360 df-starv 17361 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-rest 17511 df-topn 17512 df-topgen 17532 df-ordt 17591 df-ps 18658 df-tsr 18659 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cn 23453 df-hmeo 23982 df-xms 24547 df-ms 24548

This theorem is used by: iccpnfhmeo 25174

W3C validator