icchmeo - Metamath Proof Explorer

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

Theorem icchmeo 24908

Description: The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11118. (Revised by GG, 16-Mar-2025.)

**Hypotheses**
Ref	Expression
icchmeo.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
icchmeo.f	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))

**Assertion**
Ref	Expression
icchmeo	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾_t (𝐴[,]𝐵))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐽 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem icchmeo Dummy variables 𝑦 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		icchmeo.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))
2		iitopon 24846	. . . . . 6 ⊢ II ∈ (TopOn‘(0[,]1))
3	2	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → II ∈ (TopOn‘(0[,]1)))
4		icchmeo.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	dfii3 24850	. . . . . . . . . 10 ⊢ II = (𝐽 ↾_t (0[,]1))
6	5	eqcomi 2745	. . . . . . . . 9 ⊢ (𝐽 ↾_t (0[,]1)) = II
7	6	oveq2i 7378	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) = (II Cn II)
8	4	cnfldtop 24748	. . . . . . . . 9 ⊢ 𝐽 ∈ Top
9		cnrest2r 23252	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → (II Cn (𝐽 ↾_t (0[,]1))) ⊆ (II Cn 𝐽))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) ⊆ (II Cn 𝐽)
11	7, 10	eqsstrri 3969	. . . . . . 7 ⊢ (II Cn II) ⊆ (II Cn 𝐽)
12	3	cnmptid 23626	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
13	11, 12	sselid 3919	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn 𝐽))
14	4	cnfldtopon 24747	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐽 ∈ (TopOn‘ℂ))
16		simp2 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
17	16	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ)
18	3, 15, 17	cnmptc 23627	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐵) ∈ (II Cn 𝐽))
19	4	mpomulcn 24834	. . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
20	19	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
21		oveq12 7376	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝐵) → (𝑢 · 𝑣) = (𝑥 · 𝐵))
22	3, 13, 18, 15, 15, 20, 21	cnmpt12 23632	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (𝑥 · 𝐵)) ∈ (II Cn 𝐽))
23		1cnd 11139	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 ∈ ℂ)
24	3, 15, 23	cnmptc 23627	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 1) ∈ (II Cn 𝐽))
25	4	subcn 24832	. . . . . . . 8 ⊢ − ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
26	25	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → − ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
27	3, 24, 13, 26	cnmpt12f 23631	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn 𝐽))
28		simp1 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
29	28	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ)
30	3, 15, 29	cnmptc 23627	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ (II Cn 𝐽))
31		oveq12 7376	. . . . . 6 ⊢ ((𝑢 = (1 − 𝑥) ∧ 𝑣 = 𝐴) → (𝑢 · 𝑣) = ((1 − 𝑥) · 𝐴))
32	3, 27, 30, 15, 15, 20, 31	cnmpt12 23632	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((1 − 𝑥) · 𝐴)) ∈ (II Cn 𝐽))
33	4	addcn 24831	. . . . . 6 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
34	33	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
35	3, 22, 32, 34	cnmpt12f 23631	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ∈ (II Cn 𝐽))
36	1, 35	eqeltrid 2840	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn 𝐽))
37	1	iccf1o 13449	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) ∧ ^◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴)))))
38	37	simpld 494	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵))
39		f1of 6780	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → 𝐹:(0[,]1)⟶(𝐴[,]𝐵))
40		frn 6675	. . . . 5 ⊢ (𝐹:(0[,]1)⟶(𝐴[,]𝐵) → ran 𝐹 ⊆ (𝐴[,]𝐵))
41	38, 39, 40	3syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran 𝐹 ⊆ (𝐴[,]𝐵))
42		iccssre 13382	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
43	42	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
44		ax-resscn 11095	. . . . 5 ⊢ ℝ ⊆ ℂ
45	43, 44	sstrdi 3934	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℂ)
46		cnrest2 23251	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ran 𝐹 ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐹 ∈ (II Cn 𝐽) ↔ 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵)))))
47	14, 41, 45, 46	mp3an2i 1469	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹 ∈ (II Cn 𝐽) ↔ 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵)))))
48	36, 47	mpbid 232	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵))))
49	37	simprd 495	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))
50		resttopon 23126	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
51	14, 45, 50	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
52		cnrest2r 23252	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))) ⊆ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
53	8, 52	ax-mp 5	. . . . . . . 8 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))) ⊆ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽)
54	51	cnmptid 23626	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))))
55	53, 54	sselid 3919	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
56	51, 15, 29	cnmptc 23627	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝐴) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
57	51, 55, 56, 26	cnmpt12f 23631	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 − 𝐴)) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
58		difrp 12982	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ⁺))
59	58	biimp3a 1472	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ⁺)
60		rpcnne0 12961	. . . . . . 7 ⊢ ((𝐵 − 𝐴) ∈ ℝ⁺ → ((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0))
61	4	divccn 24840	. . . . . . 7 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / (𝐵 − 𝐴))) ∈ (𝐽 Cn 𝐽))
62	59, 60, 61	3syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ ℂ ↦ (𝑥 / (𝐵 − 𝐴))) ∈ (𝐽 Cn 𝐽))
63		oveq1 7374	. . . . . 6 ⊢ (𝑥 = (𝑦 − 𝐴) → (𝑥 / (𝐵 − 𝐴)) = ((𝑦 − 𝐴) / (𝐵 − 𝐴)))
64	51, 57, 15, 62, 63	cnmpt11 23628	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
65	49, 64	eqeltrd 2836	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
66		dfdm4 5850	. . . . . . 7 ⊢ dom 𝐹 = ran ^◡𝐹
67	66	eqimss2i 3983	. . . . . 6 ⊢ ran ^◡𝐹 ⊆ dom 𝐹
68		f1odm 6784	. . . . . . 7 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → dom 𝐹 = (0[,]1))
69	38, 68	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → dom 𝐹 = (0[,]1))
70	67, 69	sseqtrid 3964	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran ^◡𝐹 ⊆ (0[,]1))
71		unitsscn 13453	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
72	71	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ⊆ ℂ)
73		cnrest2 23251	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ran ^◡𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℂ) → (^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽) ↔ ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))))
74	14, 70, 72, 73	mp3an2i 1469	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽) ↔ ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))))
75	65, 74	mpbid 232	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1))))
76	5	oveq2i 7378	. . 3 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II) = ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))
77	75, 76	eleqtrrdi 2847	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II))
78		ishmeo 23724	. 2 ⊢ (𝐹 ∈ (IIHomeo(𝐽 ↾_t (𝐴[,]𝐵))) ↔ (𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵))) ∧ ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II)))
79	48, 77, 78	sylanbrc 584	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾_t (𝐴[,]𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ⟶wf 6494 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 − cmin 11377 / cdiv 11807 ℝ⁺crp 12942 [,]cicc 13301 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21352 Topctop 22858 TopOnctopon 22875 Cn ccn 23189 ×_t ctx 23525 Homeochmeo 23718 IIcii 24842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-ii 24844

This theorem is referenced by: xrhmph 24914

W3C validator