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Theorem icchmeo 25069

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 11180. (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 25007	. . . . . 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 25011	. . . . . . . . . 10 ⊢ II = (𝐽 ↾_t (0[,]1))
6	5	eqcomi 2778	. . . . . . . . 9 ⊢ (𝐽 ↾_t (0[,]1)) = II
7	6	oveq2i 7422	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) = (II Cn II)
8	4	cnfldtop 24909	. . . . . . . . 9 ⊢ 𝐽 ∈ Top
9		cnrest2r 23413	. . . . . . . . 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 3992	. . . . . . 7 ⊢ (II Cn II) ⊆ (II Cn 𝐽)
12	3	cnmptid 23787	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
13	11, 12	sselid 3943	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn 𝐽))
14	4	cnfldtopon 24908	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐽 ∈ (TopOn‘ℂ))
16		simp2 1153	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
17	16	recnd 11237	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ)
18	3, 15, 17	cnmptc 23788	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐵) ∈ (II Cn 𝐽))
19	4	mpomulcn 24995	. . . . . . 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 7420	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝐵) → (𝑢 · 𝑣) = (𝑥 · 𝐵))
22	3, 13, 18, 15, 15, 20, 21	cnmpt12 23793	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (𝑥 · 𝐵)) ∈ (II Cn 𝐽))
23		1cnd 11202	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 ∈ ℂ)
24	3, 15, 23	cnmptc 23788	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 1) ∈ (II Cn 𝐽))
25	4	subcn 24993	. . . . . . . 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 23792	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn 𝐽))
28		simp1 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
29	28	recnd 11237	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ)
30	3, 15, 29	cnmptc 23788	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ (II Cn 𝐽))
31		oveq12 7420	. . . . . 6 ⊢ ((𝑢 = (1 − 𝑥) ∧ 𝑣 = 𝐴) → (𝑢 · 𝑣) = ((1 − 𝑥) · 𝐴))
32	3, 27, 30, 15, 15, 20, 31	cnmpt12 23793	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((1 − 𝑥) · 𝐴)) ∈ (II Cn 𝐽))
33	4	addcn 24992	. . . . . 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 23792	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ∈ (II Cn 𝐽))
36	1, 35	eqeltrid 2873	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn 𝐽))
37	1	iccf1o 13523	. . . . . 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 499	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵))
39		f1of 6821	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → 𝐹:(0[,]1)⟶(𝐴[,]𝐵))
40		frn 6714	. . . . 5 ⊢ (𝐹:(0[,]1)⟶(𝐴[,]𝐵) → ran 𝐹 ⊆ (𝐴[,]𝐵))
41	38, 39, 40	3syl 19	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran 𝐹 ⊆ (𝐴[,]𝐵))
42		iccssre 13456	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
43	42	3adant3 1148	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
44		ax-resscn 11157	. . . . 5 ⊢ ℝ ⊆ ℂ
45	43, 44	sstrdi 3957	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℂ)
46		cnrest2 23412	. . . 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 1492	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹 ∈ (II Cn 𝐽) ↔ 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵)))))
48	36, 47	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵))))
49	37	simprd 500	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))
50		resttopon 23287	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
51	14, 45, 50	sylancr 598	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
52		cnrest2r 23413	. . . . . . . . 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 23787	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))))
55	53, 54	sselid 3943	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
56	51, 15, 29	cnmptc 23788	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝐴) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
57	51, 55, 56, 26	cnmpt12f 23792	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 − 𝐴)) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
58		difrp 13056	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ⁺))
59	58	biimp3a 1495	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ⁺)
60		rpcnne0 13035	. . . . . . 7 ⊢ ((𝐵 − 𝐴) ∈ ℝ⁺ → ((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0))
61	4	divccn 25001	. . . . . . 7 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / (𝐵 − 𝐴))) ∈ (𝐽 Cn 𝐽))
62	59, 60, 61	3syl 19	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ ℂ ↦ (𝑥 / (𝐵 − 𝐴))) ∈ (𝐽 Cn 𝐽))
63		oveq1 7418	. . . . . 6 ⊢ (𝑥 = (𝑦 − 𝐴) → (𝑥 / (𝐵 − 𝐴)) = ((𝑦 − 𝐴) / (𝐵 − 𝐴)))
64	51, 57, 15, 62, 63	cnmpt11 23789	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
65	49, 64	eqeltrd 2869	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
66		dfdm4 5886	. . . . . . 7 ⊢ dom 𝐹 = ran ^◡𝐹
67	66	eqimss2i 4006	. . . . . 6 ⊢ ran ^◡𝐹 ⊆ dom 𝐹
68		f1odm 6825	. . . . . . 7 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → dom 𝐹 = (0[,]1))
69	38, 68	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → dom 𝐹 = (0[,]1))
70	67, 69	sseqtrid 3987	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran ^◡𝐹 ⊆ (0[,]1))
71		unitsscn 13527	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
72	71	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ⊆ ℂ)
73		cnrest2 23412	. . . . 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 1492	. . . 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 235	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1))))
76	5	oveq2i 7422	. . 3 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II) = ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))
77	75, 76	eleqtrrdi 2880	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II))
78		ishmeo 23885	. 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 594	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾_t (𝐴[,]𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ⊆ wss 3913 class class class wbr 5113 ↦ cmpt 5196 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 − cmin 11441 / cdiv 11871 ℝ⁺crp 13016 [,]cicc 13375 ↾_t crest 17473 TopOpenctopn 17474 ℂ_fldccnfld 21491 Topctop 23019 TopOnctopon 23036 Cn ccn 23350 ×_t ctx 23686 Homeochmeo 23879 IIcii 25003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cn 23353 df-cnp 23354 df-tx 23688 df-hmeo 23881 df-xms 24446 df-ms 24447 df-tms 24448 df-ii 25005

This theorem is referenced by: xrhmph 25075

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