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Theorem icchmeo 24926

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 11109. (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 24864	. . . . . 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 24868	. . . . . . . . . 10 ⊢ II = (𝐽 ↾_t (0[,]1))
6	5	eqcomi 2748	. . . . . . . . 9 ⊢ (𝐽 ↾_t (0[,]1)) = II
7	6	oveq2i 7367	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) = (II Cn II)
8	4	cnfldtop 24766	. . . . . . . . 9 ⊢ 𝐽 ∈ Top
9		cnrest2r 23270	. . . . . . . . 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 3962	. . . . . . 7 ⊢ (II Cn II) ⊆ (II Cn 𝐽)
12	3	cnmptid 23644	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
13	11, 12	sselid 3913	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn 𝐽))
14	4	cnfldtopon 24765	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐽 ∈ (TopOn‘ℂ))
16		simp2 1143	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
17	16	recnd 11164	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ)
18	3, 15, 17	cnmptc 23645	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐵) ∈ (II Cn 𝐽))
19	4	mpomulcn 24852	. . . . . . 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 7365	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝐵) → (𝑢 · 𝑣) = (𝑥 · 𝐵))
22	3, 13, 18, 15, 15, 20, 21	cnmpt12 23650	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (𝑥 · 𝐵)) ∈ (II Cn 𝐽))
23		1cnd 11130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 ∈ ℂ)
24	3, 15, 23	cnmptc 23645	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 1) ∈ (II Cn 𝐽))
25	4	subcn 24850	. . . . . . . 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 23649	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn 𝐽))
28		simp1 1142	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
29	28	recnd 11164	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ)
30	3, 15, 29	cnmptc 23645	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ (II Cn 𝐽))
31		oveq12 7365	. . . . . 6 ⊢ ((𝑢 = (1 − 𝑥) ∧ 𝑣 = 𝐴) → (𝑢 · 𝑣) = ((1 − 𝑥) · 𝐴))
32	3, 27, 30, 15, 15, 20, 31	cnmpt12 23650	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((1 − 𝑥) · 𝐴)) ∈ (II Cn 𝐽))
33	4	addcn 24849	. . . . . 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 23649	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ∈ (II Cn 𝐽))
36	1, 35	eqeltrid 2843	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn 𝐽))
37	1	iccf1o 13440	. . . . . 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 495	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵))
39		f1of 6767	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → 𝐹:(0[,]1)⟶(𝐴[,]𝐵))
40		frn 6662	. . . . 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 13373	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
43	42	3adant3 1138	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
44		ax-resscn 11086	. . . . 5 ⊢ ℝ ⊆ ℂ
45	43, 44	sstrdi 3927	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℂ)
46		cnrest2 23269	. . . 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 1474	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹 ∈ (II Cn 𝐽) ↔ 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵)))))
48	36, 47	mpbid 233	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn (𝐽 ↾_t (𝐴[,]𝐵))))
49	37	simprd 496	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))
50		resttopon 23144	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
51	14, 45, 50	sylancr 593	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
52		cnrest2r 23270	. . . . . . . . 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 23644	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))))
55	53, 54	sselid 3913	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
56	51, 15, 29	cnmptc 23645	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝐴) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
57	51, 55, 56, 26	cnmpt12f 23649	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 − 𝐴)) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
58		difrp 12973	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ⁺))
59	58	biimp3a 1477	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ⁺)
60		rpcnne0 12952	. . . . . . 7 ⊢ ((𝐵 − 𝐴) ∈ ℝ⁺ → ((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0))
61	4	divccn 24858	. . . . . . 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 7363	. . . . . 6 ⊢ (𝑥 = (𝑦 − 𝐴) → (𝑥 / (𝐵 − 𝐴)) = ((𝑦 − 𝐴) / (𝐵 − 𝐴)))
64	51, 57, 15, 62, 63	cnmpt11 23646	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
65	49, 64	eqeltrd 2839	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
66		dfdm4 5837	. . . . . . 7 ⊢ dom 𝐹 = ran ^◡𝐹
67	66	eqimss2i 3976	. . . . . 6 ⊢ ran ^◡𝐹 ⊆ dom 𝐹
68		f1odm 6771	. . . . . . 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 3957	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran ^◡𝐹 ⊆ (0[,]1))
71		unitsscn 13444	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
72	71	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ⊆ ℂ)
73		cnrest2 23269	. . . . 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 1474	. . . 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 233	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1))))
76	5	oveq2i 7367	. . 3 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II) = ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))
77	75, 76	eleqtrrdi 2850	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II))
78		ishmeo 23742	. 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 589	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾_t (𝐴[,]𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ⊆ wss 3883 class class class wbr 5072 ↦ cmpt 5153 ^◡ccnv 5617 dom cdm 5618 ran crn 5619 ⟶wf 6481 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 − cmin 11368 / cdiv 11798 ℝ⁺crp 12933 [,]cicc 13292 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21347 Topctop 22876 TopOnctopon 22893 Cn ccn 23207 ×_t ctx 23543 Homeochmeo 23736 IIcii 24860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cn 23210 df-cnp 23211 df-tx 23545 df-hmeo 23738 df-xms 24303 df-ms 24304 df-tms 24305 df-ii 24862

This theorem is referenced by: xrhmph 24932

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