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Theorem icchmeo 24845

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 11155. (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 24779	. . . . . 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 24783	. . . . . . . . . 10 ⊢ II = (𝐽 ↾_t (0[,]1))
6	5	eqcomi 2739	. . . . . . . . 9 ⊢ (𝐽 ↾_t (0[,]1)) = II
7	6	oveq2i 7401	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) = (II Cn II)
8	4	cnfldtop 24678	. . . . . . . . 9 ⊢ 𝐽 ∈ Top
9		cnrest2r 23181	. . . . . . . . 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 3997	. . . . . . 7 ⊢ (II Cn II) ⊆ (II Cn 𝐽)
12	3	cnmptid 23555	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
13	11, 12	sselid 3947	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn 𝐽))
14	4	cnfldtopon 24677	. . . . . . . 8 ⊢ 𝐽 ∈ (TopOn‘ℂ)
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐽 ∈ (TopOn‘ℂ))
16		simp2 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
17	16	recnd 11209	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ)
18	3, 15, 17	cnmptc 23556	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐵) ∈ (II Cn 𝐽))
19	4	mpomulcn 24765	. . . . . . 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 7399	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝐵) → (𝑢 · 𝑣) = (𝑥 · 𝐵))
22	3, 13, 18, 15, 15, 20, 21	cnmpt12 23561	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (𝑥 · 𝐵)) ∈ (II Cn 𝐽))
23		1cnd 11176	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 ∈ ℂ)
24	3, 15, 23	cnmptc 23556	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 1) ∈ (II Cn 𝐽))
25	4	subcn 24762	. . . . . . . 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 23560	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn 𝐽))
28		simp1 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
29	28	recnd 11209	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ)
30	3, 15, 29	cnmptc 23556	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ (II Cn 𝐽))
31		oveq12 7399	. . . . . 6 ⊢ ((𝑢 = (1 − 𝑥) ∧ 𝑣 = 𝐴) → (𝑢 · 𝑣) = ((1 − 𝑥) · 𝐴))
32	3, 27, 30, 15, 15, 20, 31	cnmpt12 23561	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((1 − 𝑥) · 𝐴)) ∈ (II Cn 𝐽))
33	4	addcn 24761	. . . . . 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 23560	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ∈ (II Cn 𝐽))
36	1, 35	eqeltrid 2833	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn 𝐽))
37	1	iccf1o 13464	. . . . . 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 6803	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → 𝐹:(0[,]1)⟶(𝐴[,]𝐵))
40		frn 6698	. . . . 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 13397	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
43	42	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
44		ax-resscn 11132	. . . . 5 ⊢ ℝ ⊆ ℂ
45	43, 44	sstrdi 3962	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℂ)
46		cnrest2 23180	. . . 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 1468	. . 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 23055	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
51	14, 45, 50	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐽 ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
52		cnrest2r 23181	. . . . . . . . 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 23555	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))))
55	53, 54	sselid 3947	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
56	51, 15, 29	cnmptc 23556	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝐴) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
57	51, 55, 56, 26	cnmpt12f 23560	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 − 𝐴)) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
58		difrp 12998	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ⁺))
59	58	biimp3a 1471	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐵 − 𝐴) ∈ ℝ⁺)
60		rpcnne0 12977	. . . . . . 7 ⊢ ((𝐵 − 𝐴) ∈ ℝ⁺ → ((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0))
61	4	divccn 24771	. . . . . . 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 7397	. . . . . 6 ⊢ (𝑥 = (𝑦 − 𝐴) → (𝑥 / (𝐵 − 𝐴)) = ((𝑦 − 𝐴) / (𝐵 − 𝐴)))
64	51, 57, 15, 62, 63	cnmpt11 23557	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
65	49, 64	eqeltrd 2829	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
66		dfdm4 5862	. . . . . . 7 ⊢ dom 𝐹 = ran ^◡𝐹
67	66	eqimss2i 4011	. . . . . 6 ⊢ ran ^◡𝐹 ⊆ dom 𝐹
68		f1odm 6807	. . . . . . 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 3992	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran ^◡𝐹 ⊆ (0[,]1))
71		unitsscn 13468	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
72	71	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ⊆ ℂ)
73		cnrest2 23180	. . . . 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 1468	. . . 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 7401	. . 3 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II) = ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))
77	75, 76	eleqtrrdi 2840	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II))
78		ishmeo 23653	. 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 583	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ⊆ wss 3917 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 − cmin 11412 / cdiv 11842 ℝ⁺crp 12958 [,]cicc 13316 ↾_t crest 17390 TopOpenctopn 17391 ℂ_fldccnfld 21271 Topctop 22787 TopOnctopon 22804 Cn ccn 23118 ×_t ctx 23454 Homeochmeo 23647 IIcii 24775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-ii 24777

This theorem is referenced by: xrhmph 24852

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