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Theorem icchmeo 24971

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 11235. (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 24905	. . . . . 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 24909	. . . . . . . . . 10 ⊢ II = (𝐽 ↾_t (0[,]1))
6	5	eqcomi 2746	. . . . . . . . 9 ⊢ (𝐽 ↾_t (0[,]1)) = II
7	6	oveq2i 7442	. . . . . . . 8 ⊢ (II Cn (𝐽 ↾_t (0[,]1))) = (II Cn II)
8	4	cnfldtop 24804	. . . . . . . . 9 ⊢ 𝐽 ∈ Top
9		cnrest2r 23295	. . . . . . . . 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 4031	. . . . . . 7 ⊢ (II Cn II) ⊆ (II Cn 𝐽)
12	3	cnmptid 23669	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
13	11, 12	sselid 3981	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn 𝐽))
14	4	cnfldtopon 24803	. . . . . . . 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 11289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℂ)
18	3, 15, 17	cnmptc 23670	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐵) ∈ (II Cn 𝐽))
19	4	mpomulcn 24891	. . . . . . 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 7440	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝐵) → (𝑢 · 𝑣) = (𝑥 · 𝐵))
22	3, 13, 18, 15, 15, 20, 21	cnmpt12 23675	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ (𝑥 · 𝐵)) ∈ (II Cn 𝐽))
23		1cnd 11256	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 1 ∈ ℂ)
24	3, 15, 23	cnmptc 23670	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 1) ∈ (II Cn 𝐽))
25	4	subcn 24888	. . . . . . . 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 23674	. . . . . 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 11289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℂ)
30	3, 15, 29	cnmptc 23670	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ (II Cn 𝐽))
31		oveq12 7440	. . . . . 6 ⊢ ((𝑢 = (1 − 𝑥) ∧ 𝑣 = 𝐴) → (𝑢 · 𝑣) = ((1 − 𝑥) · 𝐴))
32	3, 27, 30, 15, 15, 20, 31	cnmpt12 23675	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((1 − 𝑥) · 𝐴)) ∈ (II Cn 𝐽))
33	4	addcn 24887	. . . . . 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 23674	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ∈ (II Cn 𝐽))
36	1, 35	eqeltrid 2845	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (II Cn 𝐽))
37	1	iccf1o 13536	. . . . . 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 6848	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) → 𝐹:(0[,]1)⟶(𝐴[,]𝐵))
40		frn 6743	. . . . 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 13469	. . . . . 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 11212	. . . . 5 ⊢ ℝ ⊆ ℂ
45	43, 44	sstrdi 3996	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴[,]𝐵) ⊆ ℂ)
46		cnrest2 23294	. . . 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 23169	. . . . . . 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 23295	. . . . . . . . 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 23669	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (𝐴[,]𝐵))))
55	53, 54	sselid 3981	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝑦) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
56	51, 15, 29	cnmptc 23670	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ 𝐴) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
57	51, 55, 56, 26	cnmpt12f 23674	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 − 𝐴)) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
58		difrp 13073	. . . . . . . 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 13053	. . . . . . 7 ⊢ ((𝐵 − 𝐴) ∈ ℝ⁺ → ((𝐵 − 𝐴) ∈ ℂ ∧ (𝐵 − 𝐴) ≠ 0))
61	4	divccn 24897	. . . . . . 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 7438	. . . . . 6 ⊢ (𝑥 = (𝑦 − 𝐴) → (𝑥 / (𝐵 − 𝐴)) = ((𝑦 − 𝐴) / (𝐵 − 𝐴)))
64	51, 57, 15, 62, 63	cnmpt11 23671	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))) ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
65	49, 64	eqeltrd 2841	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn 𝐽))
66		dfdm4 5906	. . . . . . 7 ⊢ dom 𝐹 = ran ^◡𝐹
67	66	eqimss2i 4045	. . . . . 6 ⊢ ran ^◡𝐹 ⊆ dom 𝐹
68		f1odm 6852	. . . . . . 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 4026	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ran ^◡𝐹 ⊆ (0[,]1))
71		unitsscn 13540	. . . . . 6 ⊢ (0[,]1) ⊆ ℂ
72	71	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ⊆ ℂ)
73		cnrest2 23294	. . . . 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 7442	. . 3 ⊢ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II) = ((𝐽 ↾_t (𝐴[,]𝐵)) Cn (𝐽 ↾_t (0[,]1)))
77	75, 76	eleqtrrdi 2852	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ^◡𝐹 ∈ ((𝐽 ↾_t (𝐴[,]𝐵)) Cn II))
78		ishmeo 23767	. 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ^◡ccnv 5684 dom cdm 5685 ran crn 5686 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 / cdiv 11920 ℝ⁺crp 13034 [,]cicc 13390 ↾_t crest 17465 TopOpenctopn 17466 ℂ_fldccnfld 21364 Topctop 22899 TopOnctopon 22916 Cn ccn 23232 ×_t ctx 23568 Homeochmeo 23761 IIcii 24901

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903

This theorem is referenced by: xrhmph 24978

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