xrge0iifiso - Mathbox for Thierry Arnoux

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Theorem xrge0iifiso 29981

Description: The defined bijection from the closed unit interval and the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

**Hypothesis**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))

**Assertion**
Ref	Expression
xrge0iifiso	⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Distinct variable group: 𝑥,𝐹

Proof of Theorem xrge0iifiso Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . 3 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 11974	. . 3 ⊢ < Or ℝ^*
3		soss 5053	. . 3 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. 2 ⊢ < Or (0[,]1)
5		iccssxr 12256	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
6		cnvso 5674	. . . . 5 ⊢ ( < Or ℝ^* ↔ ^◡ < Or ℝ^*)
7	2, 6	mpbi 220	. . . 4 ⊢ ^◡ < Or ℝ^*
8		sopo 5052	. . . 4 ⊢ (^◡ < Or ℝ^* → ^◡ < Po ℝ^*)
9	7, 8	ax-mp 5	. . 3 ⊢ ^◡ < Po ℝ^*
10		poss 5037	. . 3 ⊢ ((0[,]+∞) ⊆ ℝ^* → (^◡ < Po ℝ^* → ^◡ < Po (0[,]+∞)))
11	5, 9, 10	mp2 9	. 2 ⊢ ^◡ < Po (0[,]+∞)
12		xrge0iifhmeo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
13	12	xrge0iifcnv 29979	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 = +∞, 0, (exp‘-𝑧))))
14	13	simpli 474	. . 3 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
15		f1ofo 6144	. . 3 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
16	14, 15	ax-mp 5	. 2 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
17		0xr 10086	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18		1re 10039	. . . . . . . . 9 ⊢ 1 ∈ ℝ
19	18	rexri 10097	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
20		0le1 10551	. . . . . . . 8 ⊢ 0 ≤ 1
21		snunioc 12300	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
22	17, 19, 20, 21	mp3an 1424	. . . . . . 7 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
23	22	eleq2i 2693	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ 𝑤 ∈ (0[,]1))
24		elun 3753	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
25	23, 24	bitr3i 266	. . . . 5 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
26		velsn 4193	. . . . . . 7 ⊢ (𝑤 ∈ {0} ↔ 𝑤 = 0)
27		elunitrn 29943	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ ℝ)
29		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 0 < 𝑧)
30		0re 10040	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
31	30, 18	elicc2i 12239	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
32	31	simp3bi 1078	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ≤ 1)
33	32	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ≤ 1)
34		elioc2 12236	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1)))
35	17, 18, 34	mp2an 708	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1))
36	28, 29, 33, 35	syl3anbrc 1246	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ (0(,]1))
37		pnfxr 10092	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
38		0le0 11110	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
39		ltpnf 11954	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → 1 < +∞)
40	18, 39	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1 < +∞
41		iocssioo 12263	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
42	17, 37, 38, 40, 41	mp4an 709	. . . . . . . . . . . . . 14 ⊢ (0(,]1) ⊆ (0(,)+∞)
43		ioorp 12251	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
44	42, 43	sseqtri 3637	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ ℝ⁺
45	44	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0(,]1) → 𝑧 ∈ ℝ⁺)
46		relogcl 24322	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → (log‘𝑧) ∈ ℝ)
47	46	renegcld 10457	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) ∈ ℝ)
48		ltpnf 11954	. . . . . . . . . . . . . 14 ⊢ (-(log‘𝑧) ∈ ℝ → -(log‘𝑧) < +∞)
49	47, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) < +∞)
50		brcnvg 5303	. . . . . . . . . . . . . 14 ⊢ ((+∞ ∈ ℝ^* ∧ -(log‘𝑧) ∈ ℝ) → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
51	37, 47, 50	sylancr 695	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
52	49, 51	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → +∞^◡ < -(log‘𝑧))
53	45, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < -(log‘𝑧))
54	12	xrge0iifcv 29980	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → (𝐹‘𝑧) = -(log‘𝑧))
55	53, 54	breqtrrd 4681	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < (𝐹‘𝑧))
56	36, 55	syl 17	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → +∞^◡ < (𝐹‘𝑧))
57	56	ex 450	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) → (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧)))
58		breq1 4656	. . . . . . . . 9 ⊢ (𝑤 = 0 → (𝑤 < 𝑧 ↔ 0 < 𝑧))
59		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = (𝐹‘0))
60		0elunit 12290	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
61		iftrue 4092	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
62		pnfex 10093	. . . . . . . . . . . . 13 ⊢ +∞ ∈ V
63	61, 12, 62	fvmpt 6282	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
64	60, 63	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘0) = +∞
65	59, 64	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = +∞)
66	65	breq1d 4663	. . . . . . . . 9 ⊢ (𝑤 = 0 → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ +∞^◡ < (𝐹‘𝑧)))
67	58, 66	imbi12d 334	. . . . . . . 8 ⊢ (𝑤 = 0 → ((𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)) ↔ (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧))))
68	57, 67	syl5ibr 236	. . . . . . 7 ⊢ (𝑤 = 0 → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
69	26, 68	sylbi 207	. . . . . 6 ⊢ (𝑤 ∈ {0} → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
70		simpll 790	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ (0(,]1))
71	27	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ)
72	30	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 ∈ ℝ)
73	44	sseli 3599	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ⁺)
74	73	rpred 11872	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ)
75	74	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ)
76		elioc2 12236	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1)))
77	17, 18, 76	mp2an 708	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1))
78	77	simp2bi 1077	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 0 < 𝑤)
79	78	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑤)
80		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
81	72, 75, 71, 79, 80	lttrd 10198	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑧)
82	32	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ≤ 1)
83	71, 81, 82, 35	syl3anbrc 1246	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ (0(,]1))
84	70, 83	jca 554	. . . . . . . 8 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)))
85	73	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑤 ∈ ℝ⁺)
86	85	relogcld 24369	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑤) ∈ ℝ)
87	45	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑧 ∈ ℝ⁺)
88	87	relogcld 24369	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑧) ∈ ℝ)
89	86, 88	ltnegd 10605	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((log‘𝑤) < (log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
90		logltb 24346	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
91	73, 45, 90	syl2an 494	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
92		negex 10279	. . . . . . . . . . . . 13 ⊢ -(log‘𝑤) ∈ V
93		negex 10279	. . . . . . . . . . . . 13 ⊢ -(log‘𝑧) ∈ V
94	92, 93	brcnv 5305	. . . . . . . . . . . 12 ⊢ (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
96	89, 91, 95	3bitr4d 300	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
97	96	biimpd 219	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → -(log‘𝑤)^◡ < -(log‘𝑧)))
98	12	xrge0iifcv 29980	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0(,]1) → (𝐹‘𝑤) = -(log‘𝑤))
99	98, 54	breqan12d 4669	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
100	97, 99	sylibrd 249	. . . . . . . 8 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
101	84, 80, 100	sylc 65	. . . . . . 7 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
102	101	exp31 630	. . . . . 6 ⊢ (𝑤 ∈ (0(,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
103	69, 102	jaoi 394	. . . . 5 ⊢ ((𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
104	25, 103	sylbi 207	. . . 4 ⊢ (𝑤 ∈ (0[,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
105	104	imp 445	. . 3 ⊢ ((𝑤 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
106	105	rgen2a 2977	. 2 ⊢ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
107		soisoi 6578	. 2 ⊢ ((( < Or (0[,]1) ∧ ^◡ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))) → 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)))
108	4, 11, 16, 106, 107	mp4an 709	1 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∪ cun 3572 ⊆ wss 3574 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 Po wpo 5033 Or wor 5034 ^◡ccnv 5113 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 ℝ⁺crp 11832 (,)cioo 12175 (,]cioc 12176 [,]cicc 12178 expce 14792 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: xrge0iifhmeo 29982

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