Theorem xrge0iifcnv 34289

Description: Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.)

Hypothesis

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

Assertion

Ref	Expression
xrge0iifcnv	⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xrge0iifcnv

Step	Hyp	Ref	Expression
1		xrge0iifhmeo.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
2		0xr 11255	. . . . . . 7 ⊢ 0 ∈ ℝ*
3		pnfxr 11262	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4		0lepnf 13157	. . . . . . 7 ⊢ 0 ≤ +∞
5		ubicc2 13491	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
6	2, 3, 4, 5	mp3an 1488	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
7	6	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8		icossicc 13462	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
9		uncom 4111	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10		1xr 11267	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ*
11		0le1 11736	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
12		snunioc 13506	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
13	2, 10, 11, 12	mp3an 1488	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
14	9, 13	eqtr3i 2786	. . . . . . . . . . . . 13 ⊢ ((0(,]1) ∪ {0}) = (0[,]1)
15	14	eleq2i 2853	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16		elun 4106	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
17	15, 16	bitr3i 280	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18		pm2.53 864	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
19	17, 18	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20		elsni 4605	. . . . . . . . . 10 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
21	19, 20	syl6 36	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 = 0))
22	21	con1d 146	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → 𝑥 ∈ (0(,]1)))
23	22	imp 411	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24		0le0 12341	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
25		1re 11207	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
26		ltpnf 13144	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → 1 < +∞)
27	25, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1 < +∞
28		iocssioo 13465	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
29	2, 3, 24, 27, 28	mp4an 705	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ (0(,)+∞)
30		ioorp 13451	. . . . . . . . . . . . 13 ⊢ (0(,)+∞) = ℝ+
31	29, 30	sseqtri 3984	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ+
32	31	sseli 3932	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
33	32	relogcld 26764	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
34	33	renegcld 11640	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
35	34	rexrd 11258	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36		elioc1 13413	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
37	2, 10, 36	mp2an 704	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
38	37	simp3bi 1163	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39		1rp 13019	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
41	32, 40	logled 26768	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
42	38, 41	mpbid 235	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43		log1 26726	. . . . . . . . . 10 ⊢ (log‘1) = 0
44	42, 43	breqtrdi 5151	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
45	33	le0neg1d 11784	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
46	44, 45	mpbid 235	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47		ltpnf 13144	. . . . . . . . 9 ⊢ (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
48	34, 47	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49		elico1 13414	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
50	2, 3, 49	mp2an 704	. . . . . . . 8 ⊢ (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
51	35, 46, 48, 50	syl3anbrc 1360	. . . . . . 7 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
52	23, 51	syl 18	. . . . . 6 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
53	8, 52	sselid 3934	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
54	7, 53	ifclda 4522	. . . 4 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
55	54	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56		0elunit 13495	. . . . . 6 ⊢ 0 ∈ (0[,]1)
57	56	a1i 11	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58		iocssicc 13463	. . . . . 6 ⊢ (0(,]1) ⊆ (0[,]1)
59		snunico 13505	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
60	2, 3, 4, 59	mp3an 1488	. . . . . . . . . . . . 13 ⊢ ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
61	60	eleq2i 2853	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62		elun 4106	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
63	61, 62	bitr3i 280	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64		pm2.53 864	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
65	63, 64	sylbi 220	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66		elsni 4605	. . . . . . . . . 10 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
67	65, 66	syl6 36	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
68	67	con1d 146	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
69	68	imp 411	. . . . . . 7 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70		rge0ssre 13482	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
71	70	sseli 3932	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
72	71	renegcld 11640	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
73	72	reefcld 16141	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
74	73	rexrd 11258	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75		efgt0 16158	. . . . . . . . 9 ⊢ (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
76	72, 75	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77		elico1 13414	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
78	2, 3, 77	mp2an 704	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
79	78	simp2bi 1162	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
80	71	le0neg2d 11785	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
81	79, 80	mpbid 235	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82		0re 11209	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
83		efle 16173	. . . . . . . . . . 11 ⊢ ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
84	72, 82, 83	sylancl 597	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
85	81, 84	mpbid 235	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86		ef0 16144	. . . . . . . . 9 ⊢ (exp‘0) = 1
87	85, 86	breqtrdi 5151	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88		elioc1 13413	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
89	2, 10, 88	mp2an 704	. . . . . . . 8 ⊢ ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
90	74, 76, 87, 89	syl3anbrc 1360	. . . . . . 7 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
91	69, 90	syl 18	. . . . . 6 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
92	58, 91	sselid 3934	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
93	57, 92	ifclda 4522	. . . 4 ⊢ (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
94	93	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95		eqeq2 2773	. . . . . 6 ⊢ (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
96	95	bibi1d 346	. . . . 5 ⊢ (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → ((𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))) ↔ (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
97		eqeq2 2773	. . . . . 6 ⊢ ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
98	97	bibi1d 346	. . . . 5 ⊢ ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))) ↔ (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
99		simpr 489	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100		iftrue 4492	. . . . . . . 8 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101	100	eqeq2d 2772	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
102	99, 101	syl5ibrcom 250	. . . . . 6 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103		ubico 33086	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
104	82, 3, 103	mp2an 704	. . . . . . . . . 10 ⊢ ¬ +∞ ∈ (0[,)+∞)
105	104	nelir 3065	. . . . . . . . 9 ⊢ +∞ ∉ (0[,)+∞)
106		neleq1 3068	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107	106	adantl 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108	105, 107	mpbiri 261	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109		neleq1 3068	. . . . . . . 8 ⊢ (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110	108, 109	syl5ibcom 248	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111		df-nel 3063	. . . . . . . 8 ⊢ (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112		iffalse 4495	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113	112	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114	113, 52	eqeltrd 2861	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115	114	ex 417	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116	115	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
117	116	con1d 146	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞) → 𝑥 = 0))
118	111, 117	biimtrid 245	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) → 𝑥 = 0))
119	110, 118	syld 48	. . . . . 6 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → 𝑥 = 0))
120	102, 119	impbid 215	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121		eqeq2 2773	. . . . . . 7 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122	121	bibi2d 345	. . . . . 6 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123		eqeq2 2773	. . . . . . 7 ⊢ (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124	123	bibi2d 345	. . . . . 6 ⊢ (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
125	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ∈ ℝ)
126	69, 76	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 < (exp‘-𝑦))
127	125, 126	ltned 11345	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128	127	adantll 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129	128	neneqd 2961	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131	130	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = 0 → (¬ 𝑥 = (exp‘-𝑦) ↔ ¬ 0 = (exp‘-𝑦)))
132	129, 131	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133	132	imp 411	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134		simplr 780	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑦 = +∞)
135	133, 134	2falsed 379	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136		eqcom 2768	. . . . . . . . . . 11 ⊢ (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137	136	a1i 11	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138		relogeftb 26725	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
139	32, 72, 138	syl2an 607	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
140	33	recnd 11236	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
141	71	recnd 11236	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142		negcon2 11510	. . . . . . . . . . 11 ⊢ (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
143	140, 141, 142	syl2an 607	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
144	137, 139, 143	3bitr2d 310	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
145	23, 69, 144	syl2an 607	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146	145	an4s 672	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147	146	anass1rs 667	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148	122, 124, 135, 147	ifbothda 4525	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
149	96, 98, 120, 148	ifbothda 4525	. . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150	149	adantl 486	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
151	1, 55, 94, 150	f1ocnv2d 7663	. 2 ⊢ (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152	151	mptru 1575	1 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568  ⊤wtru 1569   ∈ wcel 2141   ≠ wne 2956   ∉ wnel 3062   ∪ cun 3902   ⊆ wss 3904  ifcif 4486  {csn 4588   class class class wbr 5108   ↦ cmpt 5191  ^◡ccnv 5660  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  ℂcc 11097  ℝcr 11098  0cc0 11099  1c1 11100  +∞cpnf 11239  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243  -cneg 11441  ℝ⁺crp 13015  (,)cioo 13371  (,]cioc 13372  [,)cico 13373  [,]cicc 13374  expce 16114  logclog 26695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-addf 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-seq 14037  df-exp 14097  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15103  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-limsup 15521  df-clim 15538  df-rlim 15539  df-sum 15737  df-ef 16120  df-sin 16122  df-cos 16123  df-pi 16125  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-rest 17474  df-topn 17475  df-0g 17493  df-gsum 17494  df-topgen 17495  df-pt 17496  df-prds 17499  df-xrs 17555  df-qtop 17560  df-imas 17561  df-xps 17563  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-mulg 19133  df-cntz 19386  df-cmn 19851  df-psmet 21493  df-xmet 21494  df-met 21495  df-bl 21496  df-mopn 21497  df-fbas 21498  df-fg 21499  df-cnfld 21502  df-top 23030  df-topon 23047  df-topsp 23069  df-bases 23082  df-cld 23155  df-ntr 23156  df-cls 23157  df-nei 23234  df-lp 23272  df-perf 23273  df-cn 23363  df-cnp 23364  df-haus 23451  df-tx 23698  df-hmeo 23891  df-fil 23982  df-fm 24074  df-flim 24075  df-flf 24076  df-xms 24456  df-ms 24457  df-tms 24458  df-cncf 25016  df-limc 26004  df-dv 26005  df-log 26697

This theorem is referenced by:  xrge0iifiso  34291  xrge0iifmhm  34295  xrge0pluscn  34296