Theorem xrge0iifcnv 31883

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 11022	. . . . . . 7 ⊢ 0 ∈ ℝ*
3		pnfxr 11029	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4		0lepnf 12868	. . . . . . 7 ⊢ 0 ≤ +∞
5		ubicc2 13197	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
6	2, 3, 4, 5	mp3an 1460	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
7	6	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8		icossicc 13168	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
9		uncom 4087	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10		1xr 11034	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ*
11		0le1 11498	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
12		snunioc 13212	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
13	2, 10, 11, 12	mp3an 1460	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
14	9, 13	eqtr3i 2768	. . . . . . . . . . . . 13 ⊢ ((0(,]1) ∪ {0}) = (0[,]1)
15	14	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16		elun 4083	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
17	15, 16	bitr3i 276	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18		pm2.53 848	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
19	17, 18	sylbi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20		elsni 4578	. . . . . . . . . 10 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
21	19, 20	syl6 35	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 = 0))
22	21	con1d 145	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → 𝑥 ∈ (0(,]1)))
23	22	imp 407	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24		0le0 12074	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
25		1re 10975	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
26		ltpnf 12856	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → 1 < +∞)
27	25, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1 < +∞
28		iocssioo 13171	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
29	2, 3, 24, 27, 28	mp4an 690	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ (0(,)+∞)
30		ioorp 13157	. . . . . . . . . . . . 13 ⊢ (0(,)+∞) = ℝ+
31	29, 30	sseqtri 3957	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ+
32	31	sseli 3917	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
33	32	relogcld 25778	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
34	33	renegcld 11402	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
35	34	rexrd 11025	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36		elioc1 13121	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
37	2, 10, 36	mp2an 689	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
38	37	simp3bi 1146	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39		1rp 12734	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
41	32, 40	logled 25782	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
42	38, 41	mpbid 231	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43		log1 25741	. . . . . . . . . 10 ⊢ (log‘1) = 0
44	42, 43	breqtrdi 5115	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
45	33	le0neg1d 11546	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
46	44, 45	mpbid 231	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47		ltpnf 12856	. . . . . . . . 9 ⊢ (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
48	34, 47	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49		elico1 13122	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
50	2, 3, 49	mp2an 689	. . . . . . . 8 ⊢ (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
51	35, 46, 48, 50	syl3anbrc 1342	. . . . . . 7 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
52	23, 51	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
53	8, 52	sselid 3919	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
54	7, 53	ifclda 4494	. . . 4 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
55	54	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56		0elunit 13201	. . . . . 6 ⊢ 0 ∈ (0[,]1)
57	56	a1i 11	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58		iocssicc 13169	. . . . . 6 ⊢ (0(,]1) ⊆ (0[,]1)
59		snunico 13211	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
60	2, 3, 4, 59	mp3an 1460	. . . . . . . . . . . . 13 ⊢ ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
61	60	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62		elun 4083	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
63	61, 62	bitr3i 276	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64		pm2.53 848	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
65	63, 64	sylbi 216	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66		elsni 4578	. . . . . . . . . 10 ⊢ (𝑦 ∈ {+∞} → 𝑦 = +∞)
67	65, 66	syl6 35	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
68	67	con1d 145	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
69	68	imp 407	. . . . . . 7 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70		rge0ssre 13188	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
71	70	sseli 3917	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
72	71	renegcld 11402	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
73	72	reefcld 15797	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
74	73	rexrd 11025	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75		efgt0 15812	. . . . . . . . 9 ⊢ (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
76	72, 75	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77		elico1 13122	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
78	2, 3, 77	mp2an 689	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
79	78	simp2bi 1145	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
80	71	le0neg2d 11547	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
81	79, 80	mpbid 231	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82		0re 10977	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
83		efle 15827	. . . . . . . . . . 11 ⊢ ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
84	72, 82, 83	sylancl 586	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
85	81, 84	mpbid 231	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86		ef0 15800	. . . . . . . . 9 ⊢ (exp‘0) = 1
87	85, 86	breqtrdi 5115	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88		elioc1 13121	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
89	2, 10, 88	mp2an 689	. . . . . . . 8 ⊢ ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
90	74, 76, 87, 89	syl3anbrc 1342	. . . . . . 7 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
91	69, 90	syl 17	. . . . . 6 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
92	58, 91	sselid 3919	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
93	57, 92	ifclda 4494	. . . 4 ⊢ (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
94	93	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95		eqeq2 2750	. . . . . 6 ⊢ (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
96	95	bibi1d 344	. . . . 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 2750	. . . . . 6 ⊢ ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
98	97	bibi1d 344	. . . . 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 485	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100		iftrue 4465	. . . . . . . 8 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101	100	eqeq2d 2749	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
102	99, 101	syl5ibrcom 246	. . . . . 6 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103		ubico 31096	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
104	82, 3, 103	mp2an 689	. . . . . . . . . 10 ⊢ ¬ +∞ ∈ (0[,)+∞)
105	104	nelir 3052	. . . . . . . . 9 ⊢ +∞ ∉ (0[,)+∞)
106		neleq1 3054	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107	106	adantl 482	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108	105, 107	mpbiri 257	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109		neleq1 3054	. . . . . . . 8 ⊢ (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110	108, 109	syl5ibcom 244	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111		df-nel 3050	. . . . . . . 8 ⊢ (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112		iffalse 4468	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113	112	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114	113, 52	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115	114	ex 413	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116	115	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
117	116	con1d 145	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞) → 𝑥 = 0))
118	111, 117	syl5bi 241	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) → 𝑥 = 0))
119	110, 118	syld 47	. . . . . 6 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → 𝑥 = 0))
120	102, 119	impbid 211	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121		eqeq2 2750	. . . . . . 7 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122	121	bibi2d 343	. . . . . 6 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123		eqeq2 2750	. . . . . . 7 ⊢ (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124	123	bibi2d 343	. . . . . 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 17	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 < (exp‘-𝑦))
127	125, 126	ltned 11111	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128	127	adantll 711	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129	128	neneqd 2948	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131	130	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 0 → (¬ 𝑥 = (exp‘-𝑦) ↔ ¬ 0 = (exp‘-𝑦)))
132	129, 131	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133	132	imp 407	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134		simplr 766	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑦 = +∞)
135	133, 134	2falsed 377	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136		eqcom 2745	. . . . . . . . . . 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 25740	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
139	32, 72, 138	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
140	33	recnd 11003	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
141	71	recnd 11003	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142		negcon2 11274	. . . . . . . . . . 11 ⊢ (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
143	140, 141, 142	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
144	137, 139, 143	3bitr2d 307	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
145	23, 69, 144	syl2an 596	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146	145	an4s 657	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147	146	anass1rs 652	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148	122, 124, 135, 147	ifbothda 4497	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
149	96, 98, 120, 148	ifbothda 4497	. . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150	149	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
151	1, 55, 94, 150	f1ocnv2d 7522	. 2 ⊢ (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152	151	mptru 1546	1 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049   ∪ cun 3885   ⊆ wss 3887  ifcif 4459  {csn 4561   class class class wbr 5074   ↦ cmpt 5157  ^◡ccnv 5588  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  -cneg 11206  ℝ⁺crp 12730  (,)cioo 13079  (,]cioc 13080  [,)cico 13081  [,]cicc 13082  expce 15771  logclog 25710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779  df-cos 15780  df-pi 15782  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-log 25712

This theorem is referenced by:  xrge0iifiso  31885  xrge0iifmhm  31889  xrge0pluscn  31890