Theorem xrge0iifcnv 33930

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 11228	. . . . . . 7 ⊢ 0 ∈ ℝ*
3		pnfxr 11235	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4		0lepnf 13100	. . . . . . 7 ⊢ 0 ≤ +∞
5		ubicc2 13433	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
6	2, 3, 4, 5	mp3an 1463	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
7	6	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8		icossicc 13404	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
9		uncom 4124	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10		1xr 11240	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ*
11		0le1 11708	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
12		snunioc 13448	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
13	2, 10, 11, 12	mp3an 1463	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
14	9, 13	eqtr3i 2755	. . . . . . . . . . . . 13 ⊢ ((0(,]1) ∪ {0}) = (0[,]1)
15	14	eleq2i 2821	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16		elun 4119	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
17	15, 16	bitr3i 277	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18		pm2.53 851	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
19	17, 18	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20		elsni 4609	. . . . . . . . . 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 406	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24		0le0 12294	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
25		1re 11181	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
26		ltpnf 13087	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → 1 < +∞)
27	25, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1 < +∞
28		iocssioo 13407	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
29	2, 3, 24, 27, 28	mp4an 693	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ (0(,)+∞)
30		ioorp 13393	. . . . . . . . . . . . 13 ⊢ (0(,)+∞) = ℝ+
31	29, 30	sseqtri 3998	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ+
32	31	sseli 3945	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
33	32	relogcld 26539	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
34	33	renegcld 11612	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
35	34	rexrd 11231	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36		elioc1 13355	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
37	2, 10, 36	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
38	37	simp3bi 1147	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39		1rp 12962	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
41	32, 40	logled 26543	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
42	38, 41	mpbid 232	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43		log1 26501	. . . . . . . . . 10 ⊢ (log‘1) = 0
44	42, 43	breqtrdi 5151	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
45	33	le0neg1d 11756	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
46	44, 45	mpbid 232	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47		ltpnf 13087	. . . . . . . . 9 ⊢ (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
48	34, 47	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49		elico1 13356	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
50	2, 3, 49	mp2an 692	. . . . . . . 8 ⊢ (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
51	35, 46, 48, 50	syl3anbrc 1344	. . . . . . 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 3947	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
54	7, 53	ifclda 4527	. . . 4 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
55	54	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56		0elunit 13437	. . . . . 6 ⊢ 0 ∈ (0[,]1)
57	56	a1i 11	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58		iocssicc 13405	. . . . . 6 ⊢ (0(,]1) ⊆ (0[,]1)
59		snunico 13447	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
60	2, 3, 4, 59	mp3an 1463	. . . . . . . . . . . . 13 ⊢ ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
61	60	eleq2i 2821	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62		elun 4119	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
63	61, 62	bitr3i 277	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64		pm2.53 851	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
65	63, 64	sylbi 217	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66		elsni 4609	. . . . . . . . . 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 406	. . . . . . 7 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70		rge0ssre 13424	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
71	70	sseli 3945	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
72	71	renegcld 11612	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
73	72	reefcld 16061	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
74	73	rexrd 11231	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75		efgt0 16078	. . . . . . . . 9 ⊢ (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
76	72, 75	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77		elico1 13356	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
78	2, 3, 77	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
79	78	simp2bi 1146	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
80	71	le0neg2d 11757	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
81	79, 80	mpbid 232	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82		0re 11183	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
83		efle 16093	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86		ef0 16064	. . . . . . . . 9 ⊢ (exp‘0) = 1
87	85, 86	breqtrdi 5151	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88		elioc1 13355	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
89	2, 10, 88	mp2an 692	. . . . . . . 8 ⊢ ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
90	74, 76, 87, 89	syl3anbrc 1344	. . . . . . 7 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
91	69, 90	syl 17	. . . . . 6 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
92	58, 91	sselid 3947	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
93	57, 92	ifclda 4527	. . . 4 ⊢ (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
94	93	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95		eqeq2 2742	. . . . . 6 ⊢ (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
96	95	bibi1d 343	. . . . 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 2742	. . . . . 6 ⊢ ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
98	97	bibi1d 343	. . . . 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 484	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100		iftrue 4497	. . . . . . . 8 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101	100	eqeq2d 2741	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
102	99, 101	syl5ibrcom 247	. . . . . 6 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103		ubico 32705	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
104	82, 3, 103	mp2an 692	. . . . . . . . . 10 ⊢ ¬ +∞ ∈ (0[,)+∞)
105	104	nelir 3033	. . . . . . . . 9 ⊢ +∞ ∉ (0[,)+∞)
106		neleq1 3036	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107	106	adantl 481	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108	105, 107	mpbiri 258	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109		neleq1 3036	. . . . . . . 8 ⊢ (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110	108, 109	syl5ibcom 245	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111		df-nel 3031	. . . . . . . 8 ⊢ (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112		iffalse 4500	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113	112	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114	113, 52	eqeltrd 2829	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115	114	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116	115	ad2antrr 726	. . . . . . . . 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	biimtrid 242	. . . . . . 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 212	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121		eqeq2 2742	. . . . . . 7 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122	121	bibi2d 342	. . . . . 6 ⊢ (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123		eqeq2 2742	. . . . . . 7 ⊢ (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124	123	bibi2d 342	. . . . . 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 11317	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128	127	adantll 714	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129	128	neneqd 2931	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130		eqeq1 2734	. . . . . . . . . 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 247	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133	132	imp 406	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134		simplr 768	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑦 = +∞)
135	133, 134	2falsed 376	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136		eqcom 2737	. . . . . . . . . . 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 26500	. . . . . . . . . . 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 11209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
141	71	recnd 11209	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142		negcon2 11482	. . . . . . . . . . 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 660	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147	146	anass1rs 655	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148	122, 124, 135, 147	ifbothda 4530	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
149	96, 98, 120, 148	ifbothda 4530	. . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150	149	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
151	1, 55, 94, 150	f1ocnv2d 7645	. 2 ⊢ (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152	151	mptru 1547	1 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109   ≠ wne 2926   ∉ wnel 3030   ∪ cun 3915   ⊆ wss 3917  ifcif 4491  {csn 4592   class class class wbr 5110   ↦ cmpt 5191  ^◡ccnv 5640  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076  +∞cpnf 11212  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216  -cneg 11413  ℝ⁺crp 12958  (,)cioo 13313  (,]cioc 13314  [,)cico 13315  [,]cicc 13316  expce 16034  logclog 26470

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-inf2 9601  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-pm 8805  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-ioo 13317  df-ioc 13318  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-fac 14246  df-bc 14275  df-hash 14303  df-shft 15040  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-limsup 15444  df-clim 15461  df-rlim 15462  df-sum 15660  df-ef 16040  df-sin 16042  df-cos 16043  df-pi 16045  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-fbas 21268  df-fg 21269  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cld 22913  df-ntr 22914  df-cls 22915  df-nei 22992  df-lp 23030  df-perf 23031  df-cn 23121  df-cnp 23122  df-haus 23209  df-tx 23456  df-hmeo 23649  df-fil 23740  df-fm 23832  df-flim 23833  df-flf 23834  df-xms 24215  df-ms 24216  df-tms 24217  df-cncf 24778  df-limc 25774  df-dv 25775  df-log 26472

This theorem is referenced by:  xrge0iifiso  33932  xrge0iifmhm  33936  xrge0pluscn  33937