Theorem xrge0iifcnv 34484

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 11313	. . . . . . 7 ⊢ 0 ∈ ℝ*
3		pnfxr 11320	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4		0lepnf 13217	. . . . . . 7 ⊢ 0 ≤ +∞
5		ubicc2 13551	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
6	2, 3, 4, 5	mp3an 1490	. . . . . 6 ⊢ +∞ ∈ (0[,]+∞)
7	6	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8		icossicc 13522	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
9		uncom 4105	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10		1xr 11325	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ*
11		0le1 11794	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
12		snunioc 13566	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
13	2, 10, 11, 12	mp3an 1490	. . . . . . . . . . . . . 14 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
14	9, 13	eqtr3i 2785	. . . . . . . . . . . . 13 ⊢ ((0(,]1) ∪ {0}) = (0[,]1)
15	14	eleq2i 2852	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16		elun 4100	. . . . . . . . . . . 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		pm2.53 865	. . . . . . . . . . 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 4601	. . . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24		0le0 12399	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
25		1re 11265	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
26		ltpnf 13204	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → 1 < +∞)
27	25, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1 < +∞
28		iocssioo 13525	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
29	2, 3, 24, 27, 28	mp4an 706	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ (0(,)+∞)
30		ioorp 13511	. . . . . . . . . . . . 13 ⊢ (0(,)+∞) = ℝ+
31	29, 30	sseqtri 3979	. . . . . . . . . . . 12 ⊢ (0(,]1) ⊆ ℝ+
32	31	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
33	32	relogcld 26900	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
34	33	renegcld 11698	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
35	34	rexrd 11316	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36		elioc1 13473	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
37	2, 10, 36	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
38	37	simp3bi 1165	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39		1rp 13079	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
41	32, 40	logled 26904	. . . . . . . . . . 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 26862	. . . . . . . . . 10 ⊢ (log‘1) = 0
44	42, 43	breqtrdi 5146	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
45	33	le0neg1d 11842	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
46	44, 45	mpbid 235	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47		ltpnf 13204	. . . . . . . . 9 ⊢ (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
48	34, 47	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49		elico1 13474	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
50	2, 3, 49	mp2an 705	. . . . . . . 8 ⊢ (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
51	35, 46, 48, 50	syl3anbrc 1362	. . . . . . 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 3929	. . . . 5 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
54	7, 53	ifclda 4518	. . . 4 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
55	54	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56		0elunit 13555	. . . . . 6 ⊢ 0 ∈ (0[,]1)
57	56	a1i 11	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58		iocssicc 13523	. . . . . 6 ⊢ (0(,]1) ⊆ (0[,]1)
59		snunico 13565	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
60	2, 3, 4, 59	mp3an 1490	. . . . . . . . . . . . 13 ⊢ ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
61	60	eleq2i 2852	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62		elun 4100	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
63	61, 62	bitr3i 280	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64		pm2.53 865	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
65	63, 64	sylbi 220	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66		elsni 4601	. . . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70		rge0ssre 13542	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
71	70	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
72	71	renegcld 11698	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
73	72	reefcld 16207	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
74	73	rexrd 11316	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75		efgt0 16224	. . . . . . . . 9 ⊢ (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
76	72, 75	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77		elico1 13474	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
78	2, 3, 77	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
79	78	simp2bi 1164	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
80	71	le0neg2d 11843	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
81	79, 80	mpbid 235	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82		0re 11267	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
83		efle 16239	. . . . . . . . . . 11 ⊢ ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
84	72, 82, 83	sylancl 598	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
85	81, 84	mpbid 235	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86		ef0 16210	. . . . . . . . 9 ⊢ (exp‘0) = 1
87	85, 86	breqtrdi 5146	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88		elioc1 13473	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
89	2, 10, 88	mp2an 705	. . . . . . . 8 ⊢ ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
90	74, 76, 87, 89	syl3anbrc 1362	. . . . . . 7 ⊢ (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
91	69, 90	syl 18	. . . . . 6 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
92	58, 91	sselid 3929	. . . . 5 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
93	57, 92	ifclda 4518	. . . 4 ⊢ (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
94	93	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95		eqeq2 2772	. . . . . 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 2772	. . . . . 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 490	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100		iftrue 4488	. . . . . . . 8 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101	100	eqeq2d 2771	. . . . . . 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 33286	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
104	82, 3, 103	mp2an 705	. . . . . . . . . 10 ⊢ ¬ +∞ ∈ (0[,)+∞)
105	104	nelir 3064	. . . . . . . . 9 ⊢ +∞ ∉ (0[,)+∞)
106		neleq1 3067	. . . . . . . . . 10 ⊢ (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107	106	adantl 487	. . . . . . . . 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 3067	. . . . . . . 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 3062	. . . . . . . 8 ⊢ (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112		iffalse 4491	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113	112	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114	113, 52	eqeltrd 2860	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115	114	ex 418	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116	115	ad2antrr 739	. . . . . . . . 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 2772	. . . . . . 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 2772	. . . . . . 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 11403	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128	127	adantll 727	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129	128	neneqd 2960	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130		eqeq1 2764	. . . . . . . . . 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 412	. . . . . . 7 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134		simplr 781	. . . . . . 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 2767	. . . . . . . . . . 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 26861	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
139	32, 72, 138	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
140	33	recnd 11294	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
141	71	recnd 11294	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142		negcon2 11568	. . . . . . . . . . 11 ⊢ (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
143	140, 141, 142	syl2an 608	. . . . . . . . . 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 608	. . . . . . . 8 ⊢ (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146	145	an4s 673	. . . . . . 7 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147	146	anass1rs 668	. . . . . 6 ⊢ ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148	122, 124, 135, 147	ifbothda 4521	. . . . 5 ⊢ (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
149	96, 98, 120, 148	ifbothda 4521	. . . 4 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150	149	adantl 487	. . 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 1577	1 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145   ≠ wne 2955   ∉ wnel 3061   ∪ cun 3897   ⊆ wss 3899  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5647  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7409  ℂcc 11155  ℝcr 11156  0cc0 11157  1c1 11158  +∞cpnf 11297  ℝ^*cxr 11299   < clt 11300   ≤ cle 11301  -cneg 11499  ℝ⁺crp 13075  (,)cioo 13431  (,]cioc 13432  [,)cico 13433  [,]cicc 13434  expce 16180  logclog 26831

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-inf2 9620  ax-cnex 11213  ax-resscn 11214  ax-1cn 11215  ax-icn 11216  ax-addcl 11217  ax-addrcl 11218  ax-mulcl 11219  ax-mulrcl 11220  ax-mulcom 11221  ax-addass 11222  ax-mulass 11223  ax-distr 11224  ax-i2m1 11225  ax-1ne0 11226  ax-1rid 11227  ax-rnegex 11228  ax-rrecex 11229  ax-cnre 11230  ax-pre-lttri 11231  ax-pre-lttrn 11232  ax-pre-ltadd 11233  ax-pre-mulgt0 11234  ax-pre-sup 11235  ax-addf 11236

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-isom 6537  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7677  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9977  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11500  df-neg 11501  df-div 11929  df-nn 12291  df-2 12360  df-3 12361  df-4 12362  df-5 12363  df-6 12364  df-7 12365  df-8 12366  df-9 12367  df-n0 12562  df-z 12649  df-dec 12770  df-uz 12921  df-q 13031  df-rp 13076  df-xneg 13196  df-xadd 13197  df-xmul 13198  df-ioo 13435  df-ioc 13436  df-ico 13437  df-icc 13438  df-fz 13595  df-fzo 13743  df-fl 13886  df-mod 13964  df-seq 14099  df-exp 14159  df-fac 14371  df-bc 14400  df-hash 14428  df-shft 15173  df-cj 15219  df-re 15220  df-im 15221  df-sqrt 15355  df-abs 15356  df-limsup 15591  df-clim 15608  df-rlim 15609  df-sum 15807  df-ef 16186  df-sin 16188  df-cos 16189  df-pi 16191  df-struct 17272  df-sets 17289  df-slot 17307  df-ndx 17319  df-base 17335  df-ress 17356  df-plusg 17388  df-mulr 17389  df-starv 17390  df-sca 17391  df-vsca 17392  df-ip 17393  df-tset 17394  df-ple 17395  df-ds 17397  df-unif 17398  df-hom 17399  df-cco 17400  df-rest 17540  df-topn 17541  df-0g 17559  df-gsum 17560  df-topgen 17561  df-pt 17562  df-prds 17565  df-xrs 17621  df-qtop 17626  df-imas 17627  df-xps 17629  df-mre 17703  df-mrc 17704  df-acs 17706  df-mgm 18763  df-sgrp 18855  df-mnd 18871  df-submnd 18926  df-mulg 19225  df-cntz 19478  df-cmn 19943  df-psmet 21617  df-xmet 21618  df-met 21619  df-bl 21620  df-mopn 21621  df-fbas 21622  df-fg 21623  df-cnfld 21626  df-top 23159  df-topon 23176  df-topsp 23198  df-bases 23211  df-cld 23284  df-ntr 23285  df-cls 23286  df-nei 23363  df-lp 23401  df-perf 23402  df-cn 23492  df-cnp 23493  df-haus 23580  df-tx 23828  df-hmeo 24021  df-fil 24112  df-fm 24204  df-flim 24205  df-flf 24206  df-xms 24586  df-ms 24587  df-tms 24588  df-cncf 25146  df-limc 26133  df-dv 26134  df-log 26833

This theorem is used by:  xrge0iifiso  34486  xrge0iifmhm  34490  xrge0pluscn  34491