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Theorem xrge0iifcnv 34092
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
StepHypRef Expression
1 xrge0iifhmeo.1 . . 3 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
2 0xr 11183 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11190 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13051 . . . . . . 7 0 ≤ +∞
5 ubicc2 13385 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1464 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8 icossicc 13356 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 uncom 4111 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10 1xr 11195 . . . . . . . . . . . . . . 15 1 ∈ ℝ*
11 0le1 11664 . . . . . . . . . . . . . . 15 0 ≤ 1
12 snunioc 13400 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
132, 10, 11, 12mp3an 1464 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = (0[,]1)
149, 13eqtr3i 2762 . . . . . . . . . . . . 13 ((0(,]1) ∪ {0}) = (0[,]1)
1514eleq2i 2829 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16 elun 4106 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
1715, 16bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18 pm2.53 852 . . . . . . . . . . 11 ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
1917, 18sylbi 217 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20 elsni 4598 . . . . . . . . . 10 (𝑥 ∈ {0} → 𝑥 = 0)
2119, 20syl6 35 . . . . . . . . 9 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 = 0))
2221con1d 145 . . . . . . . 8 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → 𝑥 ∈ (0(,]1)))
2322imp 406 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24 0le0 12250 . . . . . . . . . . . . . 14 0 ≤ 0
25 1re 11136 . . . . . . . . . . . . . . 15 1 ∈ ℝ
26 ltpnf 13038 . . . . . . . . . . . . . . 15 (1 ∈ ℝ → 1 < +∞)
2725, 26ax-mp 5 . . . . . . . . . . . . . 14 1 < +∞
28 iocssioo 13359 . . . . . . . . . . . . . 14 (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
292, 3, 24, 27, 28mp4an 694 . . . . . . . . . . . . 13 (0(,]1) ⊆ (0(,)+∞)
30 ioorp 13345 . . . . . . . . . . . . 13 (0(,)+∞) = ℝ+
3129, 30sseqtri 3983 . . . . . . . . . . . 12 (0(,]1) ⊆ ℝ+
3231sseli 3930 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
3332relogcld 26592 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
3433renegcld 11568 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
3534rexrd 11186 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36 elioc1 13307 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
372, 10, 36mp2an 693 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
3837simp3bi 1148 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39 1rp 12913 . . . . . . . . . . . . 13 1 ∈ ℝ+
4039a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
4132, 40logled 26596 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
4238, 41mpbid 232 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43 log1 26554 . . . . . . . . . 10 (log‘1) = 0
4442, 43breqtrdi 5140 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
4533le0neg1d 11712 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
4644, 45mpbid 232 . . . . . . . 8 (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47 ltpnf 13038 . . . . . . . . 9 (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
4834, 47syl 17 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49 elico1 13308 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
502, 3, 49mp2an 693 . . . . . . . 8 (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
5135, 46, 48, 50syl3anbrc 1345 . . . . . . 7 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
5223, 51syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
538, 52sselid 3932 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
547, 53ifclda 4516 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
5554adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56 0elunit 13389 . . . . . 6 0 ∈ (0[,]1)
5756a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58 iocssicc 13357 . . . . . 6 (0(,]1) ⊆ (0[,]1)
59 snunico 13399 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
602, 3, 4, 59mp3an 1464 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
6160eleq2i 2829 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62 elun 4106 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
6361, 62bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64 pm2.53 852 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
6563, 64sylbi 217 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66 elsni 4598 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
6765, 66syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
6867con1d 145 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
6968imp 406 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70 rge0ssre 13376 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
7170sseli 3930 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
7271renegcld 11568 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
7372reefcld 16015 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
7473rexrd 11186 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75 efgt0 16032 . . . . . . . . 9 (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
7672, 75syl 17 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77 elico1 13308 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
782, 3, 77mp2an 693 . . . . . . . . . . . 12 (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
7978simp2bi 1147 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
8071le0neg2d 11713 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
8179, 80mpbid 232 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82 0re 11138 . . . . . . . . . . 11 0 ∈ ℝ
83 efle 16047 . . . . . . . . . . 11 ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8472, 82, 83sylancl 587 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8581, 84mpbid 232 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86 ef0 16018 . . . . . . . . 9 (exp‘0) = 1
8785, 86breqtrdi 5140 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88 elioc1 13307 . . . . . . . . 9 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
892, 10, 88mp2an 693 . . . . . . . 8 ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
9074, 76, 87, 89syl3anbrc 1345 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
9169, 90syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
9258, 91sselid 3932 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
9357, 92ifclda 4516 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
9493adantl 481 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95 eqeq2 2749 . . . . . 6 (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9695bibi1d 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 2749 . . . . . 6 ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9897bibi1d 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 4486 . . . . . . . 8 (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101100eqeq2d 2748 . . . . . . 7 (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
10299, 101syl5ibrcom 247 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103 ubico 32857 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
10482, 3, 103mp2an 693 . . . . . . . . . 10 ¬ +∞ ∈ (0[,)+∞)
105104nelir 3040 . . . . . . . . 9 +∞ ∉ (0[,)+∞)
106 neleq1 3043 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107106adantl 481 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108105, 107mpbiri 258 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109 neleq1 3043 . . . . . . . 8 (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110108, 109syl5ibcom 245 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111 df-nel 3038 . . . . . . . 8 (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112 iffalse 4489 . . . . . . . . . . . . 13 (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113112adantl 481 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114113, 52eqeltrd 2837 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115114ex 412 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116115ad2antrr 727 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
117116con1d 145 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞) → 𝑥 = 0))
118111, 117biimtrid 242 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) → 𝑥 = 0))
119110, 118syld 47 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → 𝑥 = 0))
120102, 119impbid 212 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121 eqeq2 2749 . . . . . . 7 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122121bibi2d 342 . . . . . 6 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123 eqeq2 2749 . . . . . . 7 (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124123bibi2d 342 . . . . . 6 (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
12582a1i 11 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ∈ ℝ)
12669, 76syl 17 . . . . . . . . . . . 12 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 < (exp‘-𝑦))
127125, 126ltned 11273 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128127adantll 715 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129128neneqd 2938 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130 eqeq1 2741 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131130notbid 318 . . . . . . . . 9 (𝑥 = 0 → (¬ 𝑥 = (exp‘-𝑦) ↔ ¬ 0 = (exp‘-𝑦)))
132129, 131syl5ibrcom 247 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133132imp 406 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134 simplr 769 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑦 = +∞)
135133, 1342falsed 376 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136 eqcom 2744 . . . . . . . . . . 11 (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137136a1i 11 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138 relogeftb 26553 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
13932, 72, 138syl2an 597 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
14033recnd 11164 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
14171recnd 11164 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142 negcon2 11438 . . . . . . . . . . 11 (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
143140, 141, 142syl2an 597 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
144137, 139, 1433bitr2d 307 . . . . . . . . 9 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
14523, 69, 144syl2an 597 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146145an4s 661 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147146anass1rs 656 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148122, 124, 135, 147ifbothda 4519 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
14996, 98, 120, 148ifbothda 4519 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150149adantl 481 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
1511, 55, 94, 150f1ocnv2d 7613 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152151mptru 1549 1 (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037   ∪ cun 3900   ⊆ wss 3902  ifcif 4480  {csn 4581   class class class wbr 5099   ↦ cmpt 5180  ◡ccnv 5624  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  ℂcc 11028  ℝcr 11029  0cc0 11030  1c1 11031  +∞cpnf 11167  ℝ*cxr 11169   < clt 11170   ≤ cle 11171  -cneg 11369  ℝ+crp 12909  (,)cioo 13265  (,]cioc 13266  [,)cico 13267  [,]cicc 13268  expce 15988  logclog 26523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-addf 11109
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-q 12866  df-rp 12910  df-xneg 13030  df-xadd 13031  df-xmul 13032  df-ioo 13269  df-ioc 13270  df-ico 13271  df-icc 13272  df-fz 13428  df-fzo 13575  df-fl 13716  df-mod 13794  df-seq 13929  df-exp 13989  df-fac 14201  df-bc 14230  df-hash 14258  df-shft 14994  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-limsup 15398  df-clim 15415  df-rlim 15416  df-sum 15614  df-ef 15994  df-sin 15996  df-cos 15997  df-pi 15999  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-rest 17346  df-topn 17347  df-0g 17365  df-gsum 17366  df-topgen 17367  df-pt 17368  df-prds 17371  df-xrs 17427  df-qtop 17432  df-imas 17433  df-xps 17435  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-mulg 19002  df-cntz 19250  df-cmn 19715  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-fbas 21310  df-fg 21311  df-cnfld 21314  df-top 22842  df-topon 22859  df-topsp 22881  df-bases 22894  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lp 23084  df-perf 23085  df-cn 23175  df-cnp 23176  df-haus 23263  df-tx 23510  df-hmeo 23703  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-xms 24268  df-ms 24269  df-tms 24270  df-cncf 24831  df-limc 25827  df-dv 25828  df-log 26525
This theorem is referenced by:  xrge0iifiso  34094  xrge0iifmhm  34098  xrge0pluscn  34099
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