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Theorem xrge0iifcnv 32913
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 11261 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11268 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13112 . . . . . . 7 0 ≤ +∞
5 ubicc2 13442 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1462 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8 icossicc 13413 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 uncom 4154 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10 1xr 11273 . . . . . . . . . . . . . . 15 1 ∈ ℝ*
11 0le1 11737 . . . . . . . . . . . . . . 15 0 ≤ 1
12 snunioc 13457 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
132, 10, 11, 12mp3an 1462 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = (0[,]1)
149, 13eqtr3i 2763 . . . . . . . . . . . . 13 ((0(,]1) ∪ {0}) = (0[,]1)
1514eleq2i 2826 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16 elun 4149 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
1715, 16bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18 pm2.53 850 . . . . . . . . . . 11 ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
1917, 18sylbi 216 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20 elsni 4646 . . . . . . . . . 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 408 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24 0le0 12313 . . . . . . . . . . . . . 14 0 ≤ 0
25 1re 11214 . . . . . . . . . . . . . . 15 1 ∈ ℝ
26 ltpnf 13100 . . . . . . . . . . . . . . 15 (1 ∈ ℝ → 1 < +∞)
2725, 26ax-mp 5 . . . . . . . . . . . . . 14 1 < +∞
28 iocssioo 13416 . . . . . . . . . . . . . 14 (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
292, 3, 24, 27, 28mp4an 692 . . . . . . . . . . . . 13 (0(,]1) ⊆ (0(,)+∞)
30 ioorp 13402 . . . . . . . . . . . . 13 (0(,)+∞) = ℝ+
3129, 30sseqtri 4019 . . . . . . . . . . . 12 (0(,]1) ⊆ ℝ+
3231sseli 3979 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
3332relogcld 26131 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
3433renegcld 11641 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
3534rexrd 11264 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36 elioc1 13366 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥𝑥 ≤ 1)))
372, 10, 36mp2an 691 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥𝑥 ≤ 1))
3837simp3bi 1148 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39 1rp 12978 . . . . . . . . . . . . 13 1 ∈ ℝ+
4039a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
4132, 40logled 26135 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
4238, 41mpbid 231 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43 log1 26094 . . . . . . . . . 10 (log‘1) = 0
4442, 43breqtrdi 5190 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
4533le0neg1d 11785 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
4644, 45mpbid 231 . . . . . . . 8 (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47 ltpnf 13100 . . . . . . . . 9 (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
4834, 47syl 17 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49 elico1 13367 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
502, 3, 49mp2an 691 . . . . . . . 8 (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
5135, 46, 48, 50syl3anbrc 1344 . . . . . . 7 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
5223, 51syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
538, 52sselid 3981 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
547, 53ifclda 4564 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
5554adantl 483 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56 0elunit 13446 . . . . . 6 0 ∈ (0[,]1)
5756a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58 iocssicc 13414 . . . . . 6 (0(,]1) ⊆ (0[,]1)
59 snunico 13456 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
602, 3, 4, 59mp3an 1462 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
6160eleq2i 2826 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62 elun 4149 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
6361, 62bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64 pm2.53 850 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
6563, 64sylbi 216 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66 elsni 4646 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
6765, 66syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
6867con1d 145 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
6968imp 408 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70 rge0ssre 13433 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
7170sseli 3979 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
7271renegcld 11641 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
7372reefcld 16031 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
7473rexrd 11264 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75 efgt0 16046 . . . . . . . . 9 (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
7672, 75syl 17 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77 elico1 13367 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞)))
782, 3, 77mp2an 691 . . . . . . . . . . . 12 (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞))
7978simp2bi 1147 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
8071le0neg2d 11786 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
8179, 80mpbid 231 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82 0re 11216 . . . . . . . . . . 11 0 ∈ ℝ
83 efle 16061 . . . . . . . . . . 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 231 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86 ef0 16034 . . . . . . . . 9 (exp‘0) = 1
8785, 86breqtrdi 5190 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88 elioc1 13366 . . . . . . . . 9 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
892, 10, 88mp2an 691 . . . . . . . 8 ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
9074, 76, 87, 89syl3anbrc 1344 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
9169, 90syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
9258, 91sselid 3981 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
9357, 92ifclda 4564 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
9493adantl 483 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95 eqeq2 2745 . . . . . 6 (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9695bibi1d 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 2745 . . . . . 6 ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9897bibi1d 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 486 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100 iftrue 4535 . . . . . . . 8 (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101100eqeq2d 2744 . . . . . . 7 (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
10299, 101syl5ibrcom 246 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103 ubico 31986 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
10482, 3, 103mp2an 691 . . . . . . . . . 10 ¬ +∞ ∈ (0[,)+∞)
105104nelir 3050 . . . . . . . . 9 +∞ ∉ (0[,)+∞)
106 neleq1 3053 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107106adantl 483 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108105, 107mpbiri 258 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109 neleq1 3053 . . . . . . . 8 (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110108, 109syl5ibcom 244 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111 df-nel 3048 . . . . . . . 8 (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112 iffalse 4538 . . . . . . . . . . . . 13 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113112adantl 483 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114113, 52eqeltrd 2834 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115114ex 414 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116115ad2antrr 725 . . . . . . . . 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 241 . . . . . . 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 211 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121 eqeq2 2745 . . . . . . 7 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122121bibi2d 343 . . . . . 6 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123 eqeq2 2745 . . . . . . 7 (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124123bibi2d 343 . . . . . 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 11350 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128127adantll 713 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129128neneqd 2946 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130 eqeq1 2737 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131130notbid 318 . . . . . . . . 9 (𝑥 = 0 → (¬ 𝑥 = (exp‘-𝑦) ↔ ¬ 0 = (exp‘-𝑦)))
132129, 131syl5ibrcom 246 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133132imp 408 . . . . . . 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 = +∞)
135133, 1342falsed 377 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136 eqcom 2740 . . . . . . . . . . 11 (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137136a1i 11 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138 relogeftb 26093 . . . . . . . . . . 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 11242 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
14171recnd 11242 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142 negcon2 11513 . . . . . . . . . . 11 (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦𝑦 = -(log‘𝑥)))
143140, 141, 142syl2an 597 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦𝑦 = -(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 659 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147146anass1rs 654 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148122, 124, 135, 147ifbothda 4567 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
14996, 98, 120, 148ifbothda 4567 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150149adantl 483 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
1511, 55, 94, 150f1ocnv2d 7659 . 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wtru 1543  wcel 2107  wne 2941  wnel 3047  cun 3947  wss 3949  ifcif 4529  {csn 4629   class class class wbr 5149  cmpt 5232  ccnv 5676  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  cc 11108  cr 11109  0cc0 11110  1c1 11111  +∞cpnf 11245  *cxr 11247   < clt 11248  cle 11249  -cneg 11445  +crp 12974  (,)cioo 13324  (,]cioc 13325  [,)cico 13326  [,]cicc 13327  expce 16005  logclog 26063
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15014  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-sum 15633  df-ef 16011  df-sin 16013  df-cos 16014  df-pi 16016  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640  df-perf 22641  df-cn 22731  df-cnp 22732  df-haus 22819  df-tx 23066  df-hmeo 23259  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-xms 23826  df-ms 23827  df-tms 23828  df-cncf 24394  df-limc 25383  df-dv 25384  df-log 26065
This theorem is referenced by:  xrge0iifiso  32915  xrge0iifmhm  32919  xrge0pluscn  32920
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