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Theorem xrge0iifcnv 32982
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 11263 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11270 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13114 . . . . . . 7 0 ≤ +∞
5 ubicc2 13444 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1461 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8 icossicc 13415 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 uncom 4153 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10 1xr 11275 . . . . . . . . . . . . . . 15 1 ∈ ℝ*
11 0le1 11739 . . . . . . . . . . . . . . 15 0 ≤ 1
12 snunioc 13459 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
132, 10, 11, 12mp3an 1461 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = (0[,]1)
149, 13eqtr3i 2762 . . . . . . . . . . . . 13 ((0(,]1) ∪ {0}) = (0[,]1)
1514eleq2i 2825 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16 elun 4148 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
1715, 16bitr3i 276 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18 pm2.53 849 . . . . . . . . . . 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 4645 . . . . . . . . . 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 407 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24 0le0 12315 . . . . . . . . . . . . . 14 0 ≤ 0
25 1re 11216 . . . . . . . . . . . . . . 15 1 ∈ ℝ
26 ltpnf 13102 . . . . . . . . . . . . . . 15 (1 ∈ ℝ → 1 < +∞)
2725, 26ax-mp 5 . . . . . . . . . . . . . 14 1 < +∞
28 iocssioo 13418 . . . . . . . . . . . . . 14 (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
292, 3, 24, 27, 28mp4an 691 . . . . . . . . . . . . 13 (0(,]1) ⊆ (0(,)+∞)
30 ioorp 13404 . . . . . . . . . . . . 13 (0(,)+∞) = ℝ+
3129, 30sseqtri 4018 . . . . . . . . . . . 12 (0(,]1) ⊆ ℝ+
3231sseli 3978 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
3332relogcld 26138 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
3433renegcld 11643 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
3534rexrd 11266 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36 elioc1 13368 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥𝑥 ≤ 1)))
372, 10, 36mp2an 690 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥𝑥 ≤ 1))
3837simp3bi 1147 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39 1rp 12980 . . . . . . . . . . . . 13 1 ∈ ℝ+
4039a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
4132, 40logled 26142 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
4238, 41mpbid 231 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43 log1 26101 . . . . . . . . . 10 (log‘1) = 0
4442, 43breqtrdi 5189 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
4533le0neg1d 11787 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
4644, 45mpbid 231 . . . . . . . 8 (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47 ltpnf 13102 . . . . . . . . 9 (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
4834, 47syl 17 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49 elico1 13369 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
502, 3, 49mp2an 690 . . . . . . . 8 (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞))
5135, 46, 48, 50syl3anbrc 1343 . . . . . . 7 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
5223, 51syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
538, 52sselid 3980 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
547, 53ifclda 4563 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
5554adantl 482 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56 0elunit 13448 . . . . . 6 0 ∈ (0[,]1)
5756a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58 iocssicc 13416 . . . . . 6 (0(,]1) ⊆ (0[,]1)
59 snunico 13458 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
602, 3, 4, 59mp3an 1461 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
6160eleq2i 2825 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62 elun 4148 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
6361, 62bitr3i 276 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64 pm2.53 849 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
6563, 64sylbi 216 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66 elsni 4645 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
6765, 66syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
6867con1d 145 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
6968imp 407 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70 rge0ssre 13435 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
7170sseli 3978 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
7271renegcld 11643 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
7372reefcld 16033 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
7473rexrd 11266 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75 efgt0 16048 . . . . . . . . 9 (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
7672, 75syl 17 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77 elico1 13369 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞)))
782, 3, 77mp2an 690 . . . . . . . . . . . 12 (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞))
7978simp2bi 1146 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
8071le0neg2d 11788 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
8179, 80mpbid 231 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82 0re 11218 . . . . . . . . . . 11 0 ∈ ℝ
83 efle 16063 . . . . . . . . . . 11 ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8472, 82, 83sylancl 586 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8581, 84mpbid 231 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86 ef0 16036 . . . . . . . . 9 (exp‘0) = 1
8785, 86breqtrdi 5189 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88 elioc1 13368 . . . . . . . . 9 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
892, 10, 88mp2an 690 . . . . . . . 8 ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1))
9074, 76, 87, 89syl3anbrc 1343 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
9169, 90syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
9258, 91sselid 3980 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
9357, 92ifclda 4563 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
9493adantl 482 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95 eqeq2 2744 . . . . . 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 2744 . . . . . 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 485 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100 iftrue 4534 . . . . . . . 8 (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101100eqeq2d 2743 . . . . . . 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 32024 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
10482, 3, 103mp2an 690 . . . . . . . . . 10 ¬ +∞ ∈ (0[,)+∞)
105104nelir 3049 . . . . . . . . 9 +∞ ∉ (0[,)+∞)
106 neleq1 3052 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107106adantl 482 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108105, 107mpbiri 257 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109 neleq1 3052 . . . . . . . 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 3047 . . . . . . . 8 (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112 iffalse 4537 . . . . . . . . . . . . 13 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113112adantl 482 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114113, 52eqeltrd 2833 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115114ex 413 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞)))
116115ad2antrr 724 . . . . . . . . 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 2744 . . . . . . 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 2744 . . . . . . 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 11352 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128127adantll 712 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129128neneqd 2945 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130 eqeq1 2736 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131130notbid 317 . . . . . . . . 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 407 . . . . . . 7 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → ¬ 𝑥 = (exp‘-𝑦))
134 simplr 767 . . . . . . 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 2739 . . . . . . . . . . 11 (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137136a1i 11 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138 relogeftb 26100 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
13932, 72, 138syl2an 596 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
14033recnd 11244 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
14171recnd 11244 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142 negcon2 11515 . . . . . . . . . . 11 (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦𝑦 = -(log‘𝑥)))
143140, 141, 142syl2an 596 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦𝑦 = -(log‘𝑥)))
144137, 139, 1433bitr2d 306 . . . . . . . . 9 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
14523, 69, 144syl2an 596 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146145an4s 658 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147146anass1rs 653 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148122, 124, 135, 147ifbothda 4566 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
14996, 98, 120, 148ifbothda 4566 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150149adantl 482 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
1511, 55, 94, 150f1ocnv2d 7661 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ 𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152151mptru 1548 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 396  wo 845  w3a 1087   = wceq 1541  wtru 1542  wcel 2106  wne 2940  wnel 3046  cun 3946  wss 3948  ifcif 4528  {csn 4628   class class class wbr 5148  cmpt 5231  ccnv 5675  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7411  cc 11110  cr 11111  0cc0 11112  1c1 11113  +∞cpnf 11247  *cxr 11249   < clt 11250  cle 11251  -cneg 11447  +crp 12976  (,)cioo 13326  (,]cioc 13327  [,)cico 13328  [,]cicc 13329  expce 16007  logclog 26070
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-ioo 13330  df-ioc 13331  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-fac 14236  df-bc 14265  df-hash 14293  df-shft 15016  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-limsup 15417  df-clim 15434  df-rlim 15435  df-sum 15635  df-ef 16013  df-sin 16015  df-cos 16016  df-pi 16018  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-starv 17214  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-unif 17222  df-hom 17223  df-cco 17224  df-rest 17370  df-topn 17371  df-0g 17389  df-gsum 17390  df-topgen 17391  df-pt 17392  df-prds 17395  df-xrs 17450  df-qtop 17455  df-imas 17456  df-xps 17458  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-submnd 18674  df-mulg 18953  df-cntz 19183  df-cmn 19652  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-fbas 20947  df-fg 20948  df-cnfld 20951  df-top 22403  df-topon 22420  df-topsp 22442  df-bases 22456  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-lp 22647  df-perf 22648  df-cn 22738  df-cnp 22739  df-haus 22826  df-tx 23073  df-hmeo 23266  df-fil 23357  df-fm 23449  df-flim 23450  df-flf 23451  df-xms 23833  df-ms 23834  df-tms 23835  df-cncf 24401  df-limc 25390  df-dv 25391  df-log 26072
This theorem is referenced by:  xrge0iifiso  32984  xrge0iifmhm  32988  xrge0pluscn  32989
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