Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrge0iifcnv Structured version   Visualization version   GIF version

Theorem xrge0iifcnv 31418
 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 10740 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 10747 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 12582 . . . . . . 7 0 ≤ +∞
5 ubicc2 12911 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1459 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8 icossicc 12882 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 uncom 4061 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10 1xr 10752 . . . . . . . . . . . . . . 15 1 ∈ ℝ*
11 0le1 11215 . . . . . . . . . . . . . . 15 0 ≤ 1
12 snunioc 12926 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
132, 10, 11, 12mp3an 1459 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = (0[,]1)
149, 13eqtr3i 2784 . . . . . . . . . . . . 13 ((0(,]1) ∪ {0}) = (0[,]1)
1514eleq2i 2844 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16 elun 4057 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
1715, 16bitr3i 280 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18 pm2.53 848 . . . . . . . . . . 11 ((𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
1917, 18sylbi 220 . . . . . . . . . 10 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 ∈ {0}))
20 elsni 4543 . . . . . . . . . 10 (𝑥 ∈ {0} → 𝑥 = 0)
2119, 20syl6 35 . . . . . . . . 9 (𝑥 ∈ (0[,]1) → (¬ 𝑥 ∈ (0(,]1) → 𝑥 = 0))
2221con1d 147 . . . . . . . 8 (𝑥 ∈ (0[,]1) → (¬ 𝑥 = 0 → 𝑥 ∈ (0(,]1)))
2322imp 410 . . . . . . 7 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → 𝑥 ∈ (0(,]1))
24 0le0 11789 . . . . . . . . . . . . . 14 0 ≤ 0
25 1re 10693 . . . . . . . . . . . . . . 15 1 ∈ ℝ
26 ltpnf 12570 . . . . . . . . . . . . . . 15 (1 ∈ ℝ → 1 < +∞)
2725, 26ax-mp 5 . . . . . . . . . . . . . 14 1 < +∞
28 iocssioo 12885 . . . . . . . . . . . . . 14 (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
292, 3, 24, 27, 28mp4an 692 . . . . . . . . . . . . 13 (0(,]1) ⊆ (0(,)+∞)
30 ioorp 12871 . . . . . . . . . . . . 13 (0(,)+∞) = ℝ+
3129, 30sseqtri 3931 . . . . . . . . . . . 12 (0(,]1) ⊆ ℝ+
3231sseli 3891 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
3332relogcld 25328 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
3433renegcld 11119 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
3534rexrd 10743 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36 elioc1 12835 . . . . . . . . . . . . 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 1145 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39 1rp 12448 . . . . . . . . . . . . 13 1 ∈ ℝ+
4039a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
4132, 40logled 25332 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
4238, 41mpbid 235 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43 log1 25291 . . . . . . . . . 10 (log‘1) = 0
4442, 43breqtrdi 5078 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
4533le0neg1d 11263 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
4644, 45mpbid 235 . . . . . . . 8 (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47 ltpnf 12570 . . . . . . . . 9 (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
4834, 47syl 17 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49 elico1 12836 . . . . . . . . 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 1341 . . . . . . 7 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ (0[,)+∞))
5223, 51syl 17 . . . . . 6 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,)+∞))
538, 52sseldi 3893 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
547, 53ifclda 4459 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
5554adantl 485 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56 0elunit 12915 . . . . . 6 0 ∈ (0[,]1)
5756a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58 iocssicc 12883 . . . . . 6 (0(,]1) ⊆ (0[,]1)
59 snunico 12925 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
602, 3, 4, 59mp3an 1459 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
6160eleq2i 2844 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62 elun 4057 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
6361, 62bitr3i 280 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64 pm2.53 848 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
6563, 64sylbi 220 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66 elsni 4543 . . . . . . . . . 10 (𝑦 ∈ {+∞} → 𝑦 = +∞)
6765, 66syl6 35 . . . . . . . . 9 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 = +∞))
6867con1d 147 . . . . . . . 8 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 = +∞ → 𝑦 ∈ (0[,)+∞)))
6968imp 410 . . . . . . 7 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ (0[,)+∞))
70 rge0ssre 12902 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
7170sseli 3891 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
7271renegcld 11119 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
7372reefcld 15503 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
7473rexrd 10743 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75 efgt0 15518 . . . . . . . . 9 (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
7672, 75syl 17 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77 elico1 12836 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞)))
782, 3, 77mp2an 691 . . . . . . . . . . . 12 (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦𝑦 < +∞))
7978simp2bi 1144 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
8071le0neg2d 11264 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
8179, 80mpbid 235 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82 0re 10695 . . . . . . . . . . 11 0 ∈ ℝ
83 efle 15533 . . . . . . . . . . 11 ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8472, 82, 83sylancl 589 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → (-𝑦 ≤ 0 ↔ (exp‘-𝑦) ≤ (exp‘0)))
8581, 84mpbid 235 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86 ef0 15506 . . . . . . . . 9 (exp‘0) = 1
8785, 86breqtrdi 5078 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88 elioc1 12835 . . . . . . . . 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 1341 . . . . . . 7 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ (0(,]1))
9169, 90syl 17 . . . . . 6 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0(,]1))
9258, 91sseldi 3893 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
9357, 92ifclda 4459 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
9493adantl 485 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95 eqeq2 2771 . . . . . 6 (0 = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = 0 ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9695bibi1d 347 . . . . 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 2771 . . . . . 6 ((exp‘-𝑦) = if(𝑦 = +∞, 0, (exp‘-𝑦)) → (𝑥 = (exp‘-𝑦) ↔ 𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦))))
9897bibi1d 347 . . . . 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 488 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 = +∞)
100 iftrue 4430 . . . . . . . 8 (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101100eqeq2d 2770 . . . . . . 7 (𝑥 = 0 → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) ↔ 𝑦 = +∞))
10299, 101syl5ibrcom 250 . . . . . 6 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 → 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
103 ubico 30634 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
10482, 3, 103mp2an 691 . . . . . . . . . 10 ¬ +∞ ∈ (0[,)+∞)
105104nelir 3059 . . . . . . . . 9 +∞ ∉ (0[,)+∞)
106 neleq1 3061 . . . . . . . . . 10 (𝑦 = +∞ → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
107106adantl 485 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 ∉ (0[,)+∞) ↔ +∞ ∉ (0[,)+∞)))
108105, 107mpbiri 261 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → 𝑦 ∉ (0[,)+∞))
109 neleq1 3061 . . . . . . . 8 (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 ∉ (0[,)+∞) ↔ if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
110108, 109syl5ibcom 248 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞)))
111 df-nel 3057 . . . . . . . 8 (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112 iffalse 4433 . . . . . . . . . . . . 13 𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
113112adantl 485 . . . . . . . . . . . 12 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) = -(log‘𝑥))
114113, 52eqeltrd 2853 . . . . . . . . . . 11 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
115114ex 416 . . . . . . . . . 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 147 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞) → 𝑥 = 0))
118111, 117syl5bi 245 . . . . . . 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 215 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 𝑦 = +∞) → (𝑥 = 0 ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
121 eqeq2 2771 . . . . . . 7 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = +∞ ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
122121bibi2d 346 . . . . . 6 (+∞ = if(𝑥 = 0, +∞, -(log‘𝑥)) → ((𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞) ↔ (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥)))))
123 eqeq2 2771 . . . . . . 7 (-(log‘𝑥) = if(𝑥 = 0, +∞, -(log‘𝑥)) → (𝑦 = -(log‘𝑥) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
124123bibi2d 346 . . . . . 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 10828 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128127adantll 713 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129128neneqd 2957 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130 eqeq1 2763 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 = (exp‘-𝑦) ↔ 0 = (exp‘-𝑦)))
131130notbid 321 . . . . . . . . 9 (𝑥 = 0 → (¬ 𝑥 = (exp‘-𝑦) ↔ ¬ 0 = (exp‘-𝑦)))
132129, 131syl5ibrcom 250 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = 0 → ¬ 𝑥 = (exp‘-𝑦)))
133132imp 410 . . . . . . 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 380 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = +∞))
136 eqcom 2766 . . . . . . . . . . 11 (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137136a1i 11 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138 relogeftb 25290 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ -𝑦 ∈ ℝ) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
13932, 72, 138syl2an 598 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦 ↔ (exp‘-𝑦) = 𝑥))
14033recnd 10721 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
14171recnd 10721 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142 negcon2 10991 . . . . . . . . . . 11 (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦𝑦 = -(log‘𝑥)))
143140, 141, 142syl2an 598 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → ((log‘𝑥) = -𝑦𝑦 = -(log‘𝑥)))
144137, 139, 1433bitr2d 310 . . . . . . . . 9 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
14523, 69, 144syl2an 598 . . . . . . . 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 4462 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
14996, 98, 120, 148ifbothda 4462 . . . 4 ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
150149adantl 485 . . 3 ((⊤ ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 = if(𝑦 = +∞, 0, (exp‘-𝑦)) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
1511, 55, 94, 150f1ocnv2d 7401 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ 𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152151mptru 1546 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1085   = wceq 1539  ⊤wtru 1540   ∈ wcel 2112   ≠ wne 2952   ∉ wnel 3056   ∪ cun 3859   ⊆ wss 3861  ifcif 4424  {csn 4526   class class class wbr 5037   ↦ cmpt 5117  ◡ccnv 5528  –1-1-onto→wf1o 6340  ‘cfv 6341  (class class class)co 7157  ℂcc 10587  ℝcr 10588  0cc0 10589  1c1 10590  +∞cpnf 10724  ℝ*cxr 10726   < clt 10727   ≤ cle 10728  -cneg 10923  ℝ+crp 12444  (,)cioo 12793  (,]cioc 12794  [,)cico 12795  [,]cicc 12796  expce 15477  logclog 25260 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-inf2 9151  ax-cnex 10645  ax-resscn 10646  ax-1cn 10647  ax-icn 10648  ax-addcl 10649  ax-addrcl 10650  ax-mulcl 10651  ax-mulrcl 10652  ax-mulcom 10653  ax-addass 10654  ax-mulass 10655  ax-distr 10656  ax-i2m1 10657  ax-1ne0 10658  ax-1rid 10659  ax-rnegex 10660  ax-rrecex 10661  ax-cnre 10662  ax-pre-lttri 10663  ax-pre-lttrn 10664  ax-pre-ltadd 10665  ax-pre-mulgt0 10666  ax-pre-sup 10667  ax-addf 10668  ax-mulf 10669 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-iin 4890  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-se 5489  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-of 7412  df-om 7587  df-1st 7700  df-2nd 7701  df-supp 7843  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-2o 8120  df-er 8306  df-map 8425  df-pm 8426  df-ixp 8494  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-fsupp 8881  df-fi 8922  df-sup 8953  df-inf 8954  df-oi 9021  df-card 9415  df-pnf 10729  df-mnf 10730  df-xr 10731  df-ltxr 10732  df-le 10733  df-sub 10924  df-neg 10925  df-div 11350  df-nn 11689  df-2 11751  df-3 11752  df-4 11753  df-5 11754  df-6 11755  df-7 11756  df-8 11757  df-9 11758  df-n0 11949  df-z 12035  df-dec 12152  df-uz 12297  df-q 12403  df-rp 12445  df-xneg 12562  df-xadd 12563  df-xmul 12564  df-ioo 12797  df-ioc 12798  df-ico 12799  df-icc 12800  df-fz 12954  df-fzo 13097  df-fl 13225  df-mod 13301  df-seq 13433  df-exp 13494  df-fac 13698  df-bc 13727  df-hash 13755  df-shft 14488  df-cj 14520  df-re 14521  df-im 14522  df-sqrt 14656  df-abs 14657  df-limsup 14890  df-clim 14907  df-rlim 14908  df-sum 15105  df-ef 15483  df-sin 15485  df-cos 15486  df-pi 15488  df-struct 16558  df-ndx 16559  df-slot 16560  df-base 16562  df-sets 16563  df-ress 16564  df-plusg 16651  df-mulr 16652  df-starv 16653  df-sca 16654  df-vsca 16655  df-ip 16656  df-tset 16657  df-ple 16658  df-ds 16660  df-unif 16661  df-hom 16662  df-cco 16663  df-rest 16769  df-topn 16770  df-0g 16788  df-gsum 16789  df-topgen 16790  df-pt 16791  df-prds 16794  df-xrs 16848  df-qtop 16853  df-imas 16854  df-xps 16856  df-mre 16930  df-mrc 16931  df-acs 16933  df-mgm 17933  df-sgrp 17982  df-mnd 17993  df-submnd 18038  df-mulg 18307  df-cntz 18529  df-cmn 18990  df-psmet 20173  df-xmet 20174  df-met 20175  df-bl 20176  df-mopn 20177  df-fbas 20178  df-fg 20179  df-cnfld 20182  df-top 21609  df-topon 21626  df-topsp 21648  df-bases 21661  df-cld 21734  df-ntr 21735  df-cls 21736  df-nei 21813  df-lp 21851  df-perf 21852  df-cn 21942  df-cnp 21943  df-haus 22030  df-tx 22277  df-hmeo 22470  df-fil 22561  df-fm 22653  df-flim 22654  df-flf 22655  df-xms 23037  df-ms 23038  df-tms 23039  df-cncf 23594  df-limc 24580  df-dv 24581  df-log 25262 This theorem is referenced by:  xrge0iifiso  31420  xrge0iifmhm  31424  xrge0pluscn  31425
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