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Theorem xrge0iifcnv 33916
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 11197 . . . . . . 7 0 ∈ ℝ*
3 pnfxr 11204 . . . . . . 7 +∞ ∈ ℝ*
4 0lepnf 13069 . . . . . . 7 0 ≤ +∞
5 ubicc2 13402 . . . . . . 7 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
62, 3, 4, 5mp3an 1463 . . . . . 6 +∞ ∈ (0[,]+∞)
76a1i 11 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ 𝑥 = 0) → +∞ ∈ (0[,]+∞))
8 icossicc 13373 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
9 uncom 4117 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = ((0(,]1) ∪ {0})
10 1xr 11209 . . . . . . . . . . . . . . 15 1 ∈ ℝ*
11 0le1 11677 . . . . . . . . . . . . . . 15 0 ≤ 1
12 snunioc 13417 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
132, 10, 11, 12mp3an 1463 . . . . . . . . . . . . . 14 ({0} ∪ (0(,]1)) = (0[,]1)
149, 13eqtr3i 2754 . . . . . . . . . . . . 13 ((0(,]1) ∪ {0}) = (0[,]1)
1514eleq2i 2820 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ 𝑥 ∈ (0[,]1))
16 elun 4112 . . . . . . . . . . . 12 (𝑥 ∈ ((0(,]1) ∪ {0}) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
1715, 16bitr3i 277 . . . . . . . . . . 11 (𝑥 ∈ (0[,]1) ↔ (𝑥 ∈ (0(,]1) ∨ 𝑥 ∈ {0}))
18 pm2.53 851 . . . . . . . . . . 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 4602 . . . . . . . . . 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 12263 . . . . . . . . . . . . . 14 0 ≤ 0
25 1re 11150 . . . . . . . . . . . . . . 15 1 ∈ ℝ
26 ltpnf 13056 . . . . . . . . . . . . . . 15 (1 ∈ ℝ → 1 < +∞)
2725, 26ax-mp 5 . . . . . . . . . . . . . 14 1 < +∞
28 iocssioo 13376 . . . . . . . . . . . . . 14 (((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
292, 3, 24, 27, 28mp4an 693 . . . . . . . . . . . . 13 (0(,]1) ⊆ (0(,)+∞)
30 ioorp 13362 . . . . . . . . . . . . 13 (0(,)+∞) = ℝ+
3129, 30sseqtri 3992 . . . . . . . . . . . 12 (0(,]1) ⊆ ℝ+
3231sseli 3939 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ∈ ℝ+)
3332relogcld 26565 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℝ)
3433renegcld 11581 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ)
3534rexrd 11200 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) ∈ ℝ*)
36 elioc1 13324 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1)))
372, 10, 36mp2an 692 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) ↔ (𝑥 ∈ ℝ* ∧ 0 < 𝑥 ∧ 𝑥 ≤ 1))
3837simp3bi 1147 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → 𝑥 ≤ 1)
39 1rp 12931 . . . . . . . . . . . . 13 1 ∈ ℝ+
4039a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ (0(,]1) → 1 ∈ ℝ+)
4132, 40logled 26569 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (𝑥 ≤ 1 ↔ (log‘𝑥) ≤ (log‘1)))
4238, 41mpbid 232 . . . . . . . . . 10 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ (log‘1))
43 log1 26527 . . . . . . . . . 10 (log‘1) = 0
4442, 43breqtrdi 5143 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → (log‘𝑥) ≤ 0)
4533le0neg1d 11725 . . . . . . . . 9 (𝑥 ∈ (0(,]1) → ((log‘𝑥) ≤ 0 ↔ 0 ≤ -(log‘𝑥)))
4644, 45mpbid 232 . . . . . . . 8 (𝑥 ∈ (0(,]1) → 0 ≤ -(log‘𝑥))
47 ltpnf 13056 . . . . . . . . 9 (-(log‘𝑥) ∈ ℝ → -(log‘𝑥) < +∞)
4834, 47syl 17 . . . . . . . 8 (𝑥 ∈ (0(,]1) → -(log‘𝑥) < +∞)
49 elico1 13325 . . . . . . . . 9 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-(log‘𝑥) ∈ (0[,)+∞) ↔ (-(log‘𝑥) ∈ ℝ* ∧ 0 ≤ -(log‘𝑥) ∧ -(log‘𝑥) < +∞)))
502, 3, 49mp2an 692 . . . . . . . 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 3941 . . . . 5 ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) → -(log‘𝑥) ∈ (0[,]+∞))
547, 53ifclda 4520 . . . 4 (𝑥 ∈ (0[,]1) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
5554adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,]+∞))
56 0elunit 13406 . . . . . 6 0 ∈ (0[,]1)
5756a1i 11 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ 𝑦 = +∞) → 0 ∈ (0[,]1))
58 iocssicc 13374 . . . . . 6 (0(,]1) ⊆ (0[,]1)
59 snunico 13416 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → ((0[,)+∞) ∪ {+∞}) = (0[,]+∞))
602, 3, 4, 59mp3an 1463 . . . . . . . . . . . . 13 ((0[,)+∞) ∪ {+∞}) = (0[,]+∞)
6160eleq2i 2820 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ 𝑦 ∈ (0[,]+∞))
62 elun 4112 . . . . . . . . . . . 12 (𝑦 ∈ ((0[,)+∞) ∪ {+∞}) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
6361, 62bitr3i 277 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) ↔ (𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}))
64 pm2.53 851 . . . . . . . . . . 11 ((𝑦 ∈ (0[,)+∞) ∨ 𝑦 ∈ {+∞}) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
6563, 64sylbi 217 . . . . . . . . . 10 (𝑦 ∈ (0[,]+∞) → (¬ 𝑦 ∈ (0[,)+∞) → 𝑦 ∈ {+∞}))
66 elsni 4602 . . . . . . . . . 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 13393 . . . . . . . . . . . 12 (0[,)+∞) ⊆ ℝ
7170sseli 3939 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℝ)
7271renegcld 11581 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ∈ ℝ)
7372reefcld 16030 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ)
7473rexrd 11200 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ∈ ℝ*)
75 efgt0 16047 . . . . . . . . 9 (-𝑦 ∈ ℝ → 0 < (exp‘-𝑦))
7672, 75syl 17 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → 0 < (exp‘-𝑦))
77 elico1 13325 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)))
782, 3, 77mp2an 692 . . . . . . . . . . . 12 (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))
7978simp2bi 1146 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦)
8071le0neg2d 11726 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → (0 ≤ 𝑦 ↔ -𝑦 ≤ 0))
8179, 80mpbid 232 . . . . . . . . . 10 (𝑦 ∈ (0[,)+∞) → -𝑦 ≤ 0)
82 0re 11152 . . . . . . . . . . 11 0 ∈ ℝ
83 efle 16062 . . . . . . . . . . 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 232 . . . . . . . . 9 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ (exp‘0))
86 ef0 16033 . . . . . . . . 9 (exp‘0) = 1
8785, 86breqtrdi 5143 . . . . . . . 8 (𝑦 ∈ (0[,)+∞) → (exp‘-𝑦) ≤ 1)
88 elioc1 13324 . . . . . . . . 9 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((exp‘-𝑦) ∈ (0(,]1) ↔ ((exp‘-𝑦) ∈ ℝ* ∧ 0 < (exp‘-𝑦) ∧ (exp‘-𝑦) ≤ 1)))
892, 10, 88mp2an 692 . . . . . . . 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 3941 . . . . 5 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → (exp‘-𝑦) ∈ (0[,]1))
9357, 92ifclda 4520 . . . 4 (𝑦 ∈ (0[,]+∞) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
9493adantl 481 . . 3 ((⊤ ∧ 𝑦 ∈ (0[,]+∞)) → if(𝑦 = +∞, 0, (exp‘-𝑦)) ∈ (0[,]1))
95 eqeq2 2741 . . . . . 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 2741 . . . . . 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 4490 . . . . . . . 8 (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
101100eqeq2d 2740 . . . . . . 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 32748 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ¬ +∞ ∈ (0[,)+∞))
10482, 3, 103mp2an 692 . . . . . . . . . 10 ¬ +∞ ∈ (0[,)+∞)
105104nelir 3032 . . . . . . . . 9 +∞ ∉ (0[,)+∞)
106 neleq1 3035 . . . . . . . . . 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 3035 . . . . . . . 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 3030 . . . . . . . 8 (if(𝑥 = 0, +∞, -(log‘𝑥)) ∉ (0[,)+∞) ↔ ¬ if(𝑥 = 0, +∞, -(log‘𝑥)) ∈ (0[,)+∞))
112 iffalse 4493 . . . . . . . . . . . . 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 2828 . . . . . . . . . . 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 726 . . . . . . . . 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 2741 . . . . . . 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 2741 . . . . . . 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 11286 . . . . . . . . . . 11 ((𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
128127adantll 714 . . . . . . . . . 10 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → 0 ≠ (exp‘-𝑦))
129128neneqd 2930 . . . . . . . . 9 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → ¬ 0 = (exp‘-𝑦))
130 eqeq1 2733 . . . . . . . . . 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 768 . . . . . . 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 2736 . . . . . . . . . . 11 (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥)
137136a1i 11 . . . . . . . . . 10 ((𝑥 ∈ (0(,]1) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 = (exp‘-𝑦) ↔ (exp‘-𝑦) = 𝑥))
138 relogeftb 26526 . . . . . . . . . . 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 11178 . . . . . . . . . . 11 (𝑥 ∈ (0(,]1) → (log‘𝑥) ∈ ℂ)
14171recnd 11178 . . . . . . . . . . 11 (𝑦 ∈ (0[,)+∞) → 𝑦 ∈ ℂ)
142 negcon2 11451 . . . . . . . . . . 11 (((log‘𝑥) ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((log‘𝑥) = -𝑦 ↔ 𝑦 = -(log‘𝑥)))
143140, 141, 142syl2an 596 . . . . . . . . . 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 596 . . . . . . . 8 (((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 = 0) ∧ (𝑦 ∈ (0[,]+∞) ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
146145an4s 660 . . . . . . 7 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = +∞)) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
147146anass1rs 655 . . . . . 6 ((((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑥 = 0) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = -(log‘𝑥)))
148122, 124, 135, 147ifbothda 4523 . . . . 5 (((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]+∞)) ∧ ¬ 𝑦 = +∞) → (𝑥 = (exp‘-𝑦) ↔ 𝑦 = if(𝑥 = 0, +∞, -(log‘𝑥))))
14996, 98, 120, 148ifbothda 4523 . . . 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 7622 . 2 (⊤ → (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))))
152151mptru 1547 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 847   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109   ≠ wne 2925   ∉ wnel 3029   ∪ cun 3909   ⊆ wss 3911  ifcif 4484  {csn 4585   class class class wbr 5102   ↦ cmpt 5183  ◡ccnv 5630  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045  +∞cpnf 11181  ℝ*cxr 11183   < clt 11184   ≤ cle 11185  -cneg 11382  ℝ+crp 12927  (,)cioo 13282  (,]cioc 13283  [,)cico 13284  [,]cicc 13285  expce 16003  logclog 26496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ioc 13287  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-fac 14215  df-bc 14244  df-hash 14272  df-shft 15009  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-limsup 15413  df-clim 15430  df-rlim 15431  df-sum 15629  df-ef 16009  df-sin 16011  df-cos 16012  df-pi 16014  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17361  df-topn 17362  df-0g 17380  df-gsum 17381  df-topgen 17382  df-pt 17383  df-prds 17386  df-xrs 17441  df-qtop 17446  df-imas 17447  df-xps 17449  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-mulg 18982  df-cntz 19231  df-cmn 19696  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-lp 23056  df-perf 23057  df-cn 23147  df-cnp 23148  df-haus 23235  df-tx 23482  df-hmeo 23675  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-xms 24241  df-ms 24242  df-tms 24243  df-cncf 24804  df-limc 25800  df-dv 25801  df-log 26498
This theorem is referenced by:  xrge0iifiso  33918  xrge0iifmhm  33922  xrge0pluscn  33923
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