MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  logf1o2 Structured version   Visualization version   GIF version

Theorem logf1o2 24113
Description: The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part -π < ℑ(𝑧) < π. The negative reals are mapped to the numbers with imaginary part equal to π. (Contributed by Mario Carneiro, 2-May-2015.)
Hypothesis
Ref Expression
logcn.d 𝐷 = (ℂ ∖ (-∞(,]0))
Assertion
Ref Expression
logf1o2 (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π))

Proof of Theorem logf1o2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 logf1o 24032 . . . 4 log:(ℂ ∖ {0})–1-1-onto→ran log
2 f1of1 6034 . . . 4 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
31, 2ax-mp 5 . . 3 log:(ℂ ∖ {0})–1-1→ran log
4 logcn.d . . . 4 𝐷 = (ℂ ∖ (-∞(,]0))
54logdmss 24105 . . 3 𝐷 ⊆ (ℂ ∖ {0})
6 f1ores 6049 . . 3 ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷))
73, 5, 6mp2an 703 . 2 (log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷)
8 f1ofun 6037 . . . . . . 7 (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
91, 8ax-mp 5 . . . . . 6 Fun log
10 f1of 6035 . . . . . . . . 9 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
111, 10ax-mp 5 . . . . . . . 8 log:(ℂ ∖ {0})⟶ran log
1211fdmi 5951 . . . . . . 7 dom log = (ℂ ∖ {0})
135, 12sseqtr4i 3600 . . . . . 6 𝐷 ⊆ dom log
14 funimass4 6142 . . . . . 6 ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (ℑ “ (-π(,)π)) ↔ ∀𝑥𝐷 (log‘𝑥) ∈ (ℑ “ (-π(,)π))))
159, 13, 14mp2an 703 . . . . 5 ((log “ 𝐷) ⊆ (ℑ “ (-π(,)π)) ↔ ∀𝑥𝐷 (log‘𝑥) ∈ (ℑ “ (-π(,)π)))
164ellogdm 24102 . . . . . . . 8 (𝑥𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)))
1716simplbi 474 . . . . . . 7 (𝑥𝐷𝑥 ∈ ℂ)
184logdmn0 24103 . . . . . . 7 (𝑥𝐷𝑥 ≠ 0)
1917, 18logcld 24038 . . . . . 6 (𝑥𝐷 → (log‘𝑥) ∈ ℂ)
2019imcld 13729 . . . . . . 7 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
2117, 18logimcld 24039 . . . . . . . 8 (𝑥𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
2221simpld 473 . . . . . . 7 (𝑥𝐷 → -π < (ℑ‘(log‘𝑥)))
234logdmnrp 24104 . . . . . . . . . 10 (𝑥𝐷 → ¬ -𝑥 ∈ ℝ+)
24 lognegb 24057 . . . . . . . . . . . 12 ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
2517, 18, 24syl2anc 690 . . . . . . . . . . 11 (𝑥𝐷 → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
2625necon3bbid 2818 . . . . . . . . . 10 (𝑥𝐷 → (¬ -𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) ≠ π))
2723, 26mpbid 220 . . . . . . . . 9 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
2827necomd 2836 . . . . . . . 8 (𝑥𝐷 → π ≠ (ℑ‘(log‘𝑥)))
29 pire 23931 . . . . . . . . . 10 π ∈ ℝ
3029a1i 11 . . . . . . . . 9 (𝑥𝐷 → π ∈ ℝ)
3121simprd 477 . . . . . . . . 9 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
3220, 30, 31leltned 10041 . . . . . . . 8 (𝑥𝐷 → ((ℑ‘(log‘𝑥)) < π ↔ π ≠ (ℑ‘(log‘𝑥))))
3328, 32mpbird 245 . . . . . . 7 (𝑥𝐷 → (ℑ‘(log‘𝑥)) < π)
3429renegcli 10193 . . . . . . . . 9 -π ∈ ℝ
3534rexri 9948 . . . . . . . 8 -π ∈ ℝ*
3629rexri 9948 . . . . . . . 8 π ∈ ℝ*
37 elioo2 12043 . . . . . . . 8 ((-π ∈ ℝ* ∧ π ∈ ℝ*) → ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π)))
3835, 36, 37mp2an 703 . . . . . . 7 ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
3920, 22, 33, 38syl3anbrc 1238 . . . . . 6 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
40 imf 13647 . . . . . . 7 ℑ:ℂ⟶ℝ
41 ffn 5944 . . . . . . 7 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
42 elpreima 6230 . . . . . . 7 (ℑ Fn ℂ → ((log‘𝑥) ∈ (ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π))))
4340, 41, 42mp2b 10 . . . . . 6 ((log‘𝑥) ∈ (ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π)))
4419, 39, 43sylanbrc 694 . . . . 5 (𝑥𝐷 → (log‘𝑥) ∈ (ℑ “ (-π(,)π)))
4515, 44mprgbir 2910 . . . 4 (log “ 𝐷) ⊆ (ℑ “ (-π(,)π))
46 elpreima 6230 . . . . . . 7 (ℑ Fn ℂ → (𝑥 ∈ (ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π))))
4740, 41, 46mp2b 10 . . . . . 6 (𝑥 ∈ (ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)))
48 simpl 471 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ℂ)
49 eliooord 12060 . . . . . . . . . . 11 ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
5049adantl 480 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
5150simpld 473 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → -π < (ℑ‘𝑥))
5250simprd 477 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) < π)
53 imcl 13645 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
5453adantr 479 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
55 ltle 9977 . . . . . . . . . . 11 (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
5654, 29, 55sylancl 692 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
5752, 56mpd 15 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ≤ π)
58 ellogrn 24027 . . . . . . . . 9 (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
5948, 51, 57, 58syl3anbrc 1238 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
60 logef 24049 . . . . . . . 8 (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
6159, 60syl 17 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
62 efcl 14598 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
6362adantr 479 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
6454adantr 479 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℝ)
6564recnd 9924 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
66 picn 23932 . . . . . . . . . . . . . 14 π ∈ ℂ
6766a1i 11 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
68 pipos 23933 . . . . . . . . . . . . . . 15 0 < π
6929, 68gt0ne0ii 10413 . . . . . . . . . . . . . 14 π ≠ 0
7069a1i 11 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ≠ 0)
7152adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < π)
7266mulid1i 9898 . . . . . . . . . . . . . . . . . 18 (π · 1) = π
7371, 72syl6breqr 4619 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
74 1re 9895 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
7574a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 1 ∈ ℝ)
7629a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℝ)
7768a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 < π)
78 ltdivmul 10747 . . . . . . . . . . . . . . . . . 18 (((ℑ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
7964, 75, 76, 77, 78syl112anc 1321 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
8073, 79mpbird 245 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < 1)
81 1e0p1 11384 . . . . . . . . . . . . . . . 16 1 = (0 + 1)
8280, 81syl6breq 4618 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
8364recoscld 14659 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
8464resincld 14658 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
8583, 84crimd 13766 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
86 efeul 14677 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
8786ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
8887oveq1d 6542 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = (((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) / (exp‘(ℜ‘𝑥))))
8983recnd 9924 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
90 ax-icn 9851 . . . . . . . . . . . . . . . . . . . . . . . 24 i ∈ ℂ
9184recnd 9924 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
92 mulcl 9876 . . . . . . . . . . . . . . . . . . . . . . . 24 ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
9390, 91, 92sylancr 693 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
9489, 93addcld 9915 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
95 recl 13644 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
9695ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
9796recnd 9924 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
98 efcl 14598 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℜ‘𝑥) ∈ ℂ → (exp‘(ℜ‘𝑥)) ∈ ℂ)
9997, 98syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℂ)
100 efne0 14612 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℜ‘𝑥) ∈ ℂ → (exp‘(ℜ‘𝑥)) ≠ 0)
10197, 100syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ≠ 0)
10294, 99, 101divcan3d 10655 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
10388, 102eqtrd 2643 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
104 simpr 475 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ)
10596reefcld 14603 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
106104, 105, 101redivcld 10702 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
107103, 106eqeltrrd 2688 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
108107reim0d 13759 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
10985, 108eqtr3d 2645 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
110 sineq0 23994 . . . . . . . . . . . . . . . . . 18 ((ℑ‘𝑥) ∈ ℂ → ((sin‘(ℑ‘𝑥)) = 0 ↔ ((ℑ‘𝑥) / π) ∈ ℤ))
11165, 110syl 17 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((sin‘(ℑ‘𝑥)) = 0 ↔ ((ℑ‘𝑥) / π) ∈ ℤ))
112109, 111mpbid 220 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℤ)
113 0z 11221 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
114 zleltp1 11261 . . . . . . . . . . . . . . . 16 ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
115112, 113, 114sylancl 692 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
11682, 115mpbird 245 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ≤ 0)
117 df-neg 10120 . . . . . . . . . . . . . . . 16 -1 = (0 − 1)
11866mulm1i 10325 . . . . . . . . . . . . . . . . . 18 (-1 · π) = -π
11951adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -π < (ℑ‘𝑥))
120118, 119syl5eqbr 4612 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
12174renegcli 10193 . . . . . . . . . . . . . . . . . . 19 -1 ∈ ℝ
122121a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
123 ltmuldiv 10745 . . . . . . . . . . . . . . . . . 18 ((-1 ∈ ℝ ∧ (ℑ‘𝑥) ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
124122, 64, 76, 77, 123syl112anc 1321 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
125120, 124mpbid 220 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 < ((ℑ‘𝑥) / π))
126117, 125syl5eqbrr 4613 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
127 zlem1lt 11262 . . . . . . . . . . . . . . . 16 ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
128113, 112, 127sylancr 693 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
129126, 128mpbird 245 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 ≤ ((ℑ‘𝑥) / π))
13064, 76, 70redivcld 10702 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
131 0re 9896 . . . . . . . . . . . . . . 15 0 ∈ ℝ
132 letri3 9974 . . . . . . . . . . . . . . 15 ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
133130, 131, 132sylancl 692 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
134116, 129, 133mpbir2and 958 . . . . . . . . . . . . 13 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
13565, 67, 70, 134diveq0d 10657 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
136 reim0b 13653 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
137136ad2antrr 757 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
138135, 137mpbird 245 . . . . . . . . . . 11 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 𝑥 ∈ ℝ)
139138rpefcld 14620 . . . . . . . . . 10 (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ+)
140139ex 448 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ+))
1414ellogdm 24102 . . . . . . . . 9 ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ+)))
14263, 140, 141sylanbrc 694 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
143 funfvima2 6375 . . . . . . . . 9 ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
1449, 13, 143mp2an 703 . . . . . . . 8 ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
145142, 144syl 17 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
14661, 145eqeltrrd 2688 . . . . . 6 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
14747, 146sylbi 205 . . . . 5 (𝑥 ∈ (ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
148147ssriv 3571 . . . 4 (ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
14945, 148eqssi 3583 . . 3 (log “ 𝐷) = (ℑ “ (-π(,)π))
150 f1oeq3 6027 . . 3 ((log “ 𝐷) = (ℑ “ (-π(,)π)) → ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) ↔ (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π))))
151149, 150ax-mp 5 . 2 ((log ↾ 𝐷):𝐷1-1-onto→(log “ 𝐷) ↔ (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π)))
1527, 151mpbi 218 1 (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  cdif 3536  wss 3539  {csn 4124   class class class wbr 4577  ccnv 5027  dom cdm 5028  ran crn 5029  cres 5030  cima 5031  Fun wfun 5784   Fn wfn 5785  wf 5786  1-1wf1 5787  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793  ici 9794   + caddc 9795   · cmul 9797  -∞cmnf 9928  *cxr 9929   < clt 9930  cle 9931  cmin 10117  -cneg 10118   / cdiv 10533  cz 11210  +crp 11664  (,)cioo 12002  (,]cioc 12003  cre 13631  cim 13632  expce 14577  sincsin 14579  cosccos 14580  πcpi 14582  logclog 24022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ioc 12007  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-seq 12619  df-exp 12678  df-fac 12878  df-bc 12907  df-hash 12935  df-shft 13601  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-limsup 13996  df-clim 14013  df-rlim 14014  df-sum 14211  df-ef 14583  df-sin 14585  df-cos 14586  df-pi 14588  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-sca 15730  df-vsca 15731  df-ip 15732  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-hom 15739  df-cco 15740  df-rest 15852  df-topn 15853  df-0g 15871  df-gsum 15872  df-topgen 15873  df-pt 15874  df-prds 15877  df-xrs 15931  df-qtop 15936  df-imas 15937  df-xps 15939  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-mulg 17310  df-cntz 17519  df-cmn 17964  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-cnfld 19514  df-top 20463  df-bases 20464  df-topon 20465  df-topsp 20466  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lp 20692  df-perf 20693  df-cn 20783  df-cnp 20784  df-haus 20871  df-tx 21117  df-hmeo 21310  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-xms 21876  df-ms 21877  df-tms 21878  df-cncf 22420  df-limc 23353  df-dv 23354  df-log 24024
This theorem is referenced by:  efopnlem2  24120
  Copyright terms: Public domain W3C validator