Step |
Hyp |
Ref |
Expression |
1 |
|
reeff1 |
⊢ ( exp ↾ ℝ ) : ℝ –1-1→ ℝ+ |
2 |
|
f1f |
⊢ ( ( exp ↾ ℝ ) : ℝ –1-1→ ℝ+ → ( exp ↾ ℝ ) : ℝ ⟶ ℝ+ ) |
3 |
|
ffn |
⊢ ( ( exp ↾ ℝ ) : ℝ ⟶ ℝ+ → ( exp ↾ ℝ ) Fn ℝ ) |
4 |
1 2 3
|
mp2b |
⊢ ( exp ↾ ℝ ) Fn ℝ |
5 |
|
frn |
⊢ ( ( exp ↾ ℝ ) : ℝ ⟶ ℝ+ → ran ( exp ↾ ℝ ) ⊆ ℝ+ ) |
6 |
1 2 5
|
mp2b |
⊢ ran ( exp ↾ ℝ ) ⊆ ℝ+ |
7 |
|
elrp |
⊢ ( 𝑧 ∈ ℝ+ ↔ ( 𝑧 ∈ ℝ ∧ 0 < 𝑧 ) ) |
8 |
|
reclt1 |
⊢ ( ( 𝑧 ∈ ℝ ∧ 0 < 𝑧 ) → ( 𝑧 < 1 ↔ 1 < ( 1 / 𝑧 ) ) ) |
9 |
7 8
|
sylbi |
⊢ ( 𝑧 ∈ ℝ+ → ( 𝑧 < 1 ↔ 1 < ( 1 / 𝑧 ) ) ) |
10 |
|
rpre |
⊢ ( 𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ ) |
11 |
|
rpne0 |
⊢ ( 𝑧 ∈ ℝ+ → 𝑧 ≠ 0 ) |
12 |
10 11
|
rereccld |
⊢ ( 𝑧 ∈ ℝ+ → ( 1 / 𝑧 ) ∈ ℝ ) |
13 |
|
reeff1olem |
⊢ ( ( ( 1 / 𝑧 ) ∈ ℝ ∧ 1 < ( 1 / 𝑧 ) ) → ∃ 𝑦 ∈ ℝ ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) ) |
14 |
12 13
|
sylan |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 1 < ( 1 / 𝑧 ) ) → ∃ 𝑦 ∈ ℝ ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) ) |
15 |
|
eqcom |
⊢ ( ( 1 / 𝑧 ) = ( exp ‘ 𝑦 ) ↔ ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) ) |
16 |
|
rpcnne0 |
⊢ ( 𝑧 ∈ ℝ+ → ( 𝑧 ∈ ℂ ∧ 𝑧 ≠ 0 ) ) |
17 |
|
recn |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℂ ) |
18 |
|
efcl |
⊢ ( 𝑦 ∈ ℂ → ( exp ‘ 𝑦 ) ∈ ℂ ) |
19 |
17 18
|
syl |
⊢ ( 𝑦 ∈ ℝ → ( exp ‘ 𝑦 ) ∈ ℂ ) |
20 |
|
efne0 |
⊢ ( 𝑦 ∈ ℂ → ( exp ‘ 𝑦 ) ≠ 0 ) |
21 |
17 20
|
syl |
⊢ ( 𝑦 ∈ ℝ → ( exp ‘ 𝑦 ) ≠ 0 ) |
22 |
19 21
|
jca |
⊢ ( 𝑦 ∈ ℝ → ( ( exp ‘ 𝑦 ) ∈ ℂ ∧ ( exp ‘ 𝑦 ) ≠ 0 ) ) |
23 |
|
rec11r |
⊢ ( ( ( 𝑧 ∈ ℂ ∧ 𝑧 ≠ 0 ) ∧ ( ( exp ‘ 𝑦 ) ∈ ℂ ∧ ( exp ‘ 𝑦 ) ≠ 0 ) ) → ( ( 1 / 𝑧 ) = ( exp ‘ 𝑦 ) ↔ ( 1 / ( exp ‘ 𝑦 ) ) = 𝑧 ) ) |
24 |
16 22 23
|
syl2an |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑦 ∈ ℝ ) → ( ( 1 / 𝑧 ) = ( exp ‘ 𝑦 ) ↔ ( 1 / ( exp ‘ 𝑦 ) ) = 𝑧 ) ) |
25 |
|
efcan |
⊢ ( 𝑦 ∈ ℂ → ( ( exp ‘ 𝑦 ) · ( exp ‘ - 𝑦 ) ) = 1 ) |
26 |
25
|
eqcomd |
⊢ ( 𝑦 ∈ ℂ → 1 = ( ( exp ‘ 𝑦 ) · ( exp ‘ - 𝑦 ) ) ) |
27 |
|
negcl |
⊢ ( 𝑦 ∈ ℂ → - 𝑦 ∈ ℂ ) |
28 |
|
efcl |
⊢ ( - 𝑦 ∈ ℂ → ( exp ‘ - 𝑦 ) ∈ ℂ ) |
29 |
27 28
|
syl |
⊢ ( 𝑦 ∈ ℂ → ( exp ‘ - 𝑦 ) ∈ ℂ ) |
30 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
31 |
|
divmul2 |
⊢ ( ( 1 ∈ ℂ ∧ ( exp ‘ - 𝑦 ) ∈ ℂ ∧ ( ( exp ‘ 𝑦 ) ∈ ℂ ∧ ( exp ‘ 𝑦 ) ≠ 0 ) ) → ( ( 1 / ( exp ‘ 𝑦 ) ) = ( exp ‘ - 𝑦 ) ↔ 1 = ( ( exp ‘ 𝑦 ) · ( exp ‘ - 𝑦 ) ) ) ) |
32 |
30 31
|
mp3an1 |
⊢ ( ( ( exp ‘ - 𝑦 ) ∈ ℂ ∧ ( ( exp ‘ 𝑦 ) ∈ ℂ ∧ ( exp ‘ 𝑦 ) ≠ 0 ) ) → ( ( 1 / ( exp ‘ 𝑦 ) ) = ( exp ‘ - 𝑦 ) ↔ 1 = ( ( exp ‘ 𝑦 ) · ( exp ‘ - 𝑦 ) ) ) ) |
33 |
29 18 20 32
|
syl12anc |
⊢ ( 𝑦 ∈ ℂ → ( ( 1 / ( exp ‘ 𝑦 ) ) = ( exp ‘ - 𝑦 ) ↔ 1 = ( ( exp ‘ 𝑦 ) · ( exp ‘ - 𝑦 ) ) ) ) |
34 |
26 33
|
mpbird |
⊢ ( 𝑦 ∈ ℂ → ( 1 / ( exp ‘ 𝑦 ) ) = ( exp ‘ - 𝑦 ) ) |
35 |
17 34
|
syl |
⊢ ( 𝑦 ∈ ℝ → ( 1 / ( exp ‘ 𝑦 ) ) = ( exp ‘ - 𝑦 ) ) |
36 |
35
|
eqeq1d |
⊢ ( 𝑦 ∈ ℝ → ( ( 1 / ( exp ‘ 𝑦 ) ) = 𝑧 ↔ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑦 ∈ ℝ ) → ( ( 1 / ( exp ‘ 𝑦 ) ) = 𝑧 ↔ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
38 |
24 37
|
bitrd |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑦 ∈ ℝ ) → ( ( 1 / 𝑧 ) = ( exp ‘ 𝑦 ) ↔ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
39 |
15 38
|
bitr3id |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑦 ∈ ℝ ) → ( ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) ↔ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
40 |
39
|
biimpd |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑦 ∈ ℝ ) → ( ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) → ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
41 |
40
|
reximdva |
⊢ ( 𝑧 ∈ ℝ+ → ( ∃ 𝑦 ∈ ℝ ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) → ∃ 𝑦 ∈ ℝ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 1 < ( 1 / 𝑧 ) ) → ( ∃ 𝑦 ∈ ℝ ( exp ‘ 𝑦 ) = ( 1 / 𝑧 ) → ∃ 𝑦 ∈ ℝ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
43 |
14 42
|
mpd |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 1 < ( 1 / 𝑧 ) ) → ∃ 𝑦 ∈ ℝ ( exp ‘ - 𝑦 ) = 𝑧 ) |
44 |
|
renegcl |
⊢ ( 𝑦 ∈ ℝ → - 𝑦 ∈ ℝ ) |
45 |
|
infm3lem |
⊢ ( 𝑥 ∈ ℝ → ∃ 𝑦 ∈ ℝ 𝑥 = - 𝑦 ) |
46 |
|
fveqeq2 |
⊢ ( 𝑥 = - 𝑦 → ( ( exp ‘ 𝑥 ) = 𝑧 ↔ ( exp ‘ - 𝑦 ) = 𝑧 ) ) |
47 |
44 45 46
|
rexxfr |
⊢ ( ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ↔ ∃ 𝑦 ∈ ℝ ( exp ‘ - 𝑦 ) = 𝑧 ) |
48 |
43 47
|
sylibr |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 1 < ( 1 / 𝑧 ) ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
49 |
48
|
ex |
⊢ ( 𝑧 ∈ ℝ+ → ( 1 < ( 1 / 𝑧 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) ) |
50 |
9 49
|
sylbid |
⊢ ( 𝑧 ∈ ℝ+ → ( 𝑧 < 1 → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) ) |
51 |
50
|
imp |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑧 < 1 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
52 |
|
ef0 |
⊢ ( exp ‘ 0 ) = 1 |
53 |
52
|
eqeq2i |
⊢ ( 𝑧 = ( exp ‘ 0 ) ↔ 𝑧 = 1 ) |
54 |
|
0re |
⊢ 0 ∈ ℝ |
55 |
|
fveqeq2 |
⊢ ( 𝑥 = 0 → ( ( exp ‘ 𝑥 ) = 𝑧 ↔ ( exp ‘ 0 ) = 𝑧 ) ) |
56 |
55
|
rspcev |
⊢ ( ( 0 ∈ ℝ ∧ ( exp ‘ 0 ) = 𝑧 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
57 |
54 56
|
mpan |
⊢ ( ( exp ‘ 0 ) = 𝑧 → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
58 |
57
|
eqcoms |
⊢ ( 𝑧 = ( exp ‘ 0 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
59 |
53 58
|
sylbir |
⊢ ( 𝑧 = 1 → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
60 |
59
|
adantl |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 𝑧 = 1 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
61 |
|
reeff1olem |
⊢ ( ( 𝑧 ∈ ℝ ∧ 1 < 𝑧 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
62 |
10 61
|
sylan |
⊢ ( ( 𝑧 ∈ ℝ+ ∧ 1 < 𝑧 ) → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
63 |
|
1re |
⊢ 1 ∈ ℝ |
64 |
|
lttri4 |
⊢ ( ( 𝑧 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧 ) ) |
65 |
10 63 64
|
sylancl |
⊢ ( 𝑧 ∈ ℝ+ → ( 𝑧 < 1 ∨ 𝑧 = 1 ∨ 1 < 𝑧 ) ) |
66 |
51 60 62 65
|
mpjao3dan |
⊢ ( 𝑧 ∈ ℝ+ → ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
67 |
|
fvres |
⊢ ( 𝑥 ∈ ℝ → ( ( exp ↾ ℝ ) ‘ 𝑥 ) = ( exp ‘ 𝑥 ) ) |
68 |
67
|
eqeq1d |
⊢ ( 𝑥 ∈ ℝ → ( ( ( exp ↾ ℝ ) ‘ 𝑥 ) = 𝑧 ↔ ( exp ‘ 𝑥 ) = 𝑧 ) ) |
69 |
68
|
rexbiia |
⊢ ( ∃ 𝑥 ∈ ℝ ( ( exp ↾ ℝ ) ‘ 𝑥 ) = 𝑧 ↔ ∃ 𝑥 ∈ ℝ ( exp ‘ 𝑥 ) = 𝑧 ) |
70 |
66 69
|
sylibr |
⊢ ( 𝑧 ∈ ℝ+ → ∃ 𝑥 ∈ ℝ ( ( exp ↾ ℝ ) ‘ 𝑥 ) = 𝑧 ) |
71 |
|
fvelrnb |
⊢ ( ( exp ↾ ℝ ) Fn ℝ → ( 𝑧 ∈ ran ( exp ↾ ℝ ) ↔ ∃ 𝑥 ∈ ℝ ( ( exp ↾ ℝ ) ‘ 𝑥 ) = 𝑧 ) ) |
72 |
4 71
|
ax-mp |
⊢ ( 𝑧 ∈ ran ( exp ↾ ℝ ) ↔ ∃ 𝑥 ∈ ℝ ( ( exp ↾ ℝ ) ‘ 𝑥 ) = 𝑧 ) |
73 |
70 72
|
sylibr |
⊢ ( 𝑧 ∈ ℝ+ → 𝑧 ∈ ran ( exp ↾ ℝ ) ) |
74 |
73
|
ssriv |
⊢ ℝ+ ⊆ ran ( exp ↾ ℝ ) |
75 |
6 74
|
eqssi |
⊢ ran ( exp ↾ ℝ ) = ℝ+ |
76 |
|
df-fo |
⊢ ( ( exp ↾ ℝ ) : ℝ –onto→ ℝ+ ↔ ( ( exp ↾ ℝ ) Fn ℝ ∧ ran ( exp ↾ ℝ ) = ℝ+ ) ) |
77 |
4 75 76
|
mpbir2an |
⊢ ( exp ↾ ℝ ) : ℝ –onto→ ℝ+ |
78 |
|
df-f1o |
⊢ ( ( exp ↾ ℝ ) : ℝ –1-1-onto→ ℝ+ ↔ ( ( exp ↾ ℝ ) : ℝ –1-1→ ℝ+ ∧ ( exp ↾ ℝ ) : ℝ –onto→ ℝ+ ) ) |
79 |
1 77 78
|
mpbir2an |
⊢ ( exp ↾ ℝ ) : ℝ –1-1-onto→ ℝ+ |