Step |
Hyp |
Ref |
Expression |
1 |
|
xrge0iifhmeo.1 |
|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) ) |
2 |
|
0xr |
|- 0 e. RR* |
3 |
|
pnfxr |
|- +oo e. RR* |
4 |
|
0lepnf |
|- 0 <_ +oo |
5 |
|
ubicc2 |
|- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> +oo e. ( 0 [,] +oo ) ) |
6 |
2 3 4 5
|
mp3an |
|- +oo e. ( 0 [,] +oo ) |
7 |
6
|
a1i |
|- ( ( x e. ( 0 [,] 1 ) /\ x = 0 ) -> +oo e. ( 0 [,] +oo ) ) |
8 |
|
icossicc |
|- ( 0 [,) +oo ) C_ ( 0 [,] +oo ) |
9 |
|
uncom |
|- ( { 0 } u. ( 0 (,] 1 ) ) = ( ( 0 (,] 1 ) u. { 0 } ) |
10 |
|
1xr |
|- 1 e. RR* |
11 |
|
0le1 |
|- 0 <_ 1 |
12 |
|
snunioc |
|- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 ) ) |
13 |
2 10 11 12
|
mp3an |
|- ( { 0 } u. ( 0 (,] 1 ) ) = ( 0 [,] 1 ) |
14 |
9 13
|
eqtr3i |
|- ( ( 0 (,] 1 ) u. { 0 } ) = ( 0 [,] 1 ) |
15 |
14
|
eleq2i |
|- ( x e. ( ( 0 (,] 1 ) u. { 0 } ) <-> x e. ( 0 [,] 1 ) ) |
16 |
|
elun |
|- ( x e. ( ( 0 (,] 1 ) u. { 0 } ) <-> ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) ) |
17 |
15 16
|
bitr3i |
|- ( x e. ( 0 [,] 1 ) <-> ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) ) |
18 |
|
pm2.53 |
|- ( ( x e. ( 0 (,] 1 ) \/ x e. { 0 } ) -> ( -. x e. ( 0 (,] 1 ) -> x e. { 0 } ) ) |
19 |
17 18
|
sylbi |
|- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 (,] 1 ) -> x e. { 0 } ) ) |
20 |
|
elsni |
|- ( x e. { 0 } -> x = 0 ) |
21 |
19 20
|
syl6 |
|- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 (,] 1 ) -> x = 0 ) ) |
22 |
21
|
con1d |
|- ( x e. ( 0 [,] 1 ) -> ( -. x = 0 -> x e. ( 0 (,] 1 ) ) ) |
23 |
22
|
imp |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> x e. ( 0 (,] 1 ) ) |
24 |
|
0le0 |
|- 0 <_ 0 |
25 |
|
1re |
|- 1 e. RR |
26 |
|
ltpnf |
|- ( 1 e. RR -> 1 < +oo ) |
27 |
25 26
|
ax-mp |
|- 1 < +oo |
28 |
|
iocssioo |
|- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ 1 < +oo ) ) -> ( 0 (,] 1 ) C_ ( 0 (,) +oo ) ) |
29 |
2 3 24 27 28
|
mp4an |
|- ( 0 (,] 1 ) C_ ( 0 (,) +oo ) |
30 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
31 |
29 30
|
sseqtri |
|- ( 0 (,] 1 ) C_ RR+ |
32 |
31
|
sseli |
|- ( x e. ( 0 (,] 1 ) -> x e. RR+ ) |
33 |
32
|
relogcld |
|- ( x e. ( 0 (,] 1 ) -> ( log ` x ) e. RR ) |
34 |
33
|
renegcld |
|- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. RR ) |
35 |
34
|
rexrd |
|- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. RR* ) |
36 |
|
elioc1 |
|- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( x e. ( 0 (,] 1 ) <-> ( x e. RR* /\ 0 < x /\ x <_ 1 ) ) ) |
37 |
2 10 36
|
mp2an |
|- ( x e. ( 0 (,] 1 ) <-> ( x e. RR* /\ 0 < x /\ x <_ 1 ) ) |
38 |
37
|
simp3bi |
|- ( x e. ( 0 (,] 1 ) -> x <_ 1 ) |
39 |
|
1rp |
|- 1 e. RR+ |
40 |
39
|
a1i |
|- ( x e. ( 0 (,] 1 ) -> 1 e. RR+ ) |
41 |
32 40
|
logled |
|- ( x e. ( 0 (,] 1 ) -> ( x <_ 1 <-> ( log ` x ) <_ ( log ` 1 ) ) ) |
42 |
38 41
|
mpbid |
|- ( x e. ( 0 (,] 1 ) -> ( log ` x ) <_ ( log ` 1 ) ) |
43 |
|
log1 |
|- ( log ` 1 ) = 0 |
44 |
42 43
|
breqtrdi |
|- ( x e. ( 0 (,] 1 ) -> ( log ` x ) <_ 0 ) |
45 |
33
|
le0neg1d |
|- ( x e. ( 0 (,] 1 ) -> ( ( log ` x ) <_ 0 <-> 0 <_ -u ( log ` x ) ) ) |
46 |
44 45
|
mpbid |
|- ( x e. ( 0 (,] 1 ) -> 0 <_ -u ( log ` x ) ) |
47 |
|
ltpnf |
|- ( -u ( log ` x ) e. RR -> -u ( log ` x ) < +oo ) |
48 |
34 47
|
syl |
|- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) < +oo ) |
49 |
|
elico1 |
|- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( -u ( log ` x ) e. ( 0 [,) +oo ) <-> ( -u ( log ` x ) e. RR* /\ 0 <_ -u ( log ` x ) /\ -u ( log ` x ) < +oo ) ) ) |
50 |
2 3 49
|
mp2an |
|- ( -u ( log ` x ) e. ( 0 [,) +oo ) <-> ( -u ( log ` x ) e. RR* /\ 0 <_ -u ( log ` x ) /\ -u ( log ` x ) < +oo ) ) |
51 |
35 46 48 50
|
syl3anbrc |
|- ( x e. ( 0 (,] 1 ) -> -u ( log ` x ) e. ( 0 [,) +oo ) ) |
52 |
23 51
|
syl |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> -u ( log ` x ) e. ( 0 [,) +oo ) ) |
53 |
8 52
|
sselid |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> -u ( log ` x ) e. ( 0 [,] +oo ) ) |
54 |
7 53
|
ifclda |
|- ( x e. ( 0 [,] 1 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,] +oo ) ) |
55 |
54
|
adantl |
|- ( ( T. /\ x e. ( 0 [,] 1 ) ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,] +oo ) ) |
56 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
57 |
56
|
a1i |
|- ( ( y e. ( 0 [,] +oo ) /\ y = +oo ) -> 0 e. ( 0 [,] 1 ) ) |
58 |
|
iocssicc |
|- ( 0 (,] 1 ) C_ ( 0 [,] 1 ) |
59 |
|
snunico |
|- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo ) ) |
60 |
2 3 4 59
|
mp3an |
|- ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo ) |
61 |
60
|
eleq2i |
|- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> y e. ( 0 [,] +oo ) ) |
62 |
|
elun |
|- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) ) |
63 |
61 62
|
bitr3i |
|- ( y e. ( 0 [,] +oo ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) ) |
64 |
|
pm2.53 |
|- ( ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) ) |
65 |
63 64
|
sylbi |
|- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) ) |
66 |
|
elsni |
|- ( y e. { +oo } -> y = +oo ) |
67 |
65 66
|
syl6 |
|- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y = +oo ) ) |
68 |
67
|
con1d |
|- ( y e. ( 0 [,] +oo ) -> ( -. y = +oo -> y e. ( 0 [,) +oo ) ) ) |
69 |
68
|
imp |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> y e. ( 0 [,) +oo ) ) |
70 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
71 |
70
|
sseli |
|- ( y e. ( 0 [,) +oo ) -> y e. RR ) |
72 |
71
|
renegcld |
|- ( y e. ( 0 [,) +oo ) -> -u y e. RR ) |
73 |
72
|
reefcld |
|- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. RR ) |
74 |
73
|
rexrd |
|- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. RR* ) |
75 |
|
efgt0 |
|- ( -u y e. RR -> 0 < ( exp ` -u y ) ) |
76 |
72 75
|
syl |
|- ( y e. ( 0 [,) +oo ) -> 0 < ( exp ` -u y ) ) |
77 |
|
elico1 |
|- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( y e. ( 0 [,) +oo ) <-> ( y e. RR* /\ 0 <_ y /\ y < +oo ) ) ) |
78 |
2 3 77
|
mp2an |
|- ( y e. ( 0 [,) +oo ) <-> ( y e. RR* /\ 0 <_ y /\ y < +oo ) ) |
79 |
78
|
simp2bi |
|- ( y e. ( 0 [,) +oo ) -> 0 <_ y ) |
80 |
71
|
le0neg2d |
|- ( y e. ( 0 [,) +oo ) -> ( 0 <_ y <-> -u y <_ 0 ) ) |
81 |
79 80
|
mpbid |
|- ( y e. ( 0 [,) +oo ) -> -u y <_ 0 ) |
82 |
|
0re |
|- 0 e. RR |
83 |
|
efle |
|- ( ( -u y e. RR /\ 0 e. RR ) -> ( -u y <_ 0 <-> ( exp ` -u y ) <_ ( exp ` 0 ) ) ) |
84 |
72 82 83
|
sylancl |
|- ( y e. ( 0 [,) +oo ) -> ( -u y <_ 0 <-> ( exp ` -u y ) <_ ( exp ` 0 ) ) ) |
85 |
81 84
|
mpbid |
|- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) <_ ( exp ` 0 ) ) |
86 |
|
ef0 |
|- ( exp ` 0 ) = 1 |
87 |
85 86
|
breqtrdi |
|- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) <_ 1 ) |
88 |
|
elioc1 |
|- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( ( exp ` -u y ) e. ( 0 (,] 1 ) <-> ( ( exp ` -u y ) e. RR* /\ 0 < ( exp ` -u y ) /\ ( exp ` -u y ) <_ 1 ) ) ) |
89 |
2 10 88
|
mp2an |
|- ( ( exp ` -u y ) e. ( 0 (,] 1 ) <-> ( ( exp ` -u y ) e. RR* /\ 0 < ( exp ` -u y ) /\ ( exp ` -u y ) <_ 1 ) ) |
90 |
74 76 87 89
|
syl3anbrc |
|- ( y e. ( 0 [,) +oo ) -> ( exp ` -u y ) e. ( 0 (,] 1 ) ) |
91 |
69 90
|
syl |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( exp ` -u y ) e. ( 0 (,] 1 ) ) |
92 |
58 91
|
sselid |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( exp ` -u y ) e. ( 0 [,] 1 ) ) |
93 |
57 92
|
ifclda |
|- ( y e. ( 0 [,] +oo ) -> if ( y = +oo , 0 , ( exp ` -u y ) ) e. ( 0 [,] 1 ) ) |
94 |
93
|
adantl |
|- ( ( T. /\ y e. ( 0 [,] +oo ) ) -> if ( y = +oo , 0 , ( exp ` -u y ) ) e. ( 0 [,] 1 ) ) |
95 |
|
eqeq2 |
|- ( 0 = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( x = 0 <-> x = if ( y = +oo , 0 , ( exp ` -u y ) ) ) ) |
96 |
95
|
bibi1d |
|- ( 0 = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( ( x = 0 <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) <-> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) ) |
97 |
|
eqeq2 |
|- ( ( exp ` -u y ) = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( x = ( exp ` -u y ) <-> x = if ( y = +oo , 0 , ( exp ` -u y ) ) ) ) |
98 |
97
|
bibi1d |
|- ( ( exp ` -u y ) = if ( y = +oo , 0 , ( exp ` -u y ) ) -> ( ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) <-> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) ) |
99 |
|
simpr |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y = +oo ) |
100 |
|
iftrue |
|- ( x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = +oo ) |
101 |
100
|
eqeq2d |
|- ( x = 0 -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) <-> y = +oo ) ) |
102 |
99 101
|
syl5ibrcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 0 -> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
103 |
|
ubico |
|- ( ( 0 e. RR /\ +oo e. RR* ) -> -. +oo e. ( 0 [,) +oo ) ) |
104 |
82 3 103
|
mp2an |
|- -. +oo e. ( 0 [,) +oo ) |
105 |
104
|
nelir |
|- +oo e/ ( 0 [,) +oo ) |
106 |
|
neleq1 |
|- ( y = +oo -> ( y e/ ( 0 [,) +oo ) <-> +oo e/ ( 0 [,) +oo ) ) ) |
107 |
106
|
adantl |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y e/ ( 0 [,) +oo ) <-> +oo e/ ( 0 [,) +oo ) ) ) |
108 |
105 107
|
mpbiri |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y e/ ( 0 [,) +oo ) ) |
109 |
|
neleq1 |
|- ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y e/ ( 0 [,) +oo ) <-> if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) ) ) |
110 |
108 109
|
syl5ibcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) ) ) |
111 |
|
df-nel |
|- ( if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) <-> -. if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) |
112 |
|
iffalse |
|- ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) = -u ( log ` x ) ) |
113 |
112
|
adantl |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) = -u ( log ` x ) ) |
114 |
113 52
|
eqeltrd |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) |
115 |
114
|
ex |
|- ( x e. ( 0 [,] 1 ) -> ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) ) |
116 |
115
|
ad2antrr |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. x = 0 -> if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) ) ) |
117 |
116
|
con1d |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. if ( x = 0 , +oo , -u ( log ` x ) ) e. ( 0 [,) +oo ) -> x = 0 ) ) |
118 |
111 117
|
syl5bi |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( if ( x = 0 , +oo , -u ( log ` x ) ) e/ ( 0 [,) +oo ) -> x = 0 ) ) |
119 |
110 118
|
syld |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 0 , +oo , -u ( log ` x ) ) -> x = 0 ) ) |
120 |
102 119
|
impbid |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 0 <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
121 |
|
eqeq2 |
|- ( +oo = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y = +oo <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
122 |
121
|
bibi2d |
|- ( +oo = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( ( x = ( exp ` -u y ) <-> y = +oo ) <-> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) ) |
123 |
|
eqeq2 |
|- ( -u ( log ` x ) = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( y = -u ( log ` x ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
124 |
123
|
bibi2d |
|- ( -u ( log ` x ) = if ( x = 0 , +oo , -u ( log ` x ) ) -> ( ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) <-> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) ) |
125 |
82
|
a1i |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 e. RR ) |
126 |
69 76
|
syl |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 < ( exp ` -u y ) ) |
127 |
125 126
|
ltned |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 0 =/= ( exp ` -u y ) ) |
128 |
127
|
adantll |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> 0 =/= ( exp ` -u y ) ) |
129 |
128
|
neneqd |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> -. 0 = ( exp ` -u y ) ) |
130 |
|
eqeq1 |
|- ( x = 0 -> ( x = ( exp ` -u y ) <-> 0 = ( exp ` -u y ) ) ) |
131 |
130
|
notbid |
|- ( x = 0 -> ( -. x = ( exp ` -u y ) <-> -. 0 = ( exp ` -u y ) ) ) |
132 |
129 131
|
syl5ibrcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = 0 -> -. x = ( exp ` -u y ) ) ) |
133 |
132
|
imp |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> -. x = ( exp ` -u y ) ) |
134 |
|
simplr |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> -. y = +oo ) |
135 |
133 134
|
2falsed |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 0 ) -> ( x = ( exp ` -u y ) <-> y = +oo ) ) |
136 |
|
eqcom |
|- ( x = ( exp ` -u y ) <-> ( exp ` -u y ) = x ) |
137 |
136
|
a1i |
|- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( x = ( exp ` -u y ) <-> ( exp ` -u y ) = x ) ) |
138 |
|
relogeftb |
|- ( ( x e. RR+ /\ -u y e. RR ) -> ( ( log ` x ) = -u y <-> ( exp ` -u y ) = x ) ) |
139 |
32 72 138
|
syl2an |
|- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( log ` x ) = -u y <-> ( exp ` -u y ) = x ) ) |
140 |
33
|
recnd |
|- ( x e. ( 0 (,] 1 ) -> ( log ` x ) e. CC ) |
141 |
71
|
recnd |
|- ( y e. ( 0 [,) +oo ) -> y e. CC ) |
142 |
|
negcon2 |
|- ( ( ( log ` x ) e. CC /\ y e. CC ) -> ( ( log ` x ) = -u y <-> y = -u ( log ` x ) ) ) |
143 |
140 141 142
|
syl2an |
|- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( log ` x ) = -u y <-> y = -u ( log ` x ) ) ) |
144 |
137 139 143
|
3bitr2d |
|- ( ( x e. ( 0 (,] 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) ) |
145 |
23 69 144
|
syl2an |
|- ( ( ( x e. ( 0 [,] 1 ) /\ -. x = 0 ) /\ ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) ) |
146 |
145
|
an4s |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ ( -. x = 0 /\ -. y = +oo ) ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) ) |
147 |
146
|
anass1rs |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ -. x = 0 ) -> ( x = ( exp ` -u y ) <-> y = -u ( log ` x ) ) ) |
148 |
122 124 135 147
|
ifbothda |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = ( exp ` -u y ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
149 |
96 98 120 148
|
ifbothda |
|- ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) -> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
150 |
149
|
adantl |
|- ( ( T. /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) ) -> ( x = if ( y = +oo , 0 , ( exp ` -u y ) ) <-> y = if ( x = 0 , +oo , -u ( log ` x ) ) ) ) |
151 |
1 55 94 150
|
f1ocnv2d |
|- ( T. -> ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 0 , ( exp ` -u y ) ) ) ) ) |
152 |
151
|
mptru |
|- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 0 , ( exp ` -u y ) ) ) ) |