Step |
Hyp |
Ref |
Expression |
1 |
|
iccpnfhmeo.f |
|- F = ( x e. ( 0 [,] 1 ) |-> if ( x = 1 , +oo , ( x / ( 1 - 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 = 1 ) -> +oo e. ( 0 [,] +oo ) ) |
8 |
|
icossicc |
|- ( 0 [,) +oo ) C_ ( 0 [,] +oo ) |
9 |
|
1xr |
|- 1 e. RR* |
10 |
|
0le1 |
|- 0 <_ 1 |
11 |
|
snunico |
|- ( ( 0 e. RR* /\ 1 e. RR* /\ 0 <_ 1 ) -> ( ( 0 [,) 1 ) u. { 1 } ) = ( 0 [,] 1 ) ) |
12 |
2 9 10 11
|
mp3an |
|- ( ( 0 [,) 1 ) u. { 1 } ) = ( 0 [,] 1 ) |
13 |
12
|
eleq2i |
|- ( x e. ( ( 0 [,) 1 ) u. { 1 } ) <-> x e. ( 0 [,] 1 ) ) |
14 |
|
elun |
|- ( x e. ( ( 0 [,) 1 ) u. { 1 } ) <-> ( x e. ( 0 [,) 1 ) \/ x e. { 1 } ) ) |
15 |
13 14
|
bitr3i |
|- ( x e. ( 0 [,] 1 ) <-> ( x e. ( 0 [,) 1 ) \/ x e. { 1 } ) ) |
16 |
|
pm2.53 |
|- ( ( x e. ( 0 [,) 1 ) \/ x e. { 1 } ) -> ( -. x e. ( 0 [,) 1 ) -> x e. { 1 } ) ) |
17 |
15 16
|
sylbi |
|- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 [,) 1 ) -> x e. { 1 } ) ) |
18 |
|
elsni |
|- ( x e. { 1 } -> x = 1 ) |
19 |
17 18
|
syl6 |
|- ( x e. ( 0 [,] 1 ) -> ( -. x e. ( 0 [,) 1 ) -> x = 1 ) ) |
20 |
19
|
con1d |
|- ( x e. ( 0 [,] 1 ) -> ( -. x = 1 -> x e. ( 0 [,) 1 ) ) ) |
21 |
20
|
imp |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> x e. ( 0 [,) 1 ) ) |
22 |
|
eqid |
|- ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) = ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) |
23 |
22
|
icopnfcnv |
|- ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) ) |
24 |
23
|
simpli |
|- ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) |
25 |
|
f1of |
|- ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) -> ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) --> ( 0 [,) +oo ) ) |
26 |
24 25
|
ax-mp |
|- ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) --> ( 0 [,) +oo ) |
27 |
22
|
fmpt |
|- ( A. x e. ( 0 [,) 1 ) ( x / ( 1 - x ) ) e. ( 0 [,) +oo ) <-> ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) --> ( 0 [,) +oo ) ) |
28 |
26 27
|
mpbir |
|- A. x e. ( 0 [,) 1 ) ( x / ( 1 - x ) ) e. ( 0 [,) +oo ) |
29 |
28
|
rspec |
|- ( x e. ( 0 [,) 1 ) -> ( x / ( 1 - x ) ) e. ( 0 [,) +oo ) ) |
30 |
21 29
|
syl |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> ( x / ( 1 - x ) ) e. ( 0 [,) +oo ) ) |
31 |
8 30
|
sselid |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> ( x / ( 1 - x ) ) e. ( 0 [,] +oo ) ) |
32 |
7 31
|
ifclda |
|- ( x e. ( 0 [,] 1 ) -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. ( 0 [,] +oo ) ) |
33 |
32
|
adantl |
|- ( ( T. /\ x e. ( 0 [,] 1 ) ) -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. ( 0 [,] +oo ) ) |
34 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
35 |
34
|
a1i |
|- ( ( y e. ( 0 [,] +oo ) /\ y = +oo ) -> 1 e. ( 0 [,] 1 ) ) |
36 |
|
icossicc |
|- ( 0 [,) 1 ) C_ ( 0 [,] 1 ) |
37 |
|
snunico |
|- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo ) ) |
38 |
2 3 4 37
|
mp3an |
|- ( ( 0 [,) +oo ) u. { +oo } ) = ( 0 [,] +oo ) |
39 |
38
|
eleq2i |
|- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> y e. ( 0 [,] +oo ) ) |
40 |
|
elun |
|- ( y e. ( ( 0 [,) +oo ) u. { +oo } ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) ) |
41 |
39 40
|
bitr3i |
|- ( y e. ( 0 [,] +oo ) <-> ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) ) |
42 |
|
pm2.53 |
|- ( ( y e. ( 0 [,) +oo ) \/ y e. { +oo } ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) ) |
43 |
41 42
|
sylbi |
|- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y e. { +oo } ) ) |
44 |
|
elsni |
|- ( y e. { +oo } -> y = +oo ) |
45 |
43 44
|
syl6 |
|- ( y e. ( 0 [,] +oo ) -> ( -. y e. ( 0 [,) +oo ) -> y = +oo ) ) |
46 |
45
|
con1d |
|- ( y e. ( 0 [,] +oo ) -> ( -. y = +oo -> y e. ( 0 [,) +oo ) ) ) |
47 |
46
|
imp |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> y e. ( 0 [,) +oo ) ) |
48 |
|
f1ocnv |
|- ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) -> `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) +oo ) -1-1-onto-> ( 0 [,) 1 ) ) |
49 |
|
f1of |
|- ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) +oo ) -1-1-onto-> ( 0 [,) 1 ) -> `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) +oo ) --> ( 0 [,) 1 ) ) |
50 |
24 48 49
|
mp2b |
|- `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) +oo ) --> ( 0 [,) 1 ) |
51 |
23
|
simpri |
|- `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) = ( y e. ( 0 [,) +oo ) |-> ( y / ( 1 + y ) ) ) |
52 |
51
|
fmpt |
|- ( A. y e. ( 0 [,) +oo ) ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) <-> `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) +oo ) --> ( 0 [,) 1 ) ) |
53 |
50 52
|
mpbir |
|- A. y e. ( 0 [,) +oo ) ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) |
54 |
53
|
rspec |
|- ( y e. ( 0 [,) +oo ) -> ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) ) |
55 |
47 54
|
syl |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) ) |
56 |
36 55
|
sselid |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( y / ( 1 + y ) ) e. ( 0 [,] 1 ) ) |
57 |
35 56
|
ifclda |
|- ( y e. ( 0 [,] +oo ) -> if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) e. ( 0 [,] 1 ) ) |
58 |
57
|
adantl |
|- ( ( T. /\ y e. ( 0 [,] +oo ) ) -> if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) e. ( 0 [,] 1 ) ) |
59 |
|
eqeq2 |
|- ( 1 = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) -> ( x = 1 <-> x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) ) ) |
60 |
59
|
bibi1d |
|- ( 1 = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) -> ( ( x = 1 <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) <-> ( x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) ) |
61 |
|
eqeq2 |
|- ( ( y / ( 1 + y ) ) = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) -> ( x = ( y / ( 1 + y ) ) <-> x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) ) ) |
62 |
61
|
bibi1d |
|- ( ( y / ( 1 + y ) ) = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) -> ( ( x = ( y / ( 1 + y ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) <-> ( x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) ) |
63 |
|
simpr |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y = +oo ) |
64 |
|
iftrue |
|- ( x = 1 -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) = +oo ) |
65 |
64
|
eqeq2d |
|- ( x = 1 -> ( y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) <-> y = +oo ) ) |
66 |
63 65
|
syl5ibrcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 1 -> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
67 |
|
pnfnre |
|- +oo e/ RR |
68 |
|
neleq1 |
|- ( y = +oo -> ( y e/ RR <-> +oo e/ RR ) ) |
69 |
68
|
adantl |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y e/ RR <-> +oo e/ RR ) ) |
70 |
67 69
|
mpbiri |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> y e/ RR ) |
71 |
|
neleq1 |
|- ( y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> ( y e/ RR <-> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e/ RR ) ) |
72 |
70 71
|
syl5ibcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e/ RR ) ) |
73 |
|
df-nel |
|- ( if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e/ RR <-> -. if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. RR ) |
74 |
|
iffalse |
|- ( -. x = 1 -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) = ( x / ( 1 - x ) ) ) |
75 |
74
|
adantl |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) = ( x / ( 1 - x ) ) ) |
76 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
77 |
76 30
|
sselid |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> ( x / ( 1 - x ) ) e. RR ) |
78 |
75 77
|
eqeltrd |
|- ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. RR ) |
79 |
78
|
ex |
|- ( x e. ( 0 [,] 1 ) -> ( -. x = 1 -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. RR ) ) |
80 |
79
|
ad2antrr |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. x = 1 -> if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. RR ) ) |
81 |
80
|
con1d |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( -. if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e. RR -> x = 1 ) ) |
82 |
73 81
|
syl5bi |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) e/ RR -> x = 1 ) ) |
83 |
72 82
|
syld |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> x = 1 ) ) |
84 |
66 83
|
impbid |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ y = +oo ) -> ( x = 1 <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
85 |
|
eqeq2 |
|- ( +oo = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> ( y = +oo <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
86 |
85
|
bibi2d |
|- ( +oo = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> ( ( x = ( y / ( 1 + y ) ) <-> y = +oo ) <-> ( x = ( y / ( 1 + y ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) ) |
87 |
|
eqeq2 |
|- ( ( x / ( 1 - x ) ) = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> ( y = ( x / ( 1 - x ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
88 |
87
|
bibi2d |
|- ( ( x / ( 1 - x ) ) = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) -> ( ( x = ( y / ( 1 + y ) ) <-> y = ( x / ( 1 - x ) ) ) <-> ( x = ( y / ( 1 + y ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) ) |
89 |
|
0re |
|- 0 e. RR |
90 |
|
elico2 |
|- ( ( 0 e. RR /\ 1 e. RR* ) -> ( ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) <-> ( ( y / ( 1 + y ) ) e. RR /\ 0 <_ ( y / ( 1 + y ) ) /\ ( y / ( 1 + y ) ) < 1 ) ) ) |
91 |
89 9 90
|
mp2an |
|- ( ( y / ( 1 + y ) ) e. ( 0 [,) 1 ) <-> ( ( y / ( 1 + y ) ) e. RR /\ 0 <_ ( y / ( 1 + y ) ) /\ ( y / ( 1 + y ) ) < 1 ) ) |
92 |
55 91
|
sylib |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( ( y / ( 1 + y ) ) e. RR /\ 0 <_ ( y / ( 1 + y ) ) /\ ( y / ( 1 + y ) ) < 1 ) ) |
93 |
92
|
simp1d |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( y / ( 1 + y ) ) e. RR ) |
94 |
92
|
simp3d |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> ( y / ( 1 + y ) ) < 1 ) |
95 |
93 94
|
gtned |
|- ( ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) -> 1 =/= ( y / ( 1 + y ) ) ) |
96 |
95
|
adantll |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> 1 =/= ( y / ( 1 + y ) ) ) |
97 |
96
|
neneqd |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> -. 1 = ( y / ( 1 + y ) ) ) |
98 |
|
eqeq1 |
|- ( x = 1 -> ( x = ( y / ( 1 + y ) ) <-> 1 = ( y / ( 1 + y ) ) ) ) |
99 |
98
|
notbid |
|- ( x = 1 -> ( -. x = ( y / ( 1 + y ) ) <-> -. 1 = ( y / ( 1 + y ) ) ) ) |
100 |
97 99
|
syl5ibrcom |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = 1 -> -. x = ( y / ( 1 + y ) ) ) ) |
101 |
100
|
imp |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 1 ) -> -. x = ( y / ( 1 + y ) ) ) |
102 |
|
simplr |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 1 ) -> -. y = +oo ) |
103 |
101 102
|
2falsed |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ x = 1 ) -> ( x = ( y / ( 1 + y ) ) <-> y = +oo ) ) |
104 |
|
f1ocnvfvb |
|- ( ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,) +oo ) /\ x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` x ) = y <-> ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` y ) = x ) ) |
105 |
24 104
|
mp3an1 |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` x ) = y <-> ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` y ) = x ) ) |
106 |
|
simpl |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> x e. ( 0 [,) 1 ) ) |
107 |
|
ovex |
|- ( x / ( 1 - x ) ) e. _V |
108 |
22
|
fvmpt2 |
|- ( ( x e. ( 0 [,) 1 ) /\ ( x / ( 1 - x ) ) e. _V ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` x ) = ( x / ( 1 - x ) ) ) |
109 |
106 107 108
|
sylancl |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` x ) = ( x / ( 1 - x ) ) ) |
110 |
109
|
eqeq1d |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` x ) = y <-> ( x / ( 1 - x ) ) = y ) ) |
111 |
|
simpr |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> y e. ( 0 [,) +oo ) ) |
112 |
|
ovex |
|- ( y / ( 1 + y ) ) e. _V |
113 |
51
|
fvmpt2 |
|- ( ( y e. ( 0 [,) +oo ) /\ ( y / ( 1 + y ) ) e. _V ) -> ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` y ) = ( y / ( 1 + y ) ) ) |
114 |
111 112 113
|
sylancl |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` y ) = ( y / ( 1 + y ) ) ) |
115 |
114
|
eqeq1d |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( `' ( x e. ( 0 [,) 1 ) |-> ( x / ( 1 - x ) ) ) ` y ) = x <-> ( y / ( 1 + y ) ) = x ) ) |
116 |
105 110 115
|
3bitr3rd |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( ( y / ( 1 + y ) ) = x <-> ( x / ( 1 - x ) ) = y ) ) |
117 |
|
eqcom |
|- ( x = ( y / ( 1 + y ) ) <-> ( y / ( 1 + y ) ) = x ) |
118 |
|
eqcom |
|- ( y = ( x / ( 1 - x ) ) <-> ( x / ( 1 - x ) ) = y ) |
119 |
116 117 118
|
3bitr4g |
|- ( ( x e. ( 0 [,) 1 ) /\ y e. ( 0 [,) +oo ) ) -> ( x = ( y / ( 1 + y ) ) <-> y = ( x / ( 1 - x ) ) ) ) |
120 |
21 47 119
|
syl2an |
|- ( ( ( x e. ( 0 [,] 1 ) /\ -. x = 1 ) /\ ( y e. ( 0 [,] +oo ) /\ -. y = +oo ) ) -> ( x = ( y / ( 1 + y ) ) <-> y = ( x / ( 1 - x ) ) ) ) |
121 |
120
|
an4s |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ ( -. x = 1 /\ -. y = +oo ) ) -> ( x = ( y / ( 1 + y ) ) <-> y = ( x / ( 1 - x ) ) ) ) |
122 |
121
|
anass1rs |
|- ( ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) /\ -. x = 1 ) -> ( x = ( y / ( 1 + y ) ) <-> y = ( x / ( 1 - x ) ) ) ) |
123 |
86 88 103 122
|
ifbothda |
|- ( ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) /\ -. y = +oo ) -> ( x = ( y / ( 1 + y ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
124 |
60 62 84 123
|
ifbothda |
|- ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) -> ( x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
125 |
124
|
adantl |
|- ( ( T. /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] +oo ) ) ) -> ( x = if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) <-> y = if ( x = 1 , +oo , ( x / ( 1 - x ) ) ) ) ) |
126 |
1 33 58 125
|
f1ocnv2d |
|- ( T. -> ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) ) ) ) |
127 |
126
|
mptru |
|- ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,] +oo ) /\ `' F = ( y e. ( 0 [,] +oo ) |-> if ( y = +oo , 1 , ( y / ( 1 + y ) ) ) ) ) |