Step |
Hyp |
Ref |
Expression |
1 |
|
dvasin.d |
|- D = ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) |
2 |
|
df-asin |
|- arcsin = ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
3 |
2
|
reseq1i |
|- ( arcsin |` D ) = ( ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |` D ) |
4 |
|
difss |
|- ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) C_ CC |
5 |
1 4
|
eqsstri |
|- D C_ CC |
6 |
|
resmpt |
|- ( D C_ CC -> ( ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |` D ) = ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) ) |
7 |
5 6
|
ax-mp |
|- ( ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |` D ) = ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
8 |
3 7
|
eqtri |
|- ( arcsin |` D ) = ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
9 |
8
|
oveq2i |
|- ( CC _D ( arcsin |` D ) ) = ( CC _D ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) ) |
10 |
|
cnelprrecn |
|- CC e. { RR , CC } |
11 |
10
|
a1i |
|- ( T. -> CC e. { RR , CC } ) |
12 |
5
|
sseli |
|- ( x e. D -> x e. CC ) |
13 |
|
ax-icn |
|- _i e. CC |
14 |
|
mulcl |
|- ( ( _i e. CC /\ x e. CC ) -> ( _i x. x ) e. CC ) |
15 |
13 14
|
mpan |
|- ( x e. CC -> ( _i x. x ) e. CC ) |
16 |
|
ax-1cn |
|- 1 e. CC |
17 |
|
sqcl |
|- ( x e. CC -> ( x ^ 2 ) e. CC ) |
18 |
|
subcl |
|- ( ( 1 e. CC /\ ( x ^ 2 ) e. CC ) -> ( 1 - ( x ^ 2 ) ) e. CC ) |
19 |
16 17 18
|
sylancr |
|- ( x e. CC -> ( 1 - ( x ^ 2 ) ) e. CC ) |
20 |
19
|
sqrtcld |
|- ( x e. CC -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC ) |
21 |
15 20
|
addcld |
|- ( x e. CC -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
22 |
12 21
|
syl |
|- ( x e. D -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
23 |
|
asinlem |
|- ( x e. CC -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
24 |
12 23
|
syl |
|- ( x e. D -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
25 |
22 24
|
logcld |
|- ( x e. D -> ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
26 |
25
|
adantl |
|- ( ( T. /\ x e. D ) -> ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
27 |
|
ovexd |
|- ( ( T. /\ x e. D ) -> ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. _V ) |
28 |
|
simpr |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) |
29 |
|
asinlem3 |
|- ( x e. CC -> 0 <_ ( Re ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
30 |
|
rere |
|- ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> ( Re ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
31 |
30
|
breq2d |
|- ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> ( 0 <_ ( Re ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) <-> 0 <_ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
32 |
31
|
biimpac |
|- ( ( 0 <_ ( Re ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> 0 <_ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
33 |
29 32
|
sylan |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> 0 <_ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
34 |
23
|
adantr |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
35 |
28 33 34
|
ne0gt0d |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> 0 < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
36 |
|
0re |
|- 0 e. RR |
37 |
|
ltnle |
|- ( ( 0 e. RR /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> ( 0 < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <-> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
38 |
36 37
|
mpan |
|- ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> ( 0 < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <-> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
39 |
38
|
adantl |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> ( 0 < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <-> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
40 |
35 39
|
mpbid |
|- ( ( x e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) -> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) |
41 |
40
|
ex |
|- ( x e. CC -> ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
42 |
12 41
|
syl |
|- ( x e. D -> ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
43 |
|
imor |
|- ( ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR -> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) <-> ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
44 |
42 43
|
sylib |
|- ( x e. D -> ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
45 |
44
|
orcomd |
|- ( x e. D -> ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) |
46 |
45
|
olcd |
|- ( x e. D -> ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) ) |
47 |
|
3ianor |
|- ( -. ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR /\ -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) <-> ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR \/ -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
48 |
|
3orrot |
|- ( ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR \/ -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) <-> ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) |
49 |
|
3orass |
|- ( ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) <-> ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) ) |
50 |
47 48 49
|
3bitrri |
|- ( ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) <-> -. ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR /\ -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
51 |
|
mnfxr |
|- -oo e. RR* |
52 |
|
elioc2 |
|- ( ( -oo e. RR* /\ 0 e. RR ) -> ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( -oo (,] 0 ) <-> ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR /\ -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) ) |
53 |
51 36 52
|
mp2an |
|- ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( -oo (,] 0 ) <-> ( ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR /\ -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 ) ) |
54 |
50 53
|
xchbinxr |
|- ( ( -. -oo < ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) \/ ( -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) <_ 0 \/ -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. RR ) ) <-> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( -oo (,] 0 ) ) |
55 |
46 54
|
sylib |
|- ( x e. D -> -. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( -oo (,] 0 ) ) |
56 |
22 55
|
eldifd |
|- ( x e. D -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( CC \ ( -oo (,] 0 ) ) ) |
57 |
56
|
adantl |
|- ( ( T. /\ x e. D ) -> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. ( CC \ ( -oo (,] 0 ) ) ) |
58 |
|
ovexd |
|- ( ( T. /\ x e. D ) -> ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. _V ) |
59 |
|
eldifi |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> y e. CC ) |
60 |
|
eldifn |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> -. y e. ( -oo (,] 0 ) ) |
61 |
|
0xr |
|- 0 e. RR* |
62 |
|
mnflt0 |
|- -oo < 0 |
63 |
|
ubioc1 |
|- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) ) |
64 |
51 61 62 63
|
mp3an |
|- 0 e. ( -oo (,] 0 ) |
65 |
|
eleq1 |
|- ( y = 0 -> ( y e. ( -oo (,] 0 ) <-> 0 e. ( -oo (,] 0 ) ) ) |
66 |
64 65
|
mpbiri |
|- ( y = 0 -> y e. ( -oo (,] 0 ) ) |
67 |
66
|
necon3bi |
|- ( -. y e. ( -oo (,] 0 ) -> y =/= 0 ) |
68 |
60 67
|
syl |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> y =/= 0 ) |
69 |
59 68
|
logcld |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> ( log ` y ) e. CC ) |
70 |
69
|
adantl |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( log ` y ) e. CC ) |
71 |
|
ovexd |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( 1 / y ) e. _V ) |
72 |
13
|
a1i |
|- ( x e. D -> _i e. CC ) |
73 |
72 12
|
mulcld |
|- ( x e. D -> ( _i x. x ) e. CC ) |
74 |
73
|
adantl |
|- ( ( T. /\ x e. D ) -> ( _i x. x ) e. CC ) |
75 |
13
|
a1i |
|- ( ( T. /\ x e. D ) -> _i e. CC ) |
76 |
12
|
adantl |
|- ( ( T. /\ x e. D ) -> x e. CC ) |
77 |
|
1cnd |
|- ( ( T. /\ x e. D ) -> 1 e. CC ) |
78 |
|
simpr |
|- ( ( T. /\ x e. CC ) -> x e. CC ) |
79 |
|
1cnd |
|- ( ( T. /\ x e. CC ) -> 1 e. CC ) |
80 |
11
|
dvmptid |
|- ( T. -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) ) |
81 |
5
|
a1i |
|- ( T. -> D C_ CC ) |
82 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
83 |
82
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
84 |
83
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
85 |
82
|
recld2 |
|- RR e. ( Clsd ` ( TopOpen ` CCfld ) ) |
86 |
|
neg1rr |
|- -u 1 e. RR |
87 |
|
iocmnfcld |
|- ( -u 1 e. RR -> ( -oo (,] -u 1 ) e. ( Clsd ` ( topGen ` ran (,) ) ) ) |
88 |
86 87
|
ax-mp |
|- ( -oo (,] -u 1 ) e. ( Clsd ` ( topGen ` ran (,) ) ) |
89 |
|
1re |
|- 1 e. RR |
90 |
|
icopnfcld |
|- ( 1 e. RR -> ( 1 [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) ) |
91 |
89 90
|
ax-mp |
|- ( 1 [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) |
92 |
|
uncld |
|- ( ( ( -oo (,] -u 1 ) e. ( Clsd ` ( topGen ` ran (,) ) ) /\ ( 1 [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) ) -> ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( topGen ` ran (,) ) ) ) |
93 |
88 91 92
|
mp2an |
|- ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( topGen ` ran (,) ) ) |
94 |
82
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
95 |
94
|
fveq2i |
|- ( Clsd ` ( topGen ` ran (,) ) ) = ( Clsd ` ( ( TopOpen ` CCfld ) |`t RR ) ) |
96 |
93 95
|
eleqtri |
|- ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( ( TopOpen ` CCfld ) |`t RR ) ) |
97 |
|
restcldr |
|- ( ( RR e. ( Clsd ` ( TopOpen ` CCfld ) ) /\ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( ( TopOpen ` CCfld ) |`t RR ) ) ) -> ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` CCfld ) ) ) |
98 |
85 96 97
|
mp2an |
|- ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` CCfld ) ) |
99 |
83
|
toponunii |
|- CC = U. ( TopOpen ` CCfld ) |
100 |
99
|
cldopn |
|- ( ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` CCfld ) ) -> ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) e. ( TopOpen ` CCfld ) ) |
101 |
98 100
|
ax-mp |
|- ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) e. ( TopOpen ` CCfld ) |
102 |
1 101
|
eqeltri |
|- D e. ( TopOpen ` CCfld ) |
103 |
102
|
a1i |
|- ( T. -> D e. ( TopOpen ` CCfld ) ) |
104 |
11 78 79 80 81 84 82 103
|
dvmptres |
|- ( T. -> ( CC _D ( x e. D |-> x ) ) = ( x e. D |-> 1 ) ) |
105 |
13
|
a1i |
|- ( T. -> _i e. CC ) |
106 |
11 76 77 104 105
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. D |-> ( _i x. x ) ) ) = ( x e. D |-> ( _i x. 1 ) ) ) |
107 |
13
|
mulid1i |
|- ( _i x. 1 ) = _i |
108 |
107
|
mpteq2i |
|- ( x e. D |-> ( _i x. 1 ) ) = ( x e. D |-> _i ) |
109 |
106 108
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. D |-> ( _i x. x ) ) ) = ( x e. D |-> _i ) ) |
110 |
12
|
sqcld |
|- ( x e. D -> ( x ^ 2 ) e. CC ) |
111 |
16 110 18
|
sylancr |
|- ( x e. D -> ( 1 - ( x ^ 2 ) ) e. CC ) |
112 |
111
|
sqrtcld |
|- ( x e. D -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC ) |
113 |
112
|
adantl |
|- ( ( T. /\ x e. D ) -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC ) |
114 |
|
ovexd |
|- ( ( T. /\ x e. D ) -> ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. _V ) |
115 |
|
elin |
|- ( x e. ( D i^i RR ) <-> ( x e. D /\ x e. RR ) ) |
116 |
1
|
asindmre |
|- ( D i^i RR ) = ( -u 1 (,) 1 ) |
117 |
116
|
eqimssi |
|- ( D i^i RR ) C_ ( -u 1 (,) 1 ) |
118 |
117
|
sseli |
|- ( x e. ( D i^i RR ) -> x e. ( -u 1 (,) 1 ) ) |
119 |
115 118
|
sylbir |
|- ( ( x e. D /\ x e. RR ) -> x e. ( -u 1 (,) 1 ) ) |
120 |
|
incom |
|- ( ( 0 (,) +oo ) i^i ( -oo (,] 0 ) ) = ( ( -oo (,] 0 ) i^i ( 0 (,) +oo ) ) |
121 |
|
pnfxr |
|- +oo e. RR* |
122 |
|
df-ioc |
|- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } ) |
123 |
|
df-ioo |
|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } ) |
124 |
|
xrltnle |
|- ( ( 0 e. RR* /\ w e. RR* ) -> ( 0 < w <-> -. w <_ 0 ) ) |
125 |
122 123 124
|
ixxdisj |
|- ( ( -oo e. RR* /\ 0 e. RR* /\ +oo e. RR* ) -> ( ( -oo (,] 0 ) i^i ( 0 (,) +oo ) ) = (/) ) |
126 |
51 61 121 125
|
mp3an |
|- ( ( -oo (,] 0 ) i^i ( 0 (,) +oo ) ) = (/) |
127 |
120 126
|
eqtri |
|- ( ( 0 (,) +oo ) i^i ( -oo (,] 0 ) ) = (/) |
128 |
|
elioore |
|- ( x e. ( -u 1 (,) 1 ) -> x e. RR ) |
129 |
128
|
resqcld |
|- ( x e. ( -u 1 (,) 1 ) -> ( x ^ 2 ) e. RR ) |
130 |
|
resubcl |
|- ( ( 1 e. RR /\ ( x ^ 2 ) e. RR ) -> ( 1 - ( x ^ 2 ) ) e. RR ) |
131 |
89 129 130
|
sylancr |
|- ( x e. ( -u 1 (,) 1 ) -> ( 1 - ( x ^ 2 ) ) e. RR ) |
132 |
86
|
rexri |
|- -u 1 e. RR* |
133 |
|
1xr |
|- 1 e. RR* |
134 |
|
elioo2 |
|- ( ( -u 1 e. RR* /\ 1 e. RR* ) -> ( x e. ( -u 1 (,) 1 ) <-> ( x e. RR /\ -u 1 < x /\ x < 1 ) ) ) |
135 |
132 133 134
|
mp2an |
|- ( x e. ( -u 1 (,) 1 ) <-> ( x e. RR /\ -u 1 < x /\ x < 1 ) ) |
136 |
|
recn |
|- ( x e. RR -> x e. CC ) |
137 |
136
|
abscld |
|- ( x e. RR -> ( abs ` x ) e. RR ) |
138 |
136
|
absge0d |
|- ( x e. RR -> 0 <_ ( abs ` x ) ) |
139 |
|
0le1 |
|- 0 <_ 1 |
140 |
|
lt2sq |
|- ( ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( abs ` x ) < 1 <-> ( ( abs ` x ) ^ 2 ) < ( 1 ^ 2 ) ) ) |
141 |
89 139 140
|
mpanr12 |
|- ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) -> ( ( abs ` x ) < 1 <-> ( ( abs ` x ) ^ 2 ) < ( 1 ^ 2 ) ) ) |
142 |
137 138 141
|
syl2anc |
|- ( x e. RR -> ( ( abs ` x ) < 1 <-> ( ( abs ` x ) ^ 2 ) < ( 1 ^ 2 ) ) ) |
143 |
|
abslt |
|- ( ( x e. RR /\ 1 e. RR ) -> ( ( abs ` x ) < 1 <-> ( -u 1 < x /\ x < 1 ) ) ) |
144 |
89 143
|
mpan2 |
|- ( x e. RR -> ( ( abs ` x ) < 1 <-> ( -u 1 < x /\ x < 1 ) ) ) |
145 |
|
absresq |
|- ( x e. RR -> ( ( abs ` x ) ^ 2 ) = ( x ^ 2 ) ) |
146 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
147 |
146
|
a1i |
|- ( x e. RR -> ( 1 ^ 2 ) = 1 ) |
148 |
145 147
|
breq12d |
|- ( x e. RR -> ( ( ( abs ` x ) ^ 2 ) < ( 1 ^ 2 ) <-> ( x ^ 2 ) < 1 ) ) |
149 |
|
resqcl |
|- ( x e. RR -> ( x ^ 2 ) e. RR ) |
150 |
|
posdif |
|- ( ( ( x ^ 2 ) e. RR /\ 1 e. RR ) -> ( ( x ^ 2 ) < 1 <-> 0 < ( 1 - ( x ^ 2 ) ) ) ) |
151 |
149 89 150
|
sylancl |
|- ( x e. RR -> ( ( x ^ 2 ) < 1 <-> 0 < ( 1 - ( x ^ 2 ) ) ) ) |
152 |
148 151
|
bitrd |
|- ( x e. RR -> ( ( ( abs ` x ) ^ 2 ) < ( 1 ^ 2 ) <-> 0 < ( 1 - ( x ^ 2 ) ) ) ) |
153 |
142 144 152
|
3bitr3d |
|- ( x e. RR -> ( ( -u 1 < x /\ x < 1 ) <-> 0 < ( 1 - ( x ^ 2 ) ) ) ) |
154 |
153
|
biimpd |
|- ( x e. RR -> ( ( -u 1 < x /\ x < 1 ) -> 0 < ( 1 - ( x ^ 2 ) ) ) ) |
155 |
154
|
3impib |
|- ( ( x e. RR /\ -u 1 < x /\ x < 1 ) -> 0 < ( 1 - ( x ^ 2 ) ) ) |
156 |
135 155
|
sylbi |
|- ( x e. ( -u 1 (,) 1 ) -> 0 < ( 1 - ( x ^ 2 ) ) ) |
157 |
131 156
|
elrpd |
|- ( x e. ( -u 1 (,) 1 ) -> ( 1 - ( x ^ 2 ) ) e. RR+ ) |
158 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
159 |
157 158
|
eleqtrrdi |
|- ( x e. ( -u 1 (,) 1 ) -> ( 1 - ( x ^ 2 ) ) e. ( 0 (,) +oo ) ) |
160 |
|
disjel |
|- ( ( ( ( 0 (,) +oo ) i^i ( -oo (,] 0 ) ) = (/) /\ ( 1 - ( x ^ 2 ) ) e. ( 0 (,) +oo ) ) -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
161 |
127 159 160
|
sylancr |
|- ( x e. ( -u 1 (,) 1 ) -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
162 |
119 161
|
syl |
|- ( ( x e. D /\ x e. RR ) -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
163 |
|
elioc2 |
|- ( ( -oo e. RR* /\ 0 e. RR ) -> ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) <-> ( ( 1 - ( x ^ 2 ) ) e. RR /\ -oo < ( 1 - ( x ^ 2 ) ) /\ ( 1 - ( x ^ 2 ) ) <_ 0 ) ) ) |
164 |
51 36 163
|
mp2an |
|- ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) <-> ( ( 1 - ( x ^ 2 ) ) e. RR /\ -oo < ( 1 - ( x ^ 2 ) ) /\ ( 1 - ( x ^ 2 ) ) <_ 0 ) ) |
165 |
164
|
biimpi |
|- ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) -> ( ( 1 - ( x ^ 2 ) ) e. RR /\ -oo < ( 1 - ( x ^ 2 ) ) /\ ( 1 - ( x ^ 2 ) ) <_ 0 ) ) |
166 |
165
|
simp1d |
|- ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) -> ( 1 - ( x ^ 2 ) ) e. RR ) |
167 |
|
resubcl |
|- ( ( 1 e. RR /\ ( 1 - ( x ^ 2 ) ) e. RR ) -> ( 1 - ( 1 - ( x ^ 2 ) ) ) e. RR ) |
168 |
89 166 167
|
sylancr |
|- ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) -> ( 1 - ( 1 - ( x ^ 2 ) ) ) e. RR ) |
169 |
|
nncan |
|- ( ( 1 e. CC /\ ( x ^ 2 ) e. CC ) -> ( 1 - ( 1 - ( x ^ 2 ) ) ) = ( x ^ 2 ) ) |
170 |
16 169
|
mpan |
|- ( ( x ^ 2 ) e. CC -> ( 1 - ( 1 - ( x ^ 2 ) ) ) = ( x ^ 2 ) ) |
171 |
170
|
eleq1d |
|- ( ( x ^ 2 ) e. CC -> ( ( 1 - ( 1 - ( x ^ 2 ) ) ) e. RR <-> ( x ^ 2 ) e. RR ) ) |
172 |
171
|
biimpa |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( 1 - ( x ^ 2 ) ) ) e. RR ) -> ( x ^ 2 ) e. RR ) |
173 |
168 172
|
sylan2 |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( x ^ 2 ) e. RR ) |
174 |
166
|
adantl |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( 1 - ( x ^ 2 ) ) e. RR ) |
175 |
165
|
simp3d |
|- ( ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) -> ( 1 - ( x ^ 2 ) ) <_ 0 ) |
176 |
175
|
adantl |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( 1 - ( x ^ 2 ) ) <_ 0 ) |
177 |
|
letr |
|- ( ( ( 1 - ( x ^ 2 ) ) e. RR /\ 0 e. RR /\ 1 e. RR ) -> ( ( ( 1 - ( x ^ 2 ) ) <_ 0 /\ 0 <_ 1 ) -> ( 1 - ( x ^ 2 ) ) <_ 1 ) ) |
178 |
36 89 177
|
mp3an23 |
|- ( ( 1 - ( x ^ 2 ) ) e. RR -> ( ( ( 1 - ( x ^ 2 ) ) <_ 0 /\ 0 <_ 1 ) -> ( 1 - ( x ^ 2 ) ) <_ 1 ) ) |
179 |
139 178
|
mpan2i |
|- ( ( 1 - ( x ^ 2 ) ) e. RR -> ( ( 1 - ( x ^ 2 ) ) <_ 0 -> ( 1 - ( x ^ 2 ) ) <_ 1 ) ) |
180 |
174 176 179
|
sylc |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( 1 - ( x ^ 2 ) ) <_ 1 ) |
181 |
|
subge0 |
|- ( ( 1 e. RR /\ ( 1 - ( x ^ 2 ) ) e. RR ) -> ( 0 <_ ( 1 - ( 1 - ( x ^ 2 ) ) ) <-> ( 1 - ( x ^ 2 ) ) <_ 1 ) ) |
182 |
89 174 181
|
sylancr |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( 0 <_ ( 1 - ( 1 - ( x ^ 2 ) ) ) <-> ( 1 - ( x ^ 2 ) ) <_ 1 ) ) |
183 |
180 182
|
mpbird |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> 0 <_ ( 1 - ( 1 - ( x ^ 2 ) ) ) ) |
184 |
170
|
adantr |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( 1 - ( 1 - ( x ^ 2 ) ) ) = ( x ^ 2 ) ) |
185 |
183 184
|
breqtrd |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> 0 <_ ( x ^ 2 ) ) |
186 |
173 185
|
resqrtcld |
|- ( ( ( x ^ 2 ) e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( sqrt ` ( x ^ 2 ) ) e. RR ) |
187 |
17 186
|
sylan |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( sqrt ` ( x ^ 2 ) ) e. RR ) |
188 |
|
eleq1 |
|- ( x = ( sqrt ` ( x ^ 2 ) ) -> ( x e. RR <-> ( sqrt ` ( x ^ 2 ) ) e. RR ) ) |
189 |
187 188
|
syl5ibrcom |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( x = ( sqrt ` ( x ^ 2 ) ) -> x e. RR ) ) |
190 |
187
|
renegcld |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> -u ( sqrt ` ( x ^ 2 ) ) e. RR ) |
191 |
|
eleq1 |
|- ( x = -u ( sqrt ` ( x ^ 2 ) ) -> ( x e. RR <-> -u ( sqrt ` ( x ^ 2 ) ) e. RR ) ) |
192 |
190 191
|
syl5ibrcom |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( x = -u ( sqrt ` ( x ^ 2 ) ) -> x e. RR ) ) |
193 |
|
eqid |
|- ( x ^ 2 ) = ( x ^ 2 ) |
194 |
|
eqsqrtor |
|- ( ( x e. CC /\ ( x ^ 2 ) e. CC ) -> ( ( x ^ 2 ) = ( x ^ 2 ) <-> ( x = ( sqrt ` ( x ^ 2 ) ) \/ x = -u ( sqrt ` ( x ^ 2 ) ) ) ) ) |
195 |
17 194
|
mpdan |
|- ( x e. CC -> ( ( x ^ 2 ) = ( x ^ 2 ) <-> ( x = ( sqrt ` ( x ^ 2 ) ) \/ x = -u ( sqrt ` ( x ^ 2 ) ) ) ) ) |
196 |
193 195
|
mpbii |
|- ( x e. CC -> ( x = ( sqrt ` ( x ^ 2 ) ) \/ x = -u ( sqrt ` ( x ^ 2 ) ) ) ) |
197 |
196
|
adantr |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> ( x = ( sqrt ` ( x ^ 2 ) ) \/ x = -u ( sqrt ` ( x ^ 2 ) ) ) ) |
198 |
189 192 197
|
mpjaod |
|- ( ( x e. CC /\ ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) -> x e. RR ) |
199 |
198
|
stoic1a |
|- ( ( x e. CC /\ -. x e. RR ) -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
200 |
12 199
|
sylan |
|- ( ( x e. D /\ -. x e. RR ) -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
201 |
162 200
|
pm2.61dan |
|- ( x e. D -> -. ( 1 - ( x ^ 2 ) ) e. ( -oo (,] 0 ) ) |
202 |
111 201
|
eldifd |
|- ( x e. D -> ( 1 - ( x ^ 2 ) ) e. ( CC \ ( -oo (,] 0 ) ) ) |
203 |
202
|
adantl |
|- ( ( T. /\ x e. D ) -> ( 1 - ( x ^ 2 ) ) e. ( CC \ ( -oo (,] 0 ) ) ) |
204 |
|
2cnd |
|- ( x e. CC -> 2 e. CC ) |
205 |
|
id |
|- ( x e. CC -> x e. CC ) |
206 |
204 205
|
mulcld |
|- ( x e. CC -> ( 2 x. x ) e. CC ) |
207 |
206
|
negcld |
|- ( x e. CC -> -u ( 2 x. x ) e. CC ) |
208 |
207
|
adantl |
|- ( ( T. /\ x e. CC ) -> -u ( 2 x. x ) e. CC ) |
209 |
12 208
|
sylan2 |
|- ( ( T. /\ x e. D ) -> -u ( 2 x. x ) e. CC ) |
210 |
59
|
sqrtcld |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> ( sqrt ` y ) e. CC ) |
211 |
210
|
adantl |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( sqrt ` y ) e. CC ) |
212 |
|
ovexd |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( 1 / ( 2 x. ( sqrt ` y ) ) ) e. _V ) |
213 |
19
|
adantl |
|- ( ( T. /\ x e. CC ) -> ( 1 - ( x ^ 2 ) ) e. CC ) |
214 |
36
|
a1i |
|- ( ( T. /\ x e. CC ) -> 0 e. RR ) |
215 |
|
1cnd |
|- ( T. -> 1 e. CC ) |
216 |
11 215
|
dvmptc |
|- ( T. -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) ) |
217 |
17
|
adantl |
|- ( ( T. /\ x e. CC ) -> ( x ^ 2 ) e. CC ) |
218 |
|
2cn |
|- 2 e. CC |
219 |
|
mulcl |
|- ( ( 2 e. CC /\ x e. CC ) -> ( 2 x. x ) e. CC ) |
220 |
218 219
|
mpan |
|- ( x e. CC -> ( 2 x. x ) e. CC ) |
221 |
220
|
adantl |
|- ( ( T. /\ x e. CC ) -> ( 2 x. x ) e. CC ) |
222 |
|
2nn |
|- 2 e. NN |
223 |
|
dvexp |
|- ( 2 e. NN -> ( CC _D ( x e. CC |-> ( x ^ 2 ) ) ) = ( x e. CC |-> ( 2 x. ( x ^ ( 2 - 1 ) ) ) ) ) |
224 |
222 223
|
ax-mp |
|- ( CC _D ( x e. CC |-> ( x ^ 2 ) ) ) = ( x e. CC |-> ( 2 x. ( x ^ ( 2 - 1 ) ) ) ) |
225 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
226 |
225
|
oveq2i |
|- ( x ^ ( 2 - 1 ) ) = ( x ^ 1 ) |
227 |
|
exp1 |
|- ( x e. CC -> ( x ^ 1 ) = x ) |
228 |
226 227
|
syl5eq |
|- ( x e. CC -> ( x ^ ( 2 - 1 ) ) = x ) |
229 |
228
|
oveq2d |
|- ( x e. CC -> ( 2 x. ( x ^ ( 2 - 1 ) ) ) = ( 2 x. x ) ) |
230 |
229
|
mpteq2ia |
|- ( x e. CC |-> ( 2 x. ( x ^ ( 2 - 1 ) ) ) ) = ( x e. CC |-> ( 2 x. x ) ) |
231 |
224 230
|
eqtri |
|- ( CC _D ( x e. CC |-> ( x ^ 2 ) ) ) = ( x e. CC |-> ( 2 x. x ) ) |
232 |
231
|
a1i |
|- ( T. -> ( CC _D ( x e. CC |-> ( x ^ 2 ) ) ) = ( x e. CC |-> ( 2 x. x ) ) ) |
233 |
11 79 214 216 217 221 232
|
dvmptsub |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 - ( x ^ 2 ) ) ) ) = ( x e. CC |-> ( 0 - ( 2 x. x ) ) ) ) |
234 |
|
df-neg |
|- -u ( 2 x. x ) = ( 0 - ( 2 x. x ) ) |
235 |
234
|
mpteq2i |
|- ( x e. CC |-> -u ( 2 x. x ) ) = ( x e. CC |-> ( 0 - ( 2 x. x ) ) ) |
236 |
233 235
|
eqtr4di |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 - ( x ^ 2 ) ) ) ) = ( x e. CC |-> -u ( 2 x. x ) ) ) |
237 |
11 213 208 236 81 84 82 103
|
dvmptres |
|- ( T. -> ( CC _D ( x e. D |-> ( 1 - ( x ^ 2 ) ) ) ) = ( x e. D |-> -u ( 2 x. x ) ) ) |
238 |
|
eqid |
|- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) ) |
239 |
238
|
dvcnsqrt |
|- ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( sqrt ` y ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / ( 2 x. ( sqrt ` y ) ) ) ) |
240 |
239
|
a1i |
|- ( T. -> ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( sqrt ` y ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / ( 2 x. ( sqrt ` y ) ) ) ) ) |
241 |
|
fveq2 |
|- ( y = ( 1 - ( x ^ 2 ) ) -> ( sqrt ` y ) = ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) |
242 |
241
|
oveq2d |
|- ( y = ( 1 - ( x ^ 2 ) ) -> ( 2 x. ( sqrt ` y ) ) = ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
243 |
242
|
oveq2d |
|- ( y = ( 1 - ( x ^ 2 ) ) -> ( 1 / ( 2 x. ( sqrt ` y ) ) ) = ( 1 / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
244 |
11 11 203 209 211 212 237 240 241 243
|
dvmptco |
|- ( T. -> ( CC _D ( x e. D |-> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( x e. D |-> ( ( 1 / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. -u ( 2 x. x ) ) ) ) |
245 |
|
mulneg2 |
|- ( ( 2 e. CC /\ x e. CC ) -> ( 2 x. -u x ) = -u ( 2 x. x ) ) |
246 |
218 12 245
|
sylancr |
|- ( x e. D -> ( 2 x. -u x ) = -u ( 2 x. x ) ) |
247 |
246
|
oveq1d |
|- ( x e. D -> ( ( 2 x. -u x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( -u ( 2 x. x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
248 |
12
|
negcld |
|- ( x e. D -> -u x e. CC ) |
249 |
|
eldifn |
|- ( x e. ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) -> -. x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) |
250 |
249 1
|
eleq2s |
|- ( x e. D -> -. x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) |
251 |
|
id |
|- ( x = -u 1 -> x = -u 1 ) |
252 |
|
mnflt |
|- ( -u 1 e. RR -> -oo < -u 1 ) |
253 |
86 252
|
ax-mp |
|- -oo < -u 1 |
254 |
|
ubioc1 |
|- ( ( -oo e. RR* /\ -u 1 e. RR* /\ -oo < -u 1 ) -> -u 1 e. ( -oo (,] -u 1 ) ) |
255 |
51 132 253 254
|
mp3an |
|- -u 1 e. ( -oo (,] -u 1 ) |
256 |
251 255
|
eqeltrdi |
|- ( x = -u 1 -> x e. ( -oo (,] -u 1 ) ) |
257 |
|
id |
|- ( x = 1 -> x = 1 ) |
258 |
|
ltpnf |
|- ( 1 e. RR -> 1 < +oo ) |
259 |
89 258
|
ax-mp |
|- 1 < +oo |
260 |
|
lbico1 |
|- ( ( 1 e. RR* /\ +oo e. RR* /\ 1 < +oo ) -> 1 e. ( 1 [,) +oo ) ) |
261 |
133 121 259 260
|
mp3an |
|- 1 e. ( 1 [,) +oo ) |
262 |
257 261
|
eqeltrdi |
|- ( x = 1 -> x e. ( 1 [,) +oo ) ) |
263 |
256 262
|
orim12i |
|- ( ( x = -u 1 \/ x = 1 ) -> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) ) |
264 |
263
|
orcoms |
|- ( ( x = 1 \/ x = -u 1 ) -> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) ) |
265 |
|
elun |
|- ( x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) <-> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) ) |
266 |
264 265
|
sylibr |
|- ( ( x = 1 \/ x = -u 1 ) -> x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) |
267 |
250 266
|
nsyl |
|- ( x e. D -> -. ( x = 1 \/ x = -u 1 ) ) |
268 |
|
1cnd |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> 1 e. CC ) |
269 |
17
|
adantr |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( x ^ 2 ) e. CC ) |
270 |
19
|
adantr |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( 1 - ( x ^ 2 ) ) e. CC ) |
271 |
|
simpr |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) |
272 |
270 271
|
sqr00d |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( 1 - ( x ^ 2 ) ) = 0 ) |
273 |
268 269 272
|
subeq0d |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> 1 = ( x ^ 2 ) ) |
274 |
146 273
|
eqtr2id |
|- ( ( x e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( x ^ 2 ) = ( 1 ^ 2 ) ) |
275 |
274
|
ex |
|- ( x e. CC -> ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 -> ( x ^ 2 ) = ( 1 ^ 2 ) ) ) |
276 |
|
sqeqor |
|- ( ( x e. CC /\ 1 e. CC ) -> ( ( x ^ 2 ) = ( 1 ^ 2 ) <-> ( x = 1 \/ x = -u 1 ) ) ) |
277 |
16 276
|
mpan2 |
|- ( x e. CC -> ( ( x ^ 2 ) = ( 1 ^ 2 ) <-> ( x = 1 \/ x = -u 1 ) ) ) |
278 |
275 277
|
sylibd |
|- ( x e. CC -> ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) = 0 -> ( x = 1 \/ x = -u 1 ) ) ) |
279 |
278
|
necon3bd |
|- ( x e. CC -> ( -. ( x = 1 \/ x = -u 1 ) -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) ) |
280 |
12 267 279
|
sylc |
|- ( x e. D -> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) |
281 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
282 |
|
divcan5 |
|- ( ( -u x e. CC /\ ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( 2 x. -u x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
283 |
281 282
|
mp3an3 |
|- ( ( -u x e. CC /\ ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) ) -> ( ( 2 x. -u x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
284 |
248 112 280 283
|
syl12anc |
|- ( x e. D -> ( ( 2 x. -u x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
285 |
218 12 219
|
sylancr |
|- ( x e. D -> ( 2 x. x ) e. CC ) |
286 |
285
|
negcld |
|- ( x e. D -> -u ( 2 x. x ) e. CC ) |
287 |
|
mulcl |
|- ( ( 2 e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC ) -> ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
288 |
218 112 287
|
sylancr |
|- ( x e. D -> ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
289 |
|
mulne0 |
|- ( ( ( 2 e. CC /\ 2 =/= 0 ) /\ ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) ) -> ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
290 |
281 289
|
mpan |
|- ( ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) -> ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
291 |
112 280 290
|
syl2anc |
|- ( x e. D -> ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) =/= 0 ) |
292 |
286 288 291
|
divrec2d |
|- ( x e. D -> ( -u ( 2 x. x ) / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( 1 / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. -u ( 2 x. x ) ) ) |
293 |
247 284 292
|
3eqtr3rd |
|- ( x e. D -> ( ( 1 / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. -u ( 2 x. x ) ) = ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
294 |
293
|
mpteq2ia |
|- ( x e. D |-> ( ( 1 / ( 2 x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. -u ( 2 x. x ) ) ) = ( x e. D |-> ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
295 |
244 294
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. D |-> ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( x e. D |-> ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
296 |
11 74 75 109 113 114 295
|
dvmptadd |
|- ( T. -> ( CC _D ( x e. D |-> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( _i + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
297 |
|
mulcl |
|- ( ( _i e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC ) -> ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
298 |
13 20 297
|
sylancr |
|- ( x e. CC -> ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
299 |
12 298
|
syl |
|- ( x e. D -> ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) |
300 |
299 248 112 280
|
divdird |
|- ( x e. D -> ( ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + -u x ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
301 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
302 |
301
|
eqcomi |
|- -u 1 = ( _i x. _i ) |
303 |
302
|
oveq1i |
|- ( -u 1 x. x ) = ( ( _i x. _i ) x. x ) |
304 |
|
mulm1 |
|- ( x e. CC -> ( -u 1 x. x ) = -u x ) |
305 |
|
mulass |
|- ( ( _i e. CC /\ _i e. CC /\ x e. CC ) -> ( ( _i x. _i ) x. x ) = ( _i x. ( _i x. x ) ) ) |
306 |
13 13 305
|
mp3an12 |
|- ( x e. CC -> ( ( _i x. _i ) x. x ) = ( _i x. ( _i x. x ) ) ) |
307 |
303 304 306
|
3eqtr3a |
|- ( x e. CC -> -u x = ( _i x. ( _i x. x ) ) ) |
308 |
307
|
oveq1d |
|- ( x e. CC -> ( -u x + ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( _i x. ( _i x. x ) ) + ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
309 |
|
negcl |
|- ( x e. CC -> -u x e. CC ) |
310 |
298 309
|
addcomd |
|- ( x e. CC -> ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + -u x ) = ( -u x + ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
311 |
13
|
a1i |
|- ( x e. CC -> _i e. CC ) |
312 |
311 15 20
|
adddid |
|- ( x e. CC -> ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( _i x. ( _i x. x ) ) + ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
313 |
308 310 312
|
3eqtr4d |
|- ( x e. CC -> ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + -u x ) = ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
314 |
12 313
|
syl |
|- ( x e. D -> ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + -u x ) = ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
315 |
314
|
oveq1d |
|- ( x e. D -> ( ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + -u x ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
316 |
72 112 280
|
divcan4d |
|- ( x e. D -> ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = _i ) |
317 |
316
|
oveq1d |
|- ( x e. D -> ( ( ( _i x. ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( _i + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
318 |
300 315 317
|
3eqtr3rd |
|- ( x e. D -> ( _i + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
319 |
318
|
mpteq2ia |
|- ( x e. D |-> ( _i + ( -u x / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
320 |
296 319
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. D |-> ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
321 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
322 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
323 |
321 322
|
mp1i |
|- ( T. -> log : ( CC \ { 0 } ) --> ran log ) |
324 |
|
snssi |
|- ( 0 e. ( -oo (,] 0 ) -> { 0 } C_ ( -oo (,] 0 ) ) |
325 |
64 324
|
ax-mp |
|- { 0 } C_ ( -oo (,] 0 ) |
326 |
|
sscon |
|- ( { 0 } C_ ( -oo (,] 0 ) -> ( CC \ ( -oo (,] 0 ) ) C_ ( CC \ { 0 } ) ) |
327 |
325 326
|
mp1i |
|- ( T. -> ( CC \ ( -oo (,] 0 ) ) C_ ( CC \ { 0 } ) ) |
328 |
323 327
|
feqresmpt |
|- ( T. -> ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) |
329 |
328
|
oveq2d |
|- ( T. -> ( CC _D ( log |` ( CC \ ( -oo (,] 0 ) ) ) ) = ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) ) |
330 |
238
|
dvlog |
|- ( CC _D ( log |` ( CC \ ( -oo (,] 0 ) ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / y ) ) |
331 |
329 330
|
eqtr3di |
|- ( T. -> ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / y ) ) ) |
332 |
|
fveq2 |
|- ( y = ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) -> ( log ` y ) = ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
333 |
|
oveq2 |
|- ( y = ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) -> ( 1 / y ) = ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
334 |
11 11 57 58 70 71 320 331 332 333
|
dvmptco |
|- ( T. -> ( CC _D ( x e. D |-> ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) = ( x e. D |-> ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
335 |
22 24
|
reccld |
|- ( x e. D -> ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
336 |
|
mulcl |
|- ( ( _i e. CC /\ ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) e. CC ) -> ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
337 |
13 21 336
|
sylancr |
|- ( x e. CC -> ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
338 |
12 337
|
syl |
|- ( x e. D -> ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) e. CC ) |
339 |
335 338 112 280
|
divassd |
|- ( x e. D -> ( ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
340 |
338 22 24
|
divrec2d |
|- ( x e. D -> ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
341 |
72 22 24
|
divcan4d |
|- ( x e. D -> ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = _i ) |
342 |
340 341
|
eqtr3d |
|- ( x e. D -> ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = _i ) |
343 |
342
|
oveq1d |
|- ( x e. D -> ( ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
344 |
339 343
|
eqtr3d |
|- ( x e. D -> ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
345 |
344
|
mpteq2ia |
|- ( x e. D |-> ( ( 1 / ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) x. ( ( _i x. ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
346 |
334 345
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. D |-> ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) = ( x e. D |-> ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
347 |
|
negicn |
|- -u _i e. CC |
348 |
347
|
a1i |
|- ( T. -> -u _i e. CC ) |
349 |
11 26 27 346 348
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) ) = ( x e. D |-> ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) |
350 |
349
|
mptru |
|- ( CC _D ( x e. D |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) ) ) = ( x e. D |-> ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
351 |
|
divass |
|- ( ( -u _i e. CC /\ _i e. CC /\ ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) ) -> ( ( -u _i x. _i ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
352 |
347 13 351
|
mp3an12 |
|- ( ( ( sqrt ` ( 1 - ( x ^ 2 ) ) ) e. CC /\ ( sqrt ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) -> ( ( -u _i x. _i ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
353 |
112 280 352
|
syl2anc |
|- ( x e. D -> ( ( -u _i x. _i ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) |
354 |
13 13
|
mulneg1i |
|- ( -u _i x. _i ) = -u ( _i x. _i ) |
355 |
301
|
negeqi |
|- -u ( _i x. _i ) = -u -u 1 |
356 |
|
negneg1e1 |
|- -u -u 1 = 1 |
357 |
354 355 356
|
3eqtri |
|- ( -u _i x. _i ) = 1 |
358 |
357
|
oveq1i |
|- ( ( -u _i x. _i ) / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) = ( 1 / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) |
359 |
353 358
|
eqtr3di |
|- ( x e. D -> ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( 1 / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
360 |
359
|
mpteq2ia |
|- ( x e. D |-> ( -u _i x. ( _i / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( 1 / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |
361 |
9 350 360
|
3eqtri |
|- ( CC _D ( arcsin |` D ) ) = ( x e. D |-> ( 1 / ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) |