| Step |
Hyp |
Ref |
Expression |
| 1 |
|
redvabs.d |
|- D = ( RR \ { 0 } ) |
| 2 |
|
reelprrecn |
|- RR e. { RR , CC } |
| 3 |
2
|
a1i |
|- ( T. -> RR e. { RR , CC } ) |
| 4 |
|
cnelprrecn |
|- CC e. { RR , CC } |
| 5 |
4
|
a1i |
|- ( T. -> CC e. { RR , CC } ) |
| 6 |
|
dfrp2 |
|- RR+ = ( 0 (,) +oo ) |
| 7 |
|
mnfxr |
|- -oo e. RR* |
| 8 |
7
|
a1i |
|- ( T. -> -oo e. RR* ) |
| 9 |
|
0xr |
|- 0 e. RR* |
| 10 |
9
|
a1i |
|- ( T. -> 0 e. RR* ) |
| 11 |
|
pnfxr |
|- +oo e. RR* |
| 12 |
11
|
a1i |
|- ( T. -> +oo e. RR* ) |
| 13 |
8 10 12
|
iocioodisjd |
|- ( T. -> ( ( -oo (,] 0 ) i^i ( 0 (,) +oo ) ) = (/) ) |
| 14 |
13
|
mptru |
|- ( ( -oo (,] 0 ) i^i ( 0 (,) +oo ) ) = (/) |
| 15 |
14
|
ineqcomi |
|- ( ( 0 (,) +oo ) i^i ( -oo (,] 0 ) ) = (/) |
| 16 |
|
disjdif2 |
|- ( ( ( 0 (,) +oo ) i^i ( -oo (,] 0 ) ) = (/) -> ( ( 0 (,) +oo ) \ ( -oo (,] 0 ) ) = ( 0 (,) +oo ) ) |
| 17 |
15 16
|
ax-mp |
|- ( ( 0 (,) +oo ) \ ( -oo (,] 0 ) ) = ( 0 (,) +oo ) |
| 18 |
6 17
|
eqtr4i |
|- RR+ = ( ( 0 (,) +oo ) \ ( -oo (,] 0 ) ) |
| 19 |
|
ioosscn |
|- ( 0 (,) +oo ) C_ CC |
| 20 |
|
ssdif |
|- ( ( 0 (,) +oo ) C_ CC -> ( ( 0 (,) +oo ) \ ( -oo (,] 0 ) ) C_ ( CC \ ( -oo (,] 0 ) ) ) |
| 21 |
19 20
|
ax-mp |
|- ( ( 0 (,) +oo ) \ ( -oo (,] 0 ) ) C_ ( CC \ ( -oo (,] 0 ) ) |
| 22 |
18 21
|
eqsstri |
|- RR+ C_ ( CC \ ( -oo (,] 0 ) ) |
| 23 |
1
|
eleq2i |
|- ( x e. D <-> x e. ( RR \ { 0 } ) ) |
| 24 |
|
eldifsn |
|- ( x e. ( RR \ { 0 } ) <-> ( x e. RR /\ x =/= 0 ) ) |
| 25 |
23 24
|
bitri |
|- ( x e. D <-> ( x e. RR /\ x =/= 0 ) ) |
| 26 |
25
|
simplbi |
|- ( x e. D -> x e. RR ) |
| 27 |
26
|
recnd |
|- ( x e. D -> x e. CC ) |
| 28 |
27
|
adantl |
|- ( ( T. /\ x e. D ) -> x e. CC ) |
| 29 |
25
|
simprbi |
|- ( x e. D -> x =/= 0 ) |
| 30 |
29
|
adantl |
|- ( ( T. /\ x e. D ) -> x =/= 0 ) |
| 31 |
28 30
|
absrpcld |
|- ( ( T. /\ x e. D ) -> ( abs ` x ) e. RR+ ) |
| 32 |
22 31
|
sselid |
|- ( ( T. /\ x e. D ) -> ( abs ` x ) e. ( CC \ ( -oo (,] 0 ) ) ) |
| 33 |
|
negex |
|- -u 1 e. _V |
| 34 |
|
1ex |
|- 1 e. _V |
| 35 |
33 34
|
ifex |
|- if ( x < 0 , -u 1 , 1 ) e. _V |
| 36 |
35
|
a1i |
|- ( ( T. /\ x e. D ) -> if ( x < 0 , -u 1 , 1 ) e. _V ) |
| 37 |
|
eldifi |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> y e. CC ) |
| 38 |
37
|
adantl |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> y e. CC ) |
| 39 |
|
eldifn |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> -. y e. ( -oo (,] 0 ) ) |
| 40 |
|
mnflt0 |
|- -oo < 0 |
| 41 |
|
ubioc1 |
|- ( ( -oo e. RR* /\ 0 e. RR* /\ -oo < 0 ) -> 0 e. ( -oo (,] 0 ) ) |
| 42 |
7 9 40 41
|
mp3an |
|- 0 e. ( -oo (,] 0 ) |
| 43 |
|
eleq1 |
|- ( y = 0 -> ( y e. ( -oo (,] 0 ) <-> 0 e. ( -oo (,] 0 ) ) ) |
| 44 |
42 43
|
mpbiri |
|- ( y = 0 -> y e. ( -oo (,] 0 ) ) |
| 45 |
44
|
necon3bi |
|- ( -. y e. ( -oo (,] 0 ) -> y =/= 0 ) |
| 46 |
39 45
|
syl |
|- ( y e. ( CC \ ( -oo (,] 0 ) ) -> y =/= 0 ) |
| 47 |
46
|
adantl |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> y =/= 0 ) |
| 48 |
38 47
|
logcld |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( log ` y ) e. CC ) |
| 49 |
|
ovexd |
|- ( ( T. /\ y e. ( CC \ ( -oo (,] 0 ) ) ) -> ( 1 / y ) e. _V ) |
| 50 |
1
|
redvmptabs |
|- ( RR _D ( x e. D |-> ( abs ` x ) ) ) = ( x e. D |-> if ( x < 0 , -u 1 , 1 ) ) |
| 51 |
50
|
a1i |
|- ( T. -> ( RR _D ( x e. D |-> ( abs ` x ) ) ) = ( x e. D |-> if ( x < 0 , -u 1 , 1 ) ) ) |
| 52 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
| 53 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
| 54 |
52 53
|
mp1i |
|- ( T. -> log : ( CC \ { 0 } ) --> ran log ) |
| 55 |
|
eqid |
|- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) ) |
| 56 |
55
|
logdmss |
|- ( CC \ ( -oo (,] 0 ) ) C_ ( CC \ { 0 } ) |
| 57 |
56
|
a1i |
|- ( T. -> ( CC \ ( -oo (,] 0 ) ) C_ ( CC \ { 0 } ) ) |
| 58 |
54 57
|
feqresmpt |
|- ( T. -> ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) |
| 59 |
58
|
mptru |
|- ( log |` ( CC \ ( -oo (,] 0 ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) |
| 60 |
59
|
oveq2i |
|- ( CC _D ( log |` ( CC \ ( -oo (,] 0 ) ) ) ) = ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) |
| 61 |
55
|
dvlog |
|- ( CC _D ( log |` ( CC \ ( -oo (,] 0 ) ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / y ) ) |
| 62 |
60 61
|
eqtr3i |
|- ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / y ) ) |
| 63 |
62
|
a1i |
|- ( T. -> ( CC _D ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( log ` y ) ) ) = ( y e. ( CC \ ( -oo (,] 0 ) ) |-> ( 1 / y ) ) ) |
| 64 |
|
fveq2 |
|- ( y = ( abs ` x ) -> ( log ` y ) = ( log ` ( abs ` x ) ) ) |
| 65 |
|
oveq2 |
|- ( y = ( abs ` x ) -> ( 1 / y ) = ( 1 / ( abs ` x ) ) ) |
| 66 |
3 5 32 36 48 49 51 63 64 65
|
dvmptco |
|- ( T. -> ( RR _D ( x e. D |-> ( log ` ( abs ` x ) ) ) ) = ( x e. D |-> ( ( 1 / ( abs ` x ) ) x. if ( x < 0 , -u 1 , 1 ) ) ) ) |
| 67 |
66
|
mptru |
|- ( RR _D ( x e. D |-> ( log ` ( abs ` x ) ) ) ) = ( x e. D |-> ( ( 1 / ( abs ` x ) ) x. if ( x < 0 , -u 1 , 1 ) ) ) |
| 68 |
|
ovif2 |
|- ( ( 1 / ( abs ` x ) ) x. if ( x < 0 , -u 1 , 1 ) ) = if ( x < 0 , ( ( 1 / ( abs ` x ) ) x. -u 1 ) , ( ( 1 / ( abs ` x ) ) x. 1 ) ) |
| 69 |
|
simpll |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> x e. RR ) |
| 70 |
69
|
recnd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> x e. CC ) |
| 71 |
70
|
abscld |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( abs ` x ) e. RR ) |
| 72 |
71
|
recnd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( abs ` x ) e. CC ) |
| 73 |
|
simplr |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> x =/= 0 ) |
| 74 |
70 73
|
absne0d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( abs ` x ) =/= 0 ) |
| 75 |
72 74
|
reccld |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( 1 / ( abs ` x ) ) e. CC ) |
| 76 |
|
neg1cn |
|- -u 1 e. CC |
| 77 |
76
|
a1i |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> -u 1 e. CC ) |
| 78 |
75 77
|
mulcomd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. -u 1 ) = ( -u 1 x. ( 1 / ( abs ` x ) ) ) ) |
| 79 |
75
|
mulm1d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( -u 1 x. ( 1 / ( abs ` x ) ) ) = -u ( 1 / ( abs ` x ) ) ) |
| 80 |
|
1cnd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> 1 e. CC ) |
| 81 |
80 72 74
|
divneg2d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> -u ( 1 / ( abs ` x ) ) = ( 1 / -u ( abs ` x ) ) ) |
| 82 |
|
0red |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> 0 e. RR ) |
| 83 |
|
simpr |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> x < 0 ) |
| 84 |
69 82 83
|
ltled |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> x <_ 0 ) |
| 85 |
69 84
|
absnidd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( abs ` x ) = -u x ) |
| 86 |
85
|
eqcomd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> -u x = ( abs ` x ) ) |
| 87 |
70 86
|
negcon1ad |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> -u ( abs ` x ) = x ) |
| 88 |
87
|
oveq2d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( 1 / -u ( abs ` x ) ) = ( 1 / x ) ) |
| 89 |
81 88
|
eqtrd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> -u ( 1 / ( abs ` x ) ) = ( 1 / x ) ) |
| 90 |
78 79 89
|
3eqtrd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. -u 1 ) = ( 1 / x ) ) |
| 91 |
25 90
|
sylanb |
|- ( ( x e. D /\ x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. -u 1 ) = ( 1 / x ) ) |
| 92 |
|
recn |
|- ( x e. RR -> x e. CC ) |
| 93 |
92
|
abscld |
|- ( x e. RR -> ( abs ` x ) e. RR ) |
| 94 |
93
|
ad2antrr |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( abs ` x ) e. RR ) |
| 95 |
92
|
ad2antrr |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> x e. CC ) |
| 96 |
|
simplr |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> x =/= 0 ) |
| 97 |
95 96
|
absne0d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( abs ` x ) =/= 0 ) |
| 98 |
94 97
|
rereccld |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( 1 / ( abs ` x ) ) e. RR ) |
| 99 |
98
|
recnd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( 1 / ( abs ` x ) ) e. CC ) |
| 100 |
99
|
mulridd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. 1 ) = ( 1 / ( abs ` x ) ) ) |
| 101 |
|
simpll |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> x e. RR ) |
| 102 |
|
0red |
|- ( ( x e. RR /\ x =/= 0 ) -> 0 e. RR ) |
| 103 |
|
simpl |
|- ( ( x e. RR /\ x =/= 0 ) -> x e. RR ) |
| 104 |
102 103
|
lenltd |
|- ( ( x e. RR /\ x =/= 0 ) -> ( 0 <_ x <-> -. x < 0 ) ) |
| 105 |
104
|
biimpar |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> 0 <_ x ) |
| 106 |
101 105
|
absidd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( abs ` x ) = x ) |
| 107 |
106
|
oveq2d |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( 1 / ( abs ` x ) ) = ( 1 / x ) ) |
| 108 |
100 107
|
eqtrd |
|- ( ( ( x e. RR /\ x =/= 0 ) /\ -. x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. 1 ) = ( 1 / x ) ) |
| 109 |
25 108
|
sylanb |
|- ( ( x e. D /\ -. x < 0 ) -> ( ( 1 / ( abs ` x ) ) x. 1 ) = ( 1 / x ) ) |
| 110 |
91 109
|
ifeqda |
|- ( x e. D -> if ( x < 0 , ( ( 1 / ( abs ` x ) ) x. -u 1 ) , ( ( 1 / ( abs ` x ) ) x. 1 ) ) = ( 1 / x ) ) |
| 111 |
68 110
|
eqtrid |
|- ( x e. D -> ( ( 1 / ( abs ` x ) ) x. if ( x < 0 , -u 1 , 1 ) ) = ( 1 / x ) ) |
| 112 |
111
|
mpteq2ia |
|- ( x e. D |-> ( ( 1 / ( abs ` x ) ) x. if ( x < 0 , -u 1 , 1 ) ) ) = ( x e. D |-> ( 1 / x ) ) |
| 113 |
67 112
|
eqtri |
|- ( RR _D ( x e. D |-> ( log ` ( abs ` x ) ) ) ) = ( x e. D |-> ( 1 / x ) ) |