Step |
Hyp |
Ref |
Expression |
1 |
|
reccn2.t |
|- T = ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) |
2 |
|
1rp |
|- 1 e. RR+ |
3 |
|
simpl |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> A e. ( CC \ { 0 } ) ) |
4 |
|
eldifsn |
|- ( A e. ( CC \ { 0 } ) <-> ( A e. CC /\ A =/= 0 ) ) |
5 |
3 4
|
sylib |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> ( A e. CC /\ A =/= 0 ) ) |
6 |
|
absrpcl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ ) |
7 |
5 6
|
syl |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> ( abs ` A ) e. RR+ ) |
8 |
|
rpmulcl |
|- ( ( ( abs ` A ) e. RR+ /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ ) |
9 |
7 8
|
sylancom |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> ( ( abs ` A ) x. B ) e. RR+ ) |
10 |
|
ifcl |
|- ( ( 1 e. RR+ /\ ( ( abs ` A ) x. B ) e. RR+ ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) e. RR+ ) |
11 |
2 9 10
|
sylancr |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) e. RR+ ) |
12 |
7
|
rphalfcld |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> ( ( abs ` A ) / 2 ) e. RR+ ) |
13 |
11 12
|
rpmulcld |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) e. RR+ ) |
14 |
1 13
|
eqeltrid |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> T e. RR+ ) |
15 |
5
|
adantr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A e. CC /\ A =/= 0 ) ) |
16 |
15
|
simpld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> A e. CC ) |
17 |
|
simprl |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> z e. ( CC \ { 0 } ) ) |
18 |
|
eldifsn |
|- ( z e. ( CC \ { 0 } ) <-> ( z e. CC /\ z =/= 0 ) ) |
19 |
17 18
|
sylib |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( z e. CC /\ z =/= 0 ) ) |
20 |
19
|
simpld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> z e. CC ) |
21 |
16 20
|
mulcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. z ) e. CC ) |
22 |
|
mulne0 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( z e. CC /\ z =/= 0 ) ) -> ( A x. z ) =/= 0 ) |
23 |
15 19 22
|
syl2anc |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. z ) =/= 0 ) |
24 |
16 20 21 23
|
divsubdird |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A - z ) / ( A x. z ) ) = ( ( A / ( A x. z ) ) - ( z / ( A x. z ) ) ) ) |
25 |
16
|
mulid1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A x. 1 ) = A ) |
26 |
25
|
oveq1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A x. 1 ) / ( A x. z ) ) = ( A / ( A x. z ) ) ) |
27 |
|
1cnd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> 1 e. CC ) |
28 |
|
divcan5 |
|- ( ( 1 e. CC /\ ( z e. CC /\ z =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( ( A x. 1 ) / ( A x. z ) ) = ( 1 / z ) ) |
29 |
27 19 15 28
|
syl3anc |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A x. 1 ) / ( A x. z ) ) = ( 1 / z ) ) |
30 |
26 29
|
eqtr3d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A / ( A x. z ) ) = ( 1 / z ) ) |
31 |
20
|
mulid1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( z x. 1 ) = z ) |
32 |
20 16
|
mulcomd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( z x. A ) = ( A x. z ) ) |
33 |
31 32
|
oveq12d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( z x. 1 ) / ( z x. A ) ) = ( z / ( A x. z ) ) ) |
34 |
|
divcan5 |
|- ( ( 1 e. CC /\ ( A e. CC /\ A =/= 0 ) /\ ( z e. CC /\ z =/= 0 ) ) -> ( ( z x. 1 ) / ( z x. A ) ) = ( 1 / A ) ) |
35 |
27 15 19 34
|
syl3anc |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( z x. 1 ) / ( z x. A ) ) = ( 1 / A ) ) |
36 |
33 35
|
eqtr3d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( z / ( A x. z ) ) = ( 1 / A ) ) |
37 |
30 36
|
oveq12d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A / ( A x. z ) ) - ( z / ( A x. z ) ) ) = ( ( 1 / z ) - ( 1 / A ) ) ) |
38 |
24 37
|
eqtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( A - z ) / ( A x. z ) ) = ( ( 1 / z ) - ( 1 / A ) ) ) |
39 |
38
|
fveq2d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( A - z ) / ( A x. z ) ) ) = ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) ) |
40 |
16 20
|
subcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( A - z ) e. CC ) |
41 |
40 21 23
|
absdivd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( A - z ) / ( A x. z ) ) ) = ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) ) |
42 |
39 41
|
eqtr3d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) = ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) ) |
43 |
16 20
|
abssubd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) = ( abs ` ( z - A ) ) ) |
44 |
20 16
|
subcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( z - A ) e. CC ) |
45 |
44
|
abscld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( z - A ) ) e. RR ) |
46 |
43 45
|
eqeltrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) e. RR ) |
47 |
14
|
adantr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T e. RR+ ) |
48 |
47
|
rpred |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T e. RR ) |
49 |
21
|
abscld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) e. RR ) |
50 |
|
rpre |
|- ( B e. RR+ -> B e. RR ) |
51 |
50
|
ad2antlr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> B e. RR ) |
52 |
49 51
|
remulcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) e. RR ) |
53 |
|
simprr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( z - A ) ) < T ) |
54 |
43 53
|
eqbrtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < T ) |
55 |
9
|
adantr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) x. B ) e. RR+ ) |
56 |
55
|
rpred |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) x. B ) e. RR ) |
57 |
12
|
adantr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. RR+ ) |
58 |
57
|
rpred |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. RR ) |
59 |
56 58
|
remulcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) e. RR ) |
60 |
|
1re |
|- 1 e. RR |
61 |
|
min2 |
|- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ ( ( abs ` A ) x. B ) ) |
62 |
60 56 61
|
sylancr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ ( ( abs ` A ) x. B ) ) |
63 |
11
|
adantr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) e. RR+ ) |
64 |
63
|
rpred |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) e. RR ) |
65 |
64 56 57
|
lemul1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ ( ( abs ` A ) x. B ) <-> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) ) ) |
66 |
62 65
|
mpbid |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) ) |
67 |
1 66
|
eqbrtrid |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) ) |
68 |
20
|
abscld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` z ) e. RR ) |
69 |
16
|
abscld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) e. RR ) |
70 |
69
|
recnd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) e. CC ) |
71 |
70
|
2halvesd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) = ( abs ` A ) ) |
72 |
69 68
|
resubcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) e. RR ) |
73 |
16 20
|
abs2difd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) <_ ( abs ` ( A - z ) ) ) |
74 |
|
min1 |
|- ( ( 1 e. RR /\ ( ( abs ` A ) x. B ) e. RR ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ 1 ) |
75 |
60 56 74
|
sylancr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ 1 ) |
76 |
|
1red |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> 1 e. RR ) |
77 |
64 76 57
|
lemul1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) <_ 1 <-> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) <_ ( 1 x. ( ( abs ` A ) / 2 ) ) ) ) |
78 |
75 77
|
mpbid |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( if ( 1 <_ ( ( abs ` A ) x. B ) , 1 , ( ( abs ` A ) x. B ) ) x. ( ( abs ` A ) / 2 ) ) <_ ( 1 x. ( ( abs ` A ) / 2 ) ) ) |
79 |
1 78
|
eqbrtrid |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( 1 x. ( ( abs ` A ) / 2 ) ) ) |
80 |
58
|
recnd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) e. CC ) |
81 |
80
|
mulid2d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( 1 x. ( ( abs ` A ) / 2 ) ) = ( ( abs ` A ) / 2 ) ) |
82 |
79 81
|
breqtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T <_ ( ( abs ` A ) / 2 ) ) |
83 |
46 48 58 54 82
|
ltletrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < ( ( abs ` A ) / 2 ) ) |
84 |
72 46 58 73 83
|
lelttrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) - ( abs ` z ) ) < ( ( abs ` A ) / 2 ) ) |
85 |
69 68 58
|
ltsubadd2d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) - ( abs ` z ) ) < ( ( abs ` A ) / 2 ) <-> ( abs ` A ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) ) |
86 |
84 85
|
mpbid |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` A ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) |
87 |
71 86
|
eqbrtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) |
88 |
58 68 58
|
ltadd1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) / 2 ) < ( abs ` z ) <-> ( ( ( abs ` A ) / 2 ) + ( ( abs ` A ) / 2 ) ) < ( ( abs ` z ) + ( ( abs ` A ) / 2 ) ) ) ) |
89 |
87 88
|
mpbird |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` A ) / 2 ) < ( abs ` z ) ) |
90 |
58 68 55 89
|
ltmul2dd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) < ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) ) |
91 |
16 20
|
absmuld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) = ( ( abs ` A ) x. ( abs ` z ) ) ) |
92 |
91
|
oveq1d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) = ( ( ( abs ` A ) x. ( abs ` z ) ) x. B ) ) |
93 |
68
|
recnd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` z ) e. CC ) |
94 |
51
|
recnd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> B e. CC ) |
95 |
70 93 94
|
mul32d |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. ( abs ` z ) ) x. B ) = ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) ) |
96 |
92 95
|
eqtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A x. z ) ) x. B ) = ( ( ( abs ` A ) x. B ) x. ( abs ` z ) ) ) |
97 |
90 96
|
breqtrrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` A ) x. B ) x. ( ( abs ` A ) / 2 ) ) < ( ( abs ` ( A x. z ) ) x. B ) ) |
98 |
48 59 52 67 97
|
lelttrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> T < ( ( abs ` ( A x. z ) ) x. B ) ) |
99 |
46 48 52 54 98
|
lttrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A - z ) ) < ( ( abs ` ( A x. z ) ) x. B ) ) |
100 |
21 23
|
absrpcld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( A x. z ) ) e. RR+ ) |
101 |
46 51 100
|
ltdivmuld |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) < B <-> ( abs ` ( A - z ) ) < ( ( abs ` ( A x. z ) ) x. B ) ) ) |
102 |
99 101
|
mpbird |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( ( abs ` ( A - z ) ) / ( abs ` ( A x. z ) ) ) < B ) |
103 |
42 102
|
eqbrtrd |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ ( z e. ( CC \ { 0 } ) /\ ( abs ` ( z - A ) ) < T ) ) -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) |
104 |
103
|
expr |
|- ( ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) /\ z e. ( CC \ { 0 } ) ) -> ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) |
105 |
104
|
ralrimiva |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> A. z e. ( CC \ { 0 } ) ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) |
106 |
|
breq2 |
|- ( y = T -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < T ) ) |
107 |
106
|
rspceaimv |
|- ( ( T e. RR+ /\ A. z e. ( CC \ { 0 } ) ( ( abs ` ( z - A ) ) < T -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) -> E. y e. RR+ A. z e. ( CC \ { 0 } ) ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) |
108 |
14 105 107
|
syl2anc |
|- ( ( A e. ( CC \ { 0 } ) /\ B e. RR+ ) -> E. y e. RR+ A. z e. ( CC \ { 0 } ) ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) ) |