Step |
Hyp |
Ref |
Expression |
1 |
|
ftalem.1 |
|- A = ( coeff ` F ) |
2 |
|
ftalem.2 |
|- N = ( deg ` F ) |
3 |
|
ftalem.3 |
|- ( ph -> F e. ( Poly ` S ) ) |
4 |
|
ftalem.4 |
|- ( ph -> N e. NN ) |
5 |
|
ftalem2.5 |
|- U = if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) |
6 |
|
ftalem2.6 |
|- T = ( ( abs ` ( F ` 0 ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) |
7 |
1
|
coef3 |
|- ( F e. ( Poly ` S ) -> A : NN0 --> CC ) |
8 |
3 7
|
syl |
|- ( ph -> A : NN0 --> CC ) |
9 |
4
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
10 |
8 9
|
ffvelrnd |
|- ( ph -> ( A ` N ) e. CC ) |
11 |
4
|
nnne0d |
|- ( ph -> N =/= 0 ) |
12 |
2 1
|
dgreq0 |
|- ( F e. ( Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) ) |
13 |
|
fveq2 |
|- ( F = 0p -> ( deg ` F ) = ( deg ` 0p ) ) |
14 |
|
dgr0 |
|- ( deg ` 0p ) = 0 |
15 |
13 14
|
eqtrdi |
|- ( F = 0p -> ( deg ` F ) = 0 ) |
16 |
2 15
|
syl5eq |
|- ( F = 0p -> N = 0 ) |
17 |
12 16
|
syl6bir |
|- ( F e. ( Poly ` S ) -> ( ( A ` N ) = 0 -> N = 0 ) ) |
18 |
3 17
|
syl |
|- ( ph -> ( ( A ` N ) = 0 -> N = 0 ) ) |
19 |
18
|
necon3d |
|- ( ph -> ( N =/= 0 -> ( A ` N ) =/= 0 ) ) |
20 |
11 19
|
mpd |
|- ( ph -> ( A ` N ) =/= 0 ) |
21 |
10 20
|
absrpcld |
|- ( ph -> ( abs ` ( A ` N ) ) e. RR+ ) |
22 |
21
|
rphalfcld |
|- ( ph -> ( ( abs ` ( A ` N ) ) / 2 ) e. RR+ ) |
23 |
|
2fveq3 |
|- ( n = k -> ( abs ` ( A ` n ) ) = ( abs ` ( A ` k ) ) ) |
24 |
23
|
cbvsumv |
|- sum_ n e. ( 0 ... ( N - 1 ) ) ( abs ` ( A ` n ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( abs ` ( A ` k ) ) |
25 |
24
|
oveq1i |
|- ( sum_ n e. ( 0 ... ( N - 1 ) ) ( abs ` ( A ` n ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) = ( sum_ k e. ( 0 ... ( N - 1 ) ) ( abs ` ( A ` k ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) |
26 |
1 2 3 4 22 25
|
ftalem1 |
|- ( ph -> E. s e. RR A. x e. CC ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) ) |
27 |
|
plyf |
|- ( F e. ( Poly ` S ) -> F : CC --> CC ) |
28 |
3 27
|
syl |
|- ( ph -> F : CC --> CC ) |
29 |
|
0cn |
|- 0 e. CC |
30 |
|
ffvelrn |
|- ( ( F : CC --> CC /\ 0 e. CC ) -> ( F ` 0 ) e. CC ) |
31 |
28 29 30
|
sylancl |
|- ( ph -> ( F ` 0 ) e. CC ) |
32 |
31
|
abscld |
|- ( ph -> ( abs ` ( F ` 0 ) ) e. RR ) |
33 |
32 22
|
rerpdivcld |
|- ( ph -> ( ( abs ` ( F ` 0 ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) e. RR ) |
34 |
6 33
|
eqeltrid |
|- ( ph -> T e. RR ) |
35 |
34
|
adantr |
|- ( ( ph /\ s e. RR ) -> T e. RR ) |
36 |
|
simpr |
|- ( ( ph /\ s e. RR ) -> s e. RR ) |
37 |
|
1re |
|- 1 e. RR |
38 |
|
ifcl |
|- ( ( s e. RR /\ 1 e. RR ) -> if ( 1 <_ s , s , 1 ) e. RR ) |
39 |
36 37 38
|
sylancl |
|- ( ( ph /\ s e. RR ) -> if ( 1 <_ s , s , 1 ) e. RR ) |
40 |
35 39
|
ifcld |
|- ( ( ph /\ s e. RR ) -> if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) e. RR ) |
41 |
5 40
|
eqeltrid |
|- ( ( ph /\ s e. RR ) -> U e. RR ) |
42 |
|
0red |
|- ( ( ph /\ s e. RR ) -> 0 e. RR ) |
43 |
|
1red |
|- ( ( ph /\ s e. RR ) -> 1 e. RR ) |
44 |
|
0lt1 |
|- 0 < 1 |
45 |
44
|
a1i |
|- ( ( ph /\ s e. RR ) -> 0 < 1 ) |
46 |
|
max1 |
|- ( ( 1 e. RR /\ s e. RR ) -> 1 <_ if ( 1 <_ s , s , 1 ) ) |
47 |
37 36 46
|
sylancr |
|- ( ( ph /\ s e. RR ) -> 1 <_ if ( 1 <_ s , s , 1 ) ) |
48 |
|
max1 |
|- ( ( if ( 1 <_ s , s , 1 ) e. RR /\ T e. RR ) -> if ( 1 <_ s , s , 1 ) <_ if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) ) |
49 |
39 35 48
|
syl2anc |
|- ( ( ph /\ s e. RR ) -> if ( 1 <_ s , s , 1 ) <_ if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) ) |
50 |
49 5
|
breqtrrdi |
|- ( ( ph /\ s e. RR ) -> if ( 1 <_ s , s , 1 ) <_ U ) |
51 |
43 39 41 47 50
|
letrd |
|- ( ( ph /\ s e. RR ) -> 1 <_ U ) |
52 |
42 43 41 45 51
|
ltletrd |
|- ( ( ph /\ s e. RR ) -> 0 < U ) |
53 |
41 52
|
elrpd |
|- ( ( ph /\ s e. RR ) -> U e. RR+ ) |
54 |
|
max2 |
|- ( ( 1 e. RR /\ s e. RR ) -> s <_ if ( 1 <_ s , s , 1 ) ) |
55 |
37 36 54
|
sylancr |
|- ( ( ph /\ s e. RR ) -> s <_ if ( 1 <_ s , s , 1 ) ) |
56 |
36 39 41 55 50
|
letrd |
|- ( ( ph /\ s e. RR ) -> s <_ U ) |
57 |
56
|
adantr |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> s <_ U ) |
58 |
|
abscl |
|- ( x e. CC -> ( abs ` x ) e. RR ) |
59 |
|
lelttr |
|- ( ( s e. RR /\ U e. RR /\ ( abs ` x ) e. RR ) -> ( ( s <_ U /\ U < ( abs ` x ) ) -> s < ( abs ` x ) ) ) |
60 |
36 41 58 59
|
syl2an3an |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( ( s <_ U /\ U < ( abs ` x ) ) -> s < ( abs ` x ) ) ) |
61 |
57 60
|
mpand |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( U < ( abs ` x ) -> s < ( abs ` x ) ) ) |
62 |
61
|
imim1d |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> ( U < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) ) ) |
63 |
28
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> F : CC --> CC ) |
64 |
|
simprl |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> x e. CC ) |
65 |
63 64
|
ffvelrnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( F ` x ) e. CC ) |
66 |
10
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( A ` N ) e. CC ) |
67 |
9
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> N e. NN0 ) |
68 |
64 67
|
expcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( x ^ N ) e. CC ) |
69 |
66 68
|
mulcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( A ` N ) x. ( x ^ N ) ) e. CC ) |
70 |
65 69
|
subcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) e. CC ) |
71 |
70
|
abscld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) e. RR ) |
72 |
69
|
abscld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) e. RR ) |
73 |
72
|
rehalfcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) e. RR ) |
74 |
71 73 72
|
ltsub2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) <-> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) ) ) |
75 |
66 68
|
absmuld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) = ( ( abs ` ( A ` N ) ) x. ( abs ` ( x ^ N ) ) ) ) |
76 |
64 67
|
absexpd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( x ^ N ) ) = ( ( abs ` x ) ^ N ) ) |
77 |
76
|
oveq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( A ` N ) ) x. ( abs ` ( x ^ N ) ) ) = ( ( abs ` ( A ` N ) ) x. ( ( abs ` x ) ^ N ) ) ) |
78 |
75 77
|
eqtrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) = ( ( abs ` ( A ` N ) ) x. ( ( abs ` x ) ^ N ) ) ) |
79 |
78
|
oveq1d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) = ( ( ( abs ` ( A ` N ) ) x. ( ( abs ` x ) ^ N ) ) / 2 ) ) |
80 |
66
|
abscld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( A ` N ) ) e. RR ) |
81 |
80
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( A ` N ) ) e. CC ) |
82 |
58
|
ad2antrl |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` x ) e. RR ) |
83 |
82 67
|
reexpcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` x ) ^ N ) e. RR ) |
84 |
83
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` x ) ^ N ) e. CC ) |
85 |
|
2cnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 2 e. CC ) |
86 |
|
2ne0 |
|- 2 =/= 0 |
87 |
86
|
a1i |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 2 =/= 0 ) |
88 |
81 84 85 87
|
div23d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( A ` N ) ) x. ( ( abs ` x ) ^ N ) ) / 2 ) = ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) |
89 |
79 88
|
eqtrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) = ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) |
90 |
89
|
breq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) <-> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) ) |
91 |
72
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) e. CC ) |
92 |
91
|
2halvesd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) + ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) = ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) ) |
93 |
92
|
oveq1d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) + ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) = ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) ) |
94 |
73
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) e. CC ) |
95 |
94 94
|
pncand |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) + ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) = ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) |
96 |
93 95
|
eqtr3d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) = ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) |
97 |
96
|
breq1d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) <-> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) ) ) |
98 |
74 90 97
|
3bitr3d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) <-> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) ) ) |
99 |
69 65
|
subcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) e. CC ) |
100 |
69 99
|
abs2difd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) ) <_ ( abs ` ( ( ( A ` N ) x. ( x ^ N ) ) - ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) ) ) |
101 |
69 65
|
abssubd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) = ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) |
102 |
101
|
oveq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) ) = ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) ) |
103 |
69 65
|
nncand |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( A ` N ) x. ( x ^ N ) ) - ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) = ( F ` x ) ) |
104 |
103
|
fveq2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( ( ( A ` N ) x. ( x ^ N ) ) - ( ( ( A ` N ) x. ( x ^ N ) ) - ( F ` x ) ) ) ) = ( abs ` ( F ` x ) ) ) |
105 |
100 102 104
|
3brtr3d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) <_ ( abs ` ( F ` x ) ) ) |
106 |
72 71
|
resubcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) e. RR ) |
107 |
65
|
abscld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( F ` x ) ) e. RR ) |
108 |
|
ltletr |
|- ( ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) e. RR /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) e. RR /\ ( abs ` ( F ` x ) ) e. RR ) -> ( ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) <_ ( abs ` ( F ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) ) |
109 |
73 106 107 108
|
syl3anc |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) <_ ( abs ` ( F ` x ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) ) |
110 |
105 109
|
mpan2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) - ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) ) |
111 |
98 110
|
sylbid |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) -> ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) ) |
112 |
32
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( F ` 0 ) ) e. RR ) |
113 |
22
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( A ` N ) ) / 2 ) e. RR+ ) |
114 |
113
|
rpred |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( A ` N ) ) / 2 ) e. RR ) |
115 |
114 82
|
remulcld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( abs ` x ) ) e. RR ) |
116 |
89 73
|
eqeltrrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) e. RR ) |
117 |
35
|
adantr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> T e. RR ) |
118 |
41
|
adantr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> U e. RR ) |
119 |
|
max2 |
|- ( ( if ( 1 <_ s , s , 1 ) e. RR /\ T e. RR ) -> T <_ if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) ) |
120 |
39 35 119
|
syl2anc |
|- ( ( ph /\ s e. RR ) -> T <_ if ( if ( 1 <_ s , s , 1 ) <_ T , T , if ( 1 <_ s , s , 1 ) ) ) |
121 |
120 5
|
breqtrrdi |
|- ( ( ph /\ s e. RR ) -> T <_ U ) |
122 |
121
|
adantr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> T <_ U ) |
123 |
|
simprr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> U < ( abs ` x ) ) |
124 |
117 118 82 122 123
|
lelttrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> T < ( abs ` x ) ) |
125 |
6 124
|
eqbrtrrid |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( F ` 0 ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) < ( abs ` x ) ) |
126 |
112 82 113
|
ltdivmuld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( F ` 0 ) ) / ( ( abs ` ( A ` N ) ) / 2 ) ) < ( abs ` x ) <-> ( abs ` ( F ` 0 ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( abs ` x ) ) ) ) |
127 |
125 126
|
mpbid |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( F ` 0 ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( abs ` x ) ) ) |
128 |
82
|
recnd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` x ) e. CC ) |
129 |
128
|
exp1d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` x ) ^ 1 ) = ( abs ` x ) ) |
130 |
|
1red |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 1 e. RR ) |
131 |
51
|
adantr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 1 <_ U ) |
132 |
130 118 82 131 123
|
lelttrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 1 < ( abs ` x ) ) |
133 |
130 82 132
|
ltled |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> 1 <_ ( abs ` x ) ) |
134 |
4
|
ad2antrr |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> N e. NN ) |
135 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
136 |
134 135
|
eleqtrdi |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> N e. ( ZZ>= ` 1 ) ) |
137 |
82 133 136
|
leexp2ad |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` x ) ^ 1 ) <_ ( ( abs ` x ) ^ N ) ) |
138 |
129 137
|
eqbrtrrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` x ) <_ ( ( abs ` x ) ^ N ) ) |
139 |
82 83 113
|
lemul2d |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` x ) <_ ( ( abs ` x ) ^ N ) <-> ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( abs ` x ) ) <_ ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) ) |
140 |
138 139
|
mpbid |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( abs ` x ) ) <_ ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) |
141 |
112 115 116 127 140
|
ltletrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( F ` 0 ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) |
142 |
141 89
|
breqtrrd |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( abs ` ( F ` 0 ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) ) |
143 |
|
lttr |
|- ( ( ( abs ` ( F ` 0 ) ) e. RR /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) e. RR /\ ( abs ` ( F ` x ) ) e. RR ) -> ( ( ( abs ` ( F ` 0 ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |
144 |
112 73 107 143
|
syl3anc |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( F ` 0 ) ) < ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) /\ ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |
145 |
142 144
|
mpand |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( ( abs ` ( ( A ` N ) x. ( x ^ N ) ) ) / 2 ) < ( abs ` ( F ` x ) ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |
146 |
111 145
|
syld |
|- ( ( ( ph /\ s e. RR ) /\ ( x e. CC /\ U < ( abs ` x ) ) ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |
147 |
146
|
expr |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( U < ( abs ` x ) -> ( ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
148 |
147
|
a2d |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( ( U < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> ( U < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
149 |
62 148
|
syld |
|- ( ( ( ph /\ s e. RR ) /\ x e. CC ) -> ( ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> ( U < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
150 |
149
|
ralimdva |
|- ( ( ph /\ s e. RR ) -> ( A. x e. CC ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> A. x e. CC ( U < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
151 |
|
breq1 |
|- ( r = U -> ( r < ( abs ` x ) <-> U < ( abs ` x ) ) ) |
152 |
151
|
rspceaimv |
|- ( ( U e. RR+ /\ A. x e. CC ( U < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) -> E. r e. RR+ A. x e. CC ( r < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |
153 |
53 150 152
|
syl6an |
|- ( ( ph /\ s e. RR ) -> ( A. x e. CC ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> E. r e. RR+ A. x e. CC ( r < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
154 |
153
|
rexlimdva |
|- ( ph -> ( E. s e. RR A. x e. CC ( s < ( abs ` x ) -> ( abs ` ( ( F ` x ) - ( ( A ` N ) x. ( x ^ N ) ) ) ) < ( ( ( abs ` ( A ` N ) ) / 2 ) x. ( ( abs ` x ) ^ N ) ) ) -> E. r e. RR+ A. x e. CC ( r < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) ) |
155 |
26 154
|
mpd |
|- ( ph -> E. r e. RR+ A. x e. CC ( r < ( abs ` x ) -> ( abs ` ( F ` 0 ) ) < ( abs ` ( F ` x ) ) ) ) |