Step |
Hyp |
Ref |
Expression |
1 |
|
radcnvrat.g |
|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
2 |
|
radcnvrat.a |
|- ( ph -> A : NN0 --> CC ) |
3 |
|
radcnvrat.r |
|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) |
4 |
|
radcnvrat.rat |
|- D = ( k e. NN0 |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |
5 |
|
radcnvrat.z |
|- Z = ( ZZ>= ` M ) |
6 |
|
radcnvrat.m |
|- ( ph -> M e. NN0 ) |
7 |
|
radcnvrat.n0 |
|- ( ( ph /\ k e. Z ) -> ( A ` k ) =/= 0 ) |
8 |
|
radcnvrat.l |
|- ( ph -> D ~~> L ) |
9 |
|
radcnvrat.ln0 |
|- ( ph -> L =/= 0 ) |
10 |
|
xrltso |
|- < Or RR* |
11 |
10
|
a1i |
|- ( ph -> < Or RR* ) |
12 |
6
|
nn0zd |
|- ( ph -> M e. ZZ ) |
13 |
5
|
reseq2i |
|- ( D |` Z ) = ( D |` ( ZZ>= ` M ) ) |
14 |
|
nn0ex |
|- NN0 e. _V |
15 |
14
|
mptex |
|- ( k e. NN0 |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) e. _V |
16 |
4 15
|
eqeltri |
|- D e. _V |
17 |
|
climres |
|- ( ( M e. ZZ /\ D e. _V ) -> ( ( D |` ( ZZ>= ` M ) ) ~~> L <-> D ~~> L ) ) |
18 |
12 16 17
|
sylancl |
|- ( ph -> ( ( D |` ( ZZ>= ` M ) ) ~~> L <-> D ~~> L ) ) |
19 |
8 18
|
mpbird |
|- ( ph -> ( D |` ( ZZ>= ` M ) ) ~~> L ) |
20 |
13 19
|
eqbrtrid |
|- ( ph -> ( D |` Z ) ~~> L ) |
21 |
4
|
reseq1i |
|- ( D |` Z ) = ( ( k e. NN0 |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |` Z ) |
22 |
|
eluznn0 |
|- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 ) |
23 |
6 22
|
sylan |
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 ) |
24 |
23
|
ex |
|- ( ph -> ( k e. ( ZZ>= ` M ) -> k e. NN0 ) ) |
25 |
24
|
ssrdv |
|- ( ph -> ( ZZ>= ` M ) C_ NN0 ) |
26 |
5 25
|
eqsstrid |
|- ( ph -> Z C_ NN0 ) |
27 |
26
|
resmptd |
|- ( ph -> ( ( k e. NN0 |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |` Z ) = ( k e. Z |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) ) |
28 |
21 27
|
syl5eq |
|- ( ph -> ( D |` Z ) = ( k e. Z |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) ) |
29 |
|
fvexd |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) e. _V ) |
30 |
28 29
|
fvmpt2d |
|- ( ( ph /\ k e. Z ) -> ( ( D |` Z ) ` k ) = ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |
31 |
5
|
peano2uzs |
|- ( k e. Z -> ( k + 1 ) e. Z ) |
32 |
26
|
sselda |
|- ( ( ph /\ ( k + 1 ) e. Z ) -> ( k + 1 ) e. NN0 ) |
33 |
2
|
ffvelrnda |
|- ( ( ph /\ ( k + 1 ) e. NN0 ) -> ( A ` ( k + 1 ) ) e. CC ) |
34 |
32 33
|
syldan |
|- ( ( ph /\ ( k + 1 ) e. Z ) -> ( A ` ( k + 1 ) ) e. CC ) |
35 |
31 34
|
sylan2 |
|- ( ( ph /\ k e. Z ) -> ( A ` ( k + 1 ) ) e. CC ) |
36 |
26
|
sselda |
|- ( ( ph /\ k e. Z ) -> k e. NN0 ) |
37 |
2
|
ffvelrnda |
|- ( ( ph /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
38 |
36 37
|
syldan |
|- ( ( ph /\ k e. Z ) -> ( A ` k ) e. CC ) |
39 |
35 38 7
|
divcld |
|- ( ( ph /\ k e. Z ) -> ( ( A ` ( k + 1 ) ) / ( A ` k ) ) e. CC ) |
40 |
39
|
abscld |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) e. RR ) |
41 |
30 40
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( ( D |` Z ) ` k ) e. RR ) |
42 |
5 12 20 41
|
climrecl |
|- ( ph -> L e. RR ) |
43 |
42 9
|
rereccld |
|- ( ph -> ( 1 / L ) e. RR ) |
44 |
43
|
rexrd |
|- ( ph -> ( 1 / L ) e. RR* ) |
45 |
|
simpr |
|- ( ( ph /\ x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
46 |
|
elrabi |
|- ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } -> x e. RR ) |
47 |
43
|
adantr |
|- ( ( ph /\ x e. RR ) -> ( 1 / L ) e. RR ) |
48 |
|
recn |
|- ( x e. RR -> x e. CC ) |
49 |
48
|
abscld |
|- ( x e. RR -> ( abs ` x ) e. RR ) |
50 |
49
|
adantl |
|- ( ( ph /\ x e. RR ) -> ( abs ` x ) e. RR ) |
51 |
47 50
|
ltlend |
|- ( ( ph /\ x e. RR ) -> ( ( 1 / L ) < ( abs ` x ) <-> ( ( 1 / L ) <_ ( abs ` x ) /\ ( abs ` x ) =/= ( 1 / L ) ) ) ) |
52 |
51
|
simplbda |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < ( abs ` x ) ) -> ( abs ` x ) =/= ( 1 / L ) ) |
53 |
51
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) < ( abs ` x ) <-> ( ( 1 / L ) <_ ( abs ` x ) /\ ( abs ` x ) =/= ( 1 / L ) ) ) ) |
54 |
|
simpr |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( abs ` x ) =/= ( 1 / L ) ) |
55 |
54
|
biantrud |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) <_ ( abs ` x ) <-> ( ( 1 / L ) <_ ( abs ` x ) /\ ( abs ` x ) =/= ( 1 / L ) ) ) ) |
56 |
47 50
|
lenltd |
|- ( ( ph /\ x e. RR ) -> ( ( 1 / L ) <_ ( abs ` x ) <-> -. ( abs ` x ) < ( 1 / L ) ) ) |
57 |
56
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) <_ ( abs ` x ) <-> -. ( abs ` x ) < ( 1 / L ) ) ) |
58 |
53 55 57
|
3bitr2d |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) < ( abs ` x ) <-> -. ( abs ` x ) < ( 1 / L ) ) ) |
59 |
|
1cnd |
|- ( ( ph /\ x e. RR ) -> 1 e. CC ) |
60 |
50
|
recnd |
|- ( ( ph /\ x e. RR ) -> ( abs ` x ) e. CC ) |
61 |
42
|
recnd |
|- ( ph -> L e. CC ) |
62 |
61
|
adantr |
|- ( ( ph /\ x e. RR ) -> L e. CC ) |
63 |
9
|
adantr |
|- ( ( ph /\ x e. RR ) -> L =/= 0 ) |
64 |
59 60 62 63
|
divmul3d |
|- ( ( ph /\ x e. RR ) -> ( ( 1 / L ) = ( abs ` x ) <-> 1 = ( ( abs ` x ) x. L ) ) ) |
65 |
|
eqcom |
|- ( ( 1 / L ) = ( abs ` x ) <-> ( abs ` x ) = ( 1 / L ) ) |
66 |
|
eqcom |
|- ( 1 = ( ( abs ` x ) x. L ) <-> ( ( abs ` x ) x. L ) = 1 ) |
67 |
64 65 66
|
3bitr3g |
|- ( ( ph /\ x e. RR ) -> ( ( abs ` x ) = ( 1 / L ) <-> ( ( abs ` x ) x. L ) = 1 ) ) |
68 |
67
|
necon3bid |
|- ( ( ph /\ x e. RR ) -> ( ( abs ` x ) =/= ( 1 / L ) <-> ( ( abs ` x ) x. L ) =/= 1 ) ) |
69 |
68
|
biimpa |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( abs ` x ) x. L ) =/= 1 ) |
70 |
|
1red |
|- ( ( ph /\ x e. RR ) -> 1 e. RR ) |
71 |
|
fvres |
|- ( k e. Z -> ( ( D |` Z ) ` k ) = ( D ` k ) ) |
72 |
71
|
adantl |
|- ( ( ph /\ k e. Z ) -> ( ( D |` Z ) ` k ) = ( D ` k ) ) |
73 |
72 41
|
eqeltrrd |
|- ( ( ph /\ k e. Z ) -> ( D ` k ) e. RR ) |
74 |
39
|
absge0d |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |
75 |
74 30
|
breqtrrd |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( ( D |` Z ) ` k ) ) |
76 |
75 72
|
breqtrd |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( D ` k ) ) |
77 |
5 12 8 73 76
|
climge0 |
|- ( ph -> 0 <_ L ) |
78 |
42 77 9
|
ne0gt0d |
|- ( ph -> 0 < L ) |
79 |
42 78
|
elrpd |
|- ( ph -> L e. RR+ ) |
80 |
79
|
adantr |
|- ( ( ph /\ x e. RR ) -> L e. RR+ ) |
81 |
50 70 80
|
ltmuldivd |
|- ( ( ph /\ x e. RR ) -> ( ( ( abs ` x ) x. L ) < 1 <-> ( abs ` x ) < ( 1 / L ) ) ) |
82 |
81
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> ( abs ` x ) < ( 1 / L ) ) ) |
83 |
|
elun |
|- ( x e. ( ( RR i^i { 0 } ) u. ( RR \ { 0 } ) ) <-> ( x e. ( RR i^i { 0 } ) \/ x e. ( RR \ { 0 } ) ) ) |
84 |
|
inundif |
|- ( ( RR i^i { 0 } ) u. ( RR \ { 0 } ) ) = RR |
85 |
84
|
eleq2i |
|- ( x e. ( ( RR i^i { 0 } ) u. ( RR \ { 0 } ) ) <-> x e. RR ) |
86 |
83 85
|
bitr3i |
|- ( ( x e. ( RR i^i { 0 } ) \/ x e. ( RR \ { 0 } ) ) <-> x e. RR ) |
87 |
|
elin |
|- ( x e. ( RR i^i { 0 } ) <-> ( x e. RR /\ x e. { 0 } ) ) |
88 |
87
|
simprbi |
|- ( x e. ( RR i^i { 0 } ) -> x e. { 0 } ) |
89 |
|
elsni |
|- ( x e. { 0 } -> x = 0 ) |
90 |
88 89
|
syl |
|- ( x e. ( RR i^i { 0 } ) -> x = 0 ) |
91 |
|
fveq2 |
|- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) ) |
92 |
|
abs0 |
|- ( abs ` 0 ) = 0 |
93 |
91 92
|
eqtrdi |
|- ( x = 0 -> ( abs ` x ) = 0 ) |
94 |
93
|
oveq1d |
|- ( x = 0 -> ( ( abs ` x ) x. L ) = ( 0 x. L ) ) |
95 |
61
|
mul02d |
|- ( ph -> ( 0 x. L ) = 0 ) |
96 |
94 95
|
sylan9eqr |
|- ( ( ph /\ x = 0 ) -> ( ( abs ` x ) x. L ) = 0 ) |
97 |
|
0lt1 |
|- 0 < 1 |
98 |
96 97
|
eqbrtrdi |
|- ( ( ph /\ x = 0 ) -> ( ( abs ` x ) x. L ) < 1 ) |
99 |
1 2
|
radcnv0 |
|- ( ph -> 0 e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
100 |
|
eleq1 |
|- ( x = 0 -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } <-> 0 e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
101 |
99 100
|
syl5ibrcom |
|- ( ph -> ( x = 0 -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
102 |
101
|
imp |
|- ( ( ph /\ x = 0 ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
103 |
98 102
|
2thd |
|- ( ( ph /\ x = 0 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
104 |
90 103
|
sylan2 |
|- ( ( ph /\ x e. ( RR i^i { 0 } ) ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
105 |
104
|
adantlr |
|- ( ( ( ph /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ x e. ( RR i^i { 0 } ) ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
106 |
|
ax-resscn |
|- RR C_ CC |
107 |
|
ssdif |
|- ( RR C_ CC -> ( RR \ { 0 } ) C_ ( CC \ { 0 } ) ) |
108 |
106 107
|
ax-mp |
|- ( RR \ { 0 } ) C_ ( CC \ { 0 } ) |
109 |
108
|
sseli |
|- ( x e. ( RR \ { 0 } ) -> x e. ( CC \ { 0 } ) ) |
110 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
111 |
6
|
ad2antrr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> M e. NN0 ) |
112 |
|
fvexd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( G ` x ) e. _V ) |
113 |
|
eldifi |
|- ( x e. ( CC \ { 0 } ) -> x e. CC ) |
114 |
1
|
a1i |
|- ( ph -> G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) ) |
115 |
14
|
mptex |
|- ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) e. _V |
116 |
115
|
a1i |
|- ( ( ph /\ x e. CC ) -> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) e. _V ) |
117 |
114 116
|
fvmpt2d |
|- ( ( ph /\ x e. CC ) -> ( G ` x ) = ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
118 |
117
|
adantr |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( G ` x ) = ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
119 |
|
fveq2 |
|- ( n = k -> ( A ` n ) = ( A ` k ) ) |
120 |
|
oveq2 |
|- ( n = k -> ( x ^ n ) = ( x ^ k ) ) |
121 |
119 120
|
oveq12d |
|- ( n = k -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
122 |
121
|
adantl |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) /\ n = k ) -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
123 |
|
simpr |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> k e. NN0 ) |
124 |
|
ovexd |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( ( A ` k ) x. ( x ^ k ) ) e. _V ) |
125 |
118 122 123 124
|
fvmptd |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( ( G ` x ) ` k ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
126 |
37
|
adantlr |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
127 |
|
simplr |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> x e. CC ) |
128 |
127 123
|
expcld |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( x ^ k ) e. CC ) |
129 |
126 128
|
mulcld |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( ( A ` k ) x. ( x ^ k ) ) e. CC ) |
130 |
125 129
|
eqeltrd |
|- ( ( ( ph /\ x e. CC ) /\ k e. NN0 ) -> ( ( G ` x ) ` k ) e. CC ) |
131 |
113 130
|
sylanl2 |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. NN0 ) -> ( ( G ` x ) ` k ) e. CC ) |
132 |
131
|
adantlr |
|- ( ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ k e. NN0 ) -> ( ( G ` x ) ` k ) e. CC ) |
133 |
36
|
adantlr |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> k e. NN0 ) |
134 |
133 125
|
syldan |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( G ` x ) ` k ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
135 |
113 134
|
sylanl2 |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( G ` x ) ` k ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
136 |
38
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( A ` k ) e. CC ) |
137 |
113
|
adantl |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> x e. CC ) |
138 |
137
|
adantr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> x e. CC ) |
139 |
36
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> k e. NN0 ) |
140 |
138 139
|
expcld |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ k ) e. CC ) |
141 |
7
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( A ` k ) =/= 0 ) |
142 |
|
eldifsni |
|- ( x e. ( CC \ { 0 } ) -> x =/= 0 ) |
143 |
142
|
ad2antlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> x =/= 0 ) |
144 |
139
|
nn0zd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> k e. ZZ ) |
145 |
138 143 144
|
expne0d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ k ) =/= 0 ) |
146 |
136 140 141 145
|
mulne0d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( A ` k ) x. ( x ^ k ) ) =/= 0 ) |
147 |
135 146
|
eqnetrd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( G ` x ) ` k ) =/= 0 ) |
148 |
147
|
adantlr |
|- ( ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ k e. Z ) -> ( ( G ` x ) ` k ) =/= 0 ) |
149 |
|
fvoveq1 |
|- ( n = k -> ( ( G ` x ) ` ( n + 1 ) ) = ( ( G ` x ) ` ( k + 1 ) ) ) |
150 |
|
fveq2 |
|- ( n = k -> ( ( G ` x ) ` n ) = ( ( G ` x ) ` k ) ) |
151 |
149 150
|
oveq12d |
|- ( n = k -> ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) = ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) |
152 |
151
|
fveq2d |
|- ( n = k -> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) = ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) ) |
153 |
152
|
cbvmptv |
|- ( n e. Z |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) = ( k e. Z |-> ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) ) |
154 |
5
|
reseq2i |
|- ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` Z ) = ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` ( ZZ>= ` M ) ) |
155 |
26
|
adantr |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> Z C_ NN0 ) |
156 |
155
|
resmptd |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` Z ) = ( n e. Z |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ) |
157 |
154 156
|
eqtr3id |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` ( ZZ>= ` M ) ) = ( n e. Z |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ) |
158 |
12
|
adantr |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> M e. ZZ ) |
159 |
8
|
adantr |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> D ~~> L ) |
160 |
137
|
abscld |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( abs ` x ) e. RR ) |
161 |
160
|
recnd |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( abs ` x ) e. CC ) |
162 |
14
|
mptex |
|- ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) e. _V |
163 |
162
|
a1i |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) e. _V ) |
164 |
73
|
recnd |
|- ( ( ph /\ k e. Z ) -> ( D ` k ) e. CC ) |
165 |
164
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( D ` k ) e. CC ) |
166 |
|
eqidd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) = ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ) |
167 |
152
|
adantl |
|- ( ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) /\ n = k ) -> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) = ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) ) |
168 |
|
fvexd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) e. _V ) |
169 |
166 167 139 168
|
fvmptd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ` k ) = ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) ) |
170 |
117
|
adantr |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( G ` x ) = ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
171 |
|
simpr |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. Z ) /\ n = ( k + 1 ) ) -> n = ( k + 1 ) ) |
172 |
171
|
fveq2d |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. Z ) /\ n = ( k + 1 ) ) -> ( A ` n ) = ( A ` ( k + 1 ) ) ) |
173 |
171
|
oveq2d |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. Z ) /\ n = ( k + 1 ) ) -> ( x ^ n ) = ( x ^ ( k + 1 ) ) ) |
174 |
172 173
|
oveq12d |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. Z ) /\ n = ( k + 1 ) ) -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) ) |
175 |
|
1nn0 |
|- 1 e. NN0 |
176 |
175
|
a1i |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> 1 e. NN0 ) |
177 |
133 176
|
nn0addcld |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( k + 1 ) e. NN0 ) |
178 |
|
ovexd |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) e. _V ) |
179 |
170 174 177 178
|
fvmptd |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( G ` x ) ` ( k + 1 ) ) = ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) ) |
180 |
121
|
adantl |
|- ( ( ( ( ph /\ x e. CC ) /\ k e. Z ) /\ n = k ) -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
181 |
|
ovexd |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( A ` k ) x. ( x ^ k ) ) e. _V ) |
182 |
170 180 133 181
|
fvmptd |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( G ` x ) ` k ) = ( ( A ` k ) x. ( x ^ k ) ) ) |
183 |
179 182
|
oveq12d |
|- ( ( ( ph /\ x e. CC ) /\ k e. Z ) -> ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) = ( ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) / ( ( A ` k ) x. ( x ^ k ) ) ) ) |
184 |
113 183
|
sylanl2 |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) = ( ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) / ( ( A ` k ) x. ( x ^ k ) ) ) ) |
185 |
35
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( A ` ( k + 1 ) ) e. CC ) |
186 |
113 177
|
sylanl2 |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( k + 1 ) e. NN0 ) |
187 |
138 186
|
expcld |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ ( k + 1 ) ) e. CC ) |
188 |
185 136 187 140 141 145
|
divmuldivd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. ( ( x ^ ( k + 1 ) ) / ( x ^ k ) ) ) = ( ( ( A ` ( k + 1 ) ) x. ( x ^ ( k + 1 ) ) ) / ( ( A ` k ) x. ( x ^ k ) ) ) ) |
189 |
139
|
nn0cnd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> k e. CC ) |
190 |
|
1cnd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> 1 e. CC ) |
191 |
189 190
|
pncan2d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( k + 1 ) - k ) = 1 ) |
192 |
191
|
oveq2d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ ( ( k + 1 ) - k ) ) = ( x ^ 1 ) ) |
193 |
186
|
nn0zd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( k + 1 ) e. ZZ ) |
194 |
138 143 144 193
|
expsubd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ ( ( k + 1 ) - k ) ) = ( ( x ^ ( k + 1 ) ) / ( x ^ k ) ) ) |
195 |
138
|
exp1d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( x ^ 1 ) = x ) |
196 |
192 194 195
|
3eqtr3d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( x ^ ( k + 1 ) ) / ( x ^ k ) ) = x ) |
197 |
196
|
oveq2d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. ( ( x ^ ( k + 1 ) ) / ( x ^ k ) ) ) = ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. x ) ) |
198 |
184 188 197
|
3eqtr2d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) = ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. x ) ) |
199 |
198
|
fveq2d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( abs ` ( ( ( G ` x ) ` ( k + 1 ) ) / ( ( G ` x ) ` k ) ) ) = ( abs ` ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. x ) ) ) |
200 |
39
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( A ` ( k + 1 ) ) / ( A ` k ) ) e. CC ) |
201 |
200 138
|
absmuld |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( abs ` ( ( ( A ` ( k + 1 ) ) / ( A ` k ) ) x. x ) ) = ( ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) x. ( abs ` x ) ) ) |
202 |
169 199 201
|
3eqtrd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ` k ) = ( ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) x. ( abs ` x ) ) ) |
203 |
72 30
|
eqtr3d |
|- ( ( ph /\ k e. Z ) -> ( D ` k ) = ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |
204 |
203
|
adantlr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( D ` k ) = ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) ) |
205 |
204
|
eqcomd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) = ( D ` k ) ) |
206 |
205
|
oveq1d |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) x. ( abs ` x ) ) = ( ( D ` k ) x. ( abs ` x ) ) ) |
207 |
161
|
adantr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( abs ` x ) e. CC ) |
208 |
165 207
|
mulcomd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( D ` k ) x. ( abs ` x ) ) = ( ( abs ` x ) x. ( D ` k ) ) ) |
209 |
202 206 208
|
3eqtrd |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ k e. Z ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ` k ) = ( ( abs ` x ) x. ( D ` k ) ) ) |
210 |
5 158 159 161 163 165 209
|
climmulc2 |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ~~> ( ( abs ` x ) x. L ) ) |
211 |
|
climres |
|- ( ( M e. ZZ /\ ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) e. _V ) -> ( ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` ( ZZ>= ` M ) ) ~~> ( ( abs ` x ) x. L ) <-> ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ~~> ( ( abs ` x ) x. L ) ) ) |
212 |
158 162 211
|
sylancl |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` ( ZZ>= ` M ) ) ~~> ( ( abs ` x ) x. L ) <-> ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ~~> ( ( abs ` x ) x. L ) ) ) |
213 |
210 212
|
mpbird |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( ( n e. NN0 |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) |` ( ZZ>= ` M ) ) ~~> ( ( abs ` x ) x. L ) ) |
214 |
157 213
|
eqbrtrrd |
|- ( ( ph /\ x e. ( CC \ { 0 } ) ) -> ( n e. Z |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ~~> ( ( abs ` x ) x. L ) ) |
215 |
214
|
adantr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( n e. Z |-> ( abs ` ( ( ( G ` x ) ` ( n + 1 ) ) / ( ( G ` x ) ` n ) ) ) ) ~~> ( ( abs ` x ) x. L ) ) |
216 |
|
simpr |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( abs ` x ) x. L ) =/= 1 ) |
217 |
110 5 111 112 132 148 153 215 216
|
cvgdvgrat |
|- ( ( ( ph /\ x e. ( CC \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
218 |
109 217
|
sylanl2 |
|- ( ( ( ph /\ x e. ( RR \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
219 |
|
eldifi |
|- ( x e. ( RR \ { 0 } ) -> x e. RR ) |
220 |
|
fveq2 |
|- ( r = x -> ( G ` r ) = ( G ` x ) ) |
221 |
220
|
seqeq3d |
|- ( r = x -> seq 0 ( + , ( G ` r ) ) = seq 0 ( + , ( G ` x ) ) ) |
222 |
221
|
eleq1d |
|- ( r = x -> ( seq 0 ( + , ( G ` r ) ) e. dom ~~> <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
223 |
222
|
elrab3 |
|- ( x e. RR -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
224 |
219 223
|
syl |
|- ( x e. ( RR \ { 0 } ) -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
225 |
224
|
ad2antlr |
|- ( ( ( ph /\ x e. ( RR \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } <-> seq 0 ( + , ( G ` x ) ) e. dom ~~> ) ) |
226 |
218 225
|
bitr4d |
|- ( ( ( ph /\ x e. ( RR \ { 0 } ) ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
227 |
226
|
an32s |
|- ( ( ( ph /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ x e. ( RR \ { 0 } ) ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
228 |
105 227
|
jaodan |
|- ( ( ( ph /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ ( x e. ( RR i^i { 0 } ) \/ x e. ( RR \ { 0 } ) ) ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
229 |
86 228
|
sylan2br |
|- ( ( ( ph /\ ( ( abs ` x ) x. L ) =/= 1 ) /\ x e. RR ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
230 |
229
|
an32s |
|- ( ( ( ph /\ x e. RR ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( ( abs ` x ) x. L ) < 1 <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
231 |
82 230
|
bitr3d |
|- ( ( ( ph /\ x e. RR ) /\ ( ( abs ` x ) x. L ) =/= 1 ) -> ( ( abs ` x ) < ( 1 / L ) <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
232 |
69 231
|
syldan |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( abs ` x ) < ( 1 / L ) <-> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
233 |
232
|
notbid |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( -. ( abs ` x ) < ( 1 / L ) <-> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
234 |
58 233
|
bitrd |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) < ( abs ` x ) <-> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
235 |
234
|
biimpd |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( 1 / L ) < ( abs ` x ) -> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
236 |
235
|
impancom |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < ( abs ` x ) ) -> ( ( abs ` x ) =/= ( 1 / L ) -> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
237 |
52 236
|
mpd |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < ( abs ` x ) ) -> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
238 |
237
|
ex |
|- ( ( ph /\ x e. RR ) -> ( ( 1 / L ) < ( abs ` x ) -> -. x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
239 |
238
|
con2d |
|- ( ( ph /\ x e. RR ) -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } -> -. ( 1 / L ) < ( abs ` x ) ) ) |
240 |
47
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> ( 1 / L ) e. RR ) |
241 |
|
simplr |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> x e. RR ) |
242 |
50
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> ( abs ` x ) e. RR ) |
243 |
|
simpr |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> ( 1 / L ) < x ) |
244 |
241
|
leabsd |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> x <_ ( abs ` x ) ) |
245 |
240 241 242 243 244
|
ltletrd |
|- ( ( ( ph /\ x e. RR ) /\ ( 1 / L ) < x ) -> ( 1 / L ) < ( abs ` x ) ) |
246 |
245
|
ex |
|- ( ( ph /\ x e. RR ) -> ( ( 1 / L ) < x -> ( 1 / L ) < ( abs ` x ) ) ) |
247 |
239 246
|
nsyld |
|- ( ( ph /\ x e. RR ) -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } -> -. ( 1 / L ) < x ) ) |
248 |
46 247
|
sylan2 |
|- ( ( ph /\ x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) -> ( x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } -> -. ( 1 / L ) < x ) ) |
249 |
45 248
|
mpd |
|- ( ( ph /\ x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) -> -. ( 1 / L ) < x ) |
250 |
43
|
renegcld |
|- ( ph -> -u ( 1 / L ) e. RR ) |
251 |
250
|
rexrd |
|- ( ph -> -u ( 1 / L ) e. RR* ) |
252 |
|
iooss1 |
|- ( ( -u ( 1 / L ) e. RR* /\ -u ( 1 / L ) <_ x ) -> ( x (,) ( 1 / L ) ) C_ ( -u ( 1 / L ) (,) ( 1 / L ) ) ) |
253 |
251 252
|
sylan |
|- ( ( ph /\ -u ( 1 / L ) <_ x ) -> ( x (,) ( 1 / L ) ) C_ ( -u ( 1 / L ) (,) ( 1 / L ) ) ) |
254 |
253
|
adantlr |
|- ( ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) /\ -u ( 1 / L ) <_ x ) -> ( x (,) ( 1 / L ) ) C_ ( -u ( 1 / L ) (,) ( 1 / L ) ) ) |
255 |
|
eliooord |
|- ( k e. ( x (,) ( 1 / L ) ) -> ( x < k /\ k < ( 1 / L ) ) ) |
256 |
255
|
simpld |
|- ( k e. ( x (,) ( 1 / L ) ) -> x < k ) |
257 |
256
|
rgen |
|- A. k e. ( x (,) ( 1 / L ) ) x < k |
258 |
|
ioon0 |
|- ( ( x e. RR* /\ ( 1 / L ) e. RR* ) -> ( ( x (,) ( 1 / L ) ) =/= (/) <-> x < ( 1 / L ) ) ) |
259 |
44 258
|
sylan2 |
|- ( ( x e. RR* /\ ph ) -> ( ( x (,) ( 1 / L ) ) =/= (/) <-> x < ( 1 / L ) ) ) |
260 |
259
|
ancoms |
|- ( ( ph /\ x e. RR* ) -> ( ( x (,) ( 1 / L ) ) =/= (/) <-> x < ( 1 / L ) ) ) |
261 |
260
|
biimpar |
|- ( ( ( ph /\ x e. RR* ) /\ x < ( 1 / L ) ) -> ( x (,) ( 1 / L ) ) =/= (/) ) |
262 |
|
r19.2zb |
|- ( ( x (,) ( 1 / L ) ) =/= (/) <-> ( A. k e. ( x (,) ( 1 / L ) ) x < k -> E. k e. ( x (,) ( 1 / L ) ) x < k ) ) |
263 |
261 262
|
sylib |
|- ( ( ( ph /\ x e. RR* ) /\ x < ( 1 / L ) ) -> ( A. k e. ( x (,) ( 1 / L ) ) x < k -> E. k e. ( x (,) ( 1 / L ) ) x < k ) ) |
264 |
257 263
|
mpi |
|- ( ( ( ph /\ x e. RR* ) /\ x < ( 1 / L ) ) -> E. k e. ( x (,) ( 1 / L ) ) x < k ) |
265 |
264
|
anasss |
|- ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) -> E. k e. ( x (,) ( 1 / L ) ) x < k ) |
266 |
265
|
adantr |
|- ( ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) /\ -u ( 1 / L ) <_ x ) -> E. k e. ( x (,) ( 1 / L ) ) x < k ) |
267 |
|
ssrexv |
|- ( ( x (,) ( 1 / L ) ) C_ ( -u ( 1 / L ) (,) ( 1 / L ) ) -> ( E. k e. ( x (,) ( 1 / L ) ) x < k -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) ) |
268 |
254 266 267
|
sylc |
|- ( ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) /\ -u ( 1 / L ) <_ x ) -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) |
269 |
|
simplr |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> x e. RR* ) |
270 |
|
xrltnle |
|- ( ( x e. RR* /\ -u ( 1 / L ) e. RR* ) -> ( x < -u ( 1 / L ) <-> -. -u ( 1 / L ) <_ x ) ) |
271 |
|
xrltle |
|- ( ( x e. RR* /\ -u ( 1 / L ) e. RR* ) -> ( x < -u ( 1 / L ) -> x <_ -u ( 1 / L ) ) ) |
272 |
270 271
|
sylbird |
|- ( ( x e. RR* /\ -u ( 1 / L ) e. RR* ) -> ( -. -u ( 1 / L ) <_ x -> x <_ -u ( 1 / L ) ) ) |
273 |
251 272
|
sylan2 |
|- ( ( x e. RR* /\ ph ) -> ( -. -u ( 1 / L ) <_ x -> x <_ -u ( 1 / L ) ) ) |
274 |
273
|
ancoms |
|- ( ( ph /\ x e. RR* ) -> ( -. -u ( 1 / L ) <_ x -> x <_ -u ( 1 / L ) ) ) |
275 |
274
|
imp |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> x <_ -u ( 1 / L ) ) |
276 |
|
iooss1 |
|- ( ( x e. RR* /\ x <_ -u ( 1 / L ) ) -> ( -u ( 1 / L ) (,) ( 1 / L ) ) C_ ( x (,) ( 1 / L ) ) ) |
277 |
269 275 276
|
syl2anc |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> ( -u ( 1 / L ) (,) ( 1 / L ) ) C_ ( x (,) ( 1 / L ) ) ) |
278 |
277
|
sselda |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) /\ k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) ) -> k e. ( x (,) ( 1 / L ) ) ) |
279 |
278 256
|
syl |
|- ( ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) /\ k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) ) -> x < k ) |
280 |
279
|
ralrimiva |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> A. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) |
281 |
42 78
|
recgt0d |
|- ( ph -> 0 < ( 1 / L ) ) |
282 |
43 43 281 281
|
addgt0d |
|- ( ph -> 0 < ( ( 1 / L ) + ( 1 / L ) ) ) |
283 |
43
|
recnd |
|- ( ph -> ( 1 / L ) e. CC ) |
284 |
283 283
|
subnegd |
|- ( ph -> ( ( 1 / L ) - -u ( 1 / L ) ) = ( ( 1 / L ) + ( 1 / L ) ) ) |
285 |
282 284
|
breqtrrd |
|- ( ph -> 0 < ( ( 1 / L ) - -u ( 1 / L ) ) ) |
286 |
250 43
|
posdifd |
|- ( ph -> ( -u ( 1 / L ) < ( 1 / L ) <-> 0 < ( ( 1 / L ) - -u ( 1 / L ) ) ) ) |
287 |
285 286
|
mpbird |
|- ( ph -> -u ( 1 / L ) < ( 1 / L ) ) |
288 |
|
ioon0 |
|- ( ( -u ( 1 / L ) e. RR* /\ ( 1 / L ) e. RR* ) -> ( ( -u ( 1 / L ) (,) ( 1 / L ) ) =/= (/) <-> -u ( 1 / L ) < ( 1 / L ) ) ) |
289 |
251 44 288
|
syl2anc |
|- ( ph -> ( ( -u ( 1 / L ) (,) ( 1 / L ) ) =/= (/) <-> -u ( 1 / L ) < ( 1 / L ) ) ) |
290 |
287 289
|
mpbird |
|- ( ph -> ( -u ( 1 / L ) (,) ( 1 / L ) ) =/= (/) ) |
291 |
|
r19.2zb |
|- ( ( -u ( 1 / L ) (,) ( 1 / L ) ) =/= (/) <-> ( A. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) ) |
292 |
290 291
|
sylib |
|- ( ph -> ( A. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) ) |
293 |
292
|
ad2antrr |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> ( A. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) ) |
294 |
280 293
|
mpd |
|- ( ( ( ph /\ x e. RR* ) /\ -. -u ( 1 / L ) <_ x ) -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) |
295 |
294
|
adantlrr |
|- ( ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) /\ -. -u ( 1 / L ) <_ x ) -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) |
296 |
268 295
|
pm2.61dan |
|- ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) -> E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k ) |
297 |
|
elioo2 |
|- ( ( -u ( 1 / L ) e. RR* /\ ( 1 / L ) e. RR* ) -> ( x e. ( -u ( 1 / L ) (,) ( 1 / L ) ) <-> ( x e. RR /\ -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) ) |
298 |
251 44 297
|
syl2anc |
|- ( ph -> ( x e. ( -u ( 1 / L ) (,) ( 1 / L ) ) <-> ( x e. RR /\ -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) ) |
299 |
298
|
biimpa |
|- ( ( ph /\ x e. ( -u ( 1 / L ) (,) ( 1 / L ) ) ) -> ( x e. RR /\ -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) |
300 |
|
simpr |
|- ( ( ph /\ x e. RR ) -> x e. RR ) |
301 |
300 47
|
absltd |
|- ( ( ph /\ x e. RR ) -> ( ( abs ` x ) < ( 1 / L ) <-> ( -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) ) |
302 |
50
|
adantr |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) < ( 1 / L ) ) -> ( abs ` x ) e. RR ) |
303 |
|
simpr |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) < ( 1 / L ) ) -> ( abs ` x ) < ( 1 / L ) ) |
304 |
302 303
|
ltned |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) < ( 1 / L ) ) -> ( abs ` x ) =/= ( 1 / L ) ) |
305 |
232
|
biimpd |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) =/= ( 1 / L ) ) -> ( ( abs ` x ) < ( 1 / L ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
306 |
305
|
impancom |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) < ( 1 / L ) ) -> ( ( abs ` x ) =/= ( 1 / L ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
307 |
304 306
|
mpd |
|- ( ( ( ph /\ x e. RR ) /\ ( abs ` x ) < ( 1 / L ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
308 |
307
|
ex |
|- ( ( ph /\ x e. RR ) -> ( ( abs ` x ) < ( 1 / L ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
309 |
301 308
|
sylbird |
|- ( ( ph /\ x e. RR ) -> ( ( -u ( 1 / L ) < x /\ x < ( 1 / L ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
310 |
309
|
impr |
|- ( ( ph /\ ( x e. RR /\ ( -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
311 |
310
|
expcom |
|- ( ( x e. RR /\ ( -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) -> ( ph -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
312 |
311
|
3impb |
|- ( ( x e. RR /\ -u ( 1 / L ) < x /\ x < ( 1 / L ) ) -> ( ph -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
313 |
312
|
impcom |
|- ( ( ph /\ ( x e. RR /\ -u ( 1 / L ) < x /\ x < ( 1 / L ) ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
314 |
299 313
|
syldan |
|- ( ( ph /\ x e. ( -u ( 1 / L ) (,) ( 1 / L ) ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
315 |
314
|
ex |
|- ( ph -> ( x e. ( -u ( 1 / L ) (,) ( 1 / L ) ) -> x e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) ) |
316 |
315
|
ssrdv |
|- ( ph -> ( -u ( 1 / L ) (,) ( 1 / L ) ) C_ { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) |
317 |
|
ssrexv |
|- ( ( -u ( 1 / L ) (,) ( 1 / L ) ) C_ { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } -> ( E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } x < k ) ) |
318 |
316 317
|
syl |
|- ( ph -> ( E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } x < k ) ) |
319 |
318
|
adantr |
|- ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) -> ( E. k e. ( -u ( 1 / L ) (,) ( 1 / L ) ) x < k -> E. k e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } x < k ) ) |
320 |
296 319
|
mpd |
|- ( ( ph /\ ( x e. RR* /\ x < ( 1 / L ) ) ) -> E. k e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } x < k ) |
321 |
11 44 249 320
|
eqsupd |
|- ( ph -> sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) = ( 1 / L ) ) |
322 |
3 321
|
syl5eq |
|- ( ph -> R = ( 1 / L ) ) |