Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem61.a |
|- ( ph -> A e. RR ) |
2 |
|
fourierdlem61.b |
|- ( ph -> B e. RR ) |
3 |
|
fourierdlem61.altb |
|- ( ph -> A < B ) |
4 |
|
fourierdlem61.f |
|- ( ph -> F : ( A (,) B ) --> RR ) |
5 |
|
fourierdlem61.y |
|- ( ph -> Y e. ( F limCC A ) ) |
6 |
|
fourierdlem61.g |
|- G = ( RR _D F ) |
7 |
|
fourierdlem61.domg |
|- ( ph -> dom G = ( A (,) B ) ) |
8 |
|
fourierdlem61.e |
|- ( ph -> E e. ( G limCC A ) ) |
9 |
|
fourierdlem61.h |
|- H = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( ( F ` ( A + s ) ) - Y ) / s ) ) |
10 |
|
fourierdlem61.n |
|- N = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) |
11 |
|
fourierdlem61.d |
|- D = ( s e. ( 0 (,) ( B - A ) ) |-> s ) |
12 |
|
0red |
|- ( ph -> 0 e. RR ) |
13 |
2 1
|
resubcld |
|- ( ph -> ( B - A ) e. RR ) |
14 |
13
|
rexrd |
|- ( ph -> ( B - A ) e. RR* ) |
15 |
1 2
|
posdifd |
|- ( ph -> ( A < B <-> 0 < ( B - A ) ) ) |
16 |
3 15
|
mpbid |
|- ( ph -> 0 < ( B - A ) ) |
17 |
4
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> F : ( A (,) B ) --> RR ) |
18 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
19 |
18
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> A e. RR* ) |
20 |
2
|
rexrd |
|- ( ph -> B e. RR* ) |
21 |
20
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> B e. RR* ) |
22 |
1
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> A e. RR ) |
23 |
|
elioore |
|- ( s e. ( 0 (,) ( B - A ) ) -> s e. RR ) |
24 |
23
|
adantl |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> s e. RR ) |
25 |
22 24
|
readdcld |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) e. RR ) |
26 |
1
|
recnd |
|- ( ph -> A e. CC ) |
27 |
26
|
addid1d |
|- ( ph -> ( A + 0 ) = A ) |
28 |
27
|
eqcomd |
|- ( ph -> A = ( A + 0 ) ) |
29 |
28
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> A = ( A + 0 ) ) |
30 |
|
0red |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> 0 e. RR ) |
31 |
|
0xr |
|- 0 e. RR* |
32 |
31
|
a1i |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> 0 e. RR* ) |
33 |
14
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( B - A ) e. RR* ) |
34 |
|
simpr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> s e. ( 0 (,) ( B - A ) ) ) |
35 |
32 33 34
|
ioogtlbd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> 0 < s ) |
36 |
30 24 22 35
|
ltadd2dd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + 0 ) < ( A + s ) ) |
37 |
29 36
|
eqbrtrd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> A < ( A + s ) ) |
38 |
13
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( B - A ) e. RR ) |
39 |
32 33 34
|
iooltubd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> s < ( B - A ) ) |
40 |
24 38 22 39
|
ltadd2dd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) < ( A + ( B - A ) ) ) |
41 |
2
|
recnd |
|- ( ph -> B e. CC ) |
42 |
26 41
|
pncan3d |
|- ( ph -> ( A + ( B - A ) ) = B ) |
43 |
42
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + ( B - A ) ) = B ) |
44 |
40 43
|
breqtrd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) < B ) |
45 |
19 21 25 37 44
|
eliood |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) e. ( A (,) B ) ) |
46 |
17 45
|
ffvelrnd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( F ` ( A + s ) ) e. RR ) |
47 |
|
ioossre |
|- ( A (,) B ) C_ RR |
48 |
47
|
a1i |
|- ( ph -> ( A (,) B ) C_ RR ) |
49 |
|
ax-resscn |
|- RR C_ CC |
50 |
48 49
|
sstrdi |
|- ( ph -> ( A (,) B ) C_ CC ) |
51 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
52 |
51 20 1 3
|
lptioo1cn |
|- ( ph -> A e. ( ( limPt ` ( TopOpen ` CCfld ) ) ` ( A (,) B ) ) ) |
53 |
4 50 52 5
|
limcrecl |
|- ( ph -> Y e. RR ) |
54 |
53
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> Y e. RR ) |
55 |
46 54
|
resubcld |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( F ` ( A + s ) ) - Y ) e. RR ) |
56 |
55 10
|
fmptd |
|- ( ph -> N : ( 0 (,) ( B - A ) ) --> RR ) |
57 |
24 11
|
fmptd |
|- ( ph -> D : ( 0 (,) ( B - A ) ) --> RR ) |
58 |
10
|
oveq2i |
|- ( RR _D N ) = ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) |
59 |
58
|
a1i |
|- ( ph -> ( RR _D N ) = ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) ) |
60 |
59
|
dmeqd |
|- ( ph -> dom ( RR _D N ) = dom ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) ) |
61 |
|
reelprrecn |
|- RR e. { RR , CC } |
62 |
61
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
63 |
46
|
recnd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( F ` ( A + s ) ) e. CC ) |
64 |
|
dvfre |
|- ( ( F : ( A (,) B ) --> RR /\ ( A (,) B ) C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR ) |
65 |
4 48 64
|
syl2anc |
|- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR ) |
66 |
6
|
a1i |
|- ( ph -> G = ( RR _D F ) ) |
67 |
66
|
feq1d |
|- ( ph -> ( G : dom ( RR _D F ) --> RR <-> ( RR _D F ) : dom ( RR _D F ) --> RR ) ) |
68 |
65 67
|
mpbird |
|- ( ph -> G : dom ( RR _D F ) --> RR ) |
69 |
68
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> G : dom ( RR _D F ) --> RR ) |
70 |
66
|
eqcomd |
|- ( ph -> ( RR _D F ) = G ) |
71 |
70
|
dmeqd |
|- ( ph -> dom ( RR _D F ) = dom G ) |
72 |
71 7
|
eqtr2d |
|- ( ph -> ( A (,) B ) = dom ( RR _D F ) ) |
73 |
72
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A (,) B ) = dom ( RR _D F ) ) |
74 |
45 73
|
eleqtrd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) e. dom ( RR _D F ) ) |
75 |
69 74
|
ffvelrnd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( G ` ( A + s ) ) e. RR ) |
76 |
|
1red |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> 1 e. RR ) |
77 |
4
|
ffvelrnda |
|- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. RR ) |
78 |
77
|
recnd |
|- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. CC ) |
79 |
72
|
feq2d |
|- ( ph -> ( G : ( A (,) B ) --> RR <-> G : dom ( RR _D F ) --> RR ) ) |
80 |
68 79
|
mpbird |
|- ( ph -> G : ( A (,) B ) --> RR ) |
81 |
80
|
ffvelrnda |
|- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) e. RR ) |
82 |
26
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> A e. CC ) |
83 |
26
|
adantr |
|- ( ( ph /\ s e. RR ) -> A e. CC ) |
84 |
|
0red |
|- ( ( ph /\ s e. RR ) -> 0 e. RR ) |
85 |
62 26
|
dvmptc |
|- ( ph -> ( RR _D ( s e. RR |-> A ) ) = ( s e. RR |-> 0 ) ) |
86 |
|
ioossre |
|- ( 0 (,) ( B - A ) ) C_ RR |
87 |
86
|
a1i |
|- ( ph -> ( 0 (,) ( B - A ) ) C_ RR ) |
88 |
|
tgioo4 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
89 |
|
iooretop |
|- ( 0 (,) ( B - A ) ) e. ( topGen ` ran (,) ) |
90 |
89
|
a1i |
|- ( ph -> ( 0 (,) ( B - A ) ) e. ( topGen ` ran (,) ) ) |
91 |
62 83 84 85 87 88 51 90
|
dvmptres |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> A ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> 0 ) ) |
92 |
24
|
recnd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> s e. CC ) |
93 |
|
recn |
|- ( s e. RR -> s e. CC ) |
94 |
93
|
adantl |
|- ( ( ph /\ s e. RR ) -> s e. CC ) |
95 |
|
1red |
|- ( ( ph /\ s e. RR ) -> 1 e. RR ) |
96 |
62
|
dvmptid |
|- ( ph -> ( RR _D ( s e. RR |-> s ) ) = ( s e. RR |-> 1 ) ) |
97 |
62 94 95 96 87 88 51 90
|
dvmptres |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> s ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ) |
98 |
62 82 30 91 92 76 97
|
dvmptadd |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( 0 + 1 ) ) ) |
99 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
100 |
99
|
mpteq2i |
|- ( s e. ( 0 (,) ( B - A ) ) |-> ( 0 + 1 ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) |
101 |
98 100
|
eqtrdi |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ) |
102 |
4
|
feqmptd |
|- ( ph -> F = ( x e. ( A (,) B ) |-> ( F ` x ) ) ) |
103 |
102
|
eqcomd |
|- ( ph -> ( x e. ( A (,) B ) |-> ( F ` x ) ) = F ) |
104 |
103
|
oveq2d |
|- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) = ( RR _D F ) ) |
105 |
80
|
feqmptd |
|- ( ph -> G = ( x e. ( A (,) B ) |-> ( G ` x ) ) ) |
106 |
104 70 105
|
3eqtrd |
|- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) = ( x e. ( A (,) B ) |-> ( G ` x ) ) ) |
107 |
|
fveq2 |
|- ( x = ( A + s ) -> ( F ` x ) = ( F ` ( A + s ) ) ) |
108 |
|
fveq2 |
|- ( x = ( A + s ) -> ( G ` x ) = ( G ` ( A + s ) ) ) |
109 |
62 62 45 76 78 81 101 106 107 108
|
dvmptco |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( F ` ( A + s ) ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( G ` ( A + s ) ) x. 1 ) ) ) |
110 |
75
|
recnd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( G ` ( A + s ) ) e. CC ) |
111 |
110
|
mulid1d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( G ` ( A + s ) ) x. 1 ) = ( G ` ( A + s ) ) ) |
112 |
111
|
mpteq2dva |
|- ( ph -> ( s e. ( 0 (,) ( B - A ) ) |-> ( ( G ` ( A + s ) ) x. 1 ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
113 |
109 112
|
eqtrd |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( F ` ( A + s ) ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
114 |
|
limccl |
|- ( F limCC A ) C_ CC |
115 |
114 5
|
sselid |
|- ( ph -> Y e. CC ) |
116 |
115
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> Y e. CC ) |
117 |
115
|
adantr |
|- ( ( ph /\ s e. RR ) -> Y e. CC ) |
118 |
62 115
|
dvmptc |
|- ( ph -> ( RR _D ( s e. RR |-> Y ) ) = ( s e. RR |-> 0 ) ) |
119 |
62 117 84 118 87 88 51 90
|
dvmptres |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> Y ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> 0 ) ) |
120 |
62 63 75 113 116 30 119
|
dvmptsub |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( G ` ( A + s ) ) - 0 ) ) ) |
121 |
110
|
subid1d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( G ` ( A + s ) ) - 0 ) = ( G ` ( A + s ) ) ) |
122 |
121
|
mpteq2dva |
|- ( ph -> ( s e. ( 0 (,) ( B - A ) ) |-> ( ( G ` ( A + s ) ) - 0 ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
123 |
120 122
|
eqtrd |
|- ( ph -> ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
124 |
123
|
dmeqd |
|- ( ph -> dom ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) ) = dom ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
125 |
75
|
ralrimiva |
|- ( ph -> A. s e. ( 0 (,) ( B - A ) ) ( G ` ( A + s ) ) e. RR ) |
126 |
|
dmmptg |
|- ( A. s e. ( 0 (,) ( B - A ) ) ( G ` ( A + s ) ) e. RR -> dom ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) = ( 0 (,) ( B - A ) ) ) |
127 |
125 126
|
syl |
|- ( ph -> dom ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) = ( 0 (,) ( B - A ) ) ) |
128 |
60 124 127
|
3eqtrd |
|- ( ph -> dom ( RR _D N ) = ( 0 (,) ( B - A ) ) ) |
129 |
11
|
a1i |
|- ( ph -> D = ( s e. ( 0 (,) ( B - A ) ) |-> s ) ) |
130 |
129
|
oveq2d |
|- ( ph -> ( RR _D D ) = ( RR _D ( s e. ( 0 (,) ( B - A ) ) |-> s ) ) ) |
131 |
130 97
|
eqtrd |
|- ( ph -> ( RR _D D ) = ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ) |
132 |
131
|
dmeqd |
|- ( ph -> dom ( RR _D D ) = dom ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ) |
133 |
76
|
ralrimiva |
|- ( ph -> A. s e. ( 0 (,) ( B - A ) ) 1 e. RR ) |
134 |
|
dmmptg |
|- ( A. s e. ( 0 (,) ( B - A ) ) 1 e. RR -> dom ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) = ( 0 (,) ( B - A ) ) ) |
135 |
133 134
|
syl |
|- ( ph -> dom ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) = ( 0 (,) ( B - A ) ) ) |
136 |
132 135
|
eqtrd |
|- ( ph -> dom ( RR _D D ) = ( 0 (,) ( B - A ) ) ) |
137 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> ( F ` ( A + s ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( F ` ( A + s ) ) ) |
138 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> Y ) = ( s e. ( 0 (,) ( B - A ) ) |-> Y ) |
139 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) |
140 |
45
|
adantrr |
|- ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) =/= A ) ) -> ( A + s ) e. ( A (,) B ) ) |
141 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> A ) = ( s e. ( 0 (,) ( B - A ) ) |-> A ) |
142 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> s ) = ( s e. ( 0 (,) ( B - A ) ) |-> s ) |
143 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) |
144 |
87 49
|
sstrdi |
|- ( ph -> ( 0 (,) ( B - A ) ) C_ CC ) |
145 |
12
|
recnd |
|- ( ph -> 0 e. CC ) |
146 |
141 144 26 145
|
constlimc |
|- ( ph -> A e. ( ( s e. ( 0 (,) ( B - A ) ) |-> A ) limCC 0 ) ) |
147 |
144 142 145
|
idlimc |
|- ( ph -> 0 e. ( ( s e. ( 0 (,) ( B - A ) ) |-> s ) limCC 0 ) ) |
148 |
141 142 143 82 92 146 147
|
addlimc |
|- ( ph -> ( A + 0 ) e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) limCC 0 ) ) |
149 |
28 148
|
eqeltrd |
|- ( ph -> A e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( A + s ) ) limCC 0 ) ) |
150 |
102
|
oveq1d |
|- ( ph -> ( F limCC A ) = ( ( x e. ( A (,) B ) |-> ( F ` x ) ) limCC A ) ) |
151 |
5 150
|
eleqtrd |
|- ( ph -> Y e. ( ( x e. ( A (,) B ) |-> ( F ` x ) ) limCC A ) ) |
152 |
|
simplrr |
|- ( ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) /\ -. ( F ` ( A + s ) ) = Y ) -> ( A + s ) = A ) |
153 |
22 37
|
gtned |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( A + s ) =/= A ) |
154 |
153
|
neneqd |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> -. ( A + s ) = A ) |
155 |
154
|
adantrr |
|- ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) -> -. ( A + s ) = A ) |
156 |
155
|
adantr |
|- ( ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) /\ -. ( F ` ( A + s ) ) = Y ) -> -. ( A + s ) = A ) |
157 |
152 156
|
condan |
|- ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) -> ( F ` ( A + s ) ) = Y ) |
158 |
140 78 149 151 107 157
|
limcco |
|- ( ph -> Y e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( F ` ( A + s ) ) ) limCC 0 ) ) |
159 |
138 144 115 145
|
constlimc |
|- ( ph -> Y e. ( ( s e. ( 0 (,) ( B - A ) ) |-> Y ) limCC 0 ) ) |
160 |
137 138 139 63 116 158 159
|
sublimc |
|- ( ph -> ( Y - Y ) e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) limCC 0 ) ) |
161 |
115
|
subidd |
|- ( ph -> ( Y - Y ) = 0 ) |
162 |
10
|
eqcomi |
|- ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) = N |
163 |
162
|
oveq1i |
|- ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) limCC 0 ) = ( N limCC 0 ) |
164 |
163
|
a1i |
|- ( ph -> ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( F ` ( A + s ) ) - Y ) ) limCC 0 ) = ( N limCC 0 ) ) |
165 |
160 161 164
|
3eltr3d |
|- ( ph -> 0 e. ( N limCC 0 ) ) |
166 |
144 11 145
|
idlimc |
|- ( ph -> 0 e. ( D limCC 0 ) ) |
167 |
|
lbioo |
|- -. 0 e. ( 0 (,) ( B - A ) ) |
168 |
167
|
a1i |
|- ( ph -> -. 0 e. ( 0 (,) ( B - A ) ) ) |
169 |
|
mptresid |
|- ( _I |` ( 0 (,) ( B - A ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> s ) |
170 |
129 169
|
eqtr4di |
|- ( ph -> D = ( _I |` ( 0 (,) ( B - A ) ) ) ) |
171 |
170
|
rneqd |
|- ( ph -> ran D = ran ( _I |` ( 0 (,) ( B - A ) ) ) ) |
172 |
|
rnresi |
|- ran ( _I |` ( 0 (,) ( B - A ) ) ) = ( 0 (,) ( B - A ) ) |
173 |
171 172
|
eqtr2di |
|- ( ph -> ( 0 (,) ( B - A ) ) = ran D ) |
174 |
168 173
|
neleqtrd |
|- ( ph -> -. 0 e. ran D ) |
175 |
|
0ne1 |
|- 0 =/= 1 |
176 |
175
|
neii |
|- -. 0 = 1 |
177 |
|
elsng |
|- ( 0 e. RR -> ( 0 e. { 1 } <-> 0 = 1 ) ) |
178 |
12 177
|
syl |
|- ( ph -> ( 0 e. { 1 } <-> 0 = 1 ) ) |
179 |
176 178
|
mtbiri |
|- ( ph -> -. 0 e. { 1 } ) |
180 |
131
|
rneqd |
|- ( ph -> ran ( RR _D D ) = ran ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ) |
181 |
|
eqid |
|- ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) = ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) |
182 |
31
|
a1i |
|- ( ph -> 0 e. RR* ) |
183 |
|
ioon0 |
|- ( ( 0 e. RR* /\ ( B - A ) e. RR* ) -> ( ( 0 (,) ( B - A ) ) =/= (/) <-> 0 < ( B - A ) ) ) |
184 |
182 14 183
|
syl2anc |
|- ( ph -> ( ( 0 (,) ( B - A ) ) =/= (/) <-> 0 < ( B - A ) ) ) |
185 |
16 184
|
mpbird |
|- ( ph -> ( 0 (,) ( B - A ) ) =/= (/) ) |
186 |
181 185
|
rnmptc |
|- ( ph -> ran ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) = { 1 } ) |
187 |
180 186
|
eqtr2d |
|- ( ph -> { 1 } = ran ( RR _D D ) ) |
188 |
179 187
|
neleqtrd |
|- ( ph -> -. 0 e. ran ( RR _D D ) ) |
189 |
81
|
recnd |
|- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) e. CC ) |
190 |
105
|
oveq1d |
|- ( ph -> ( G limCC A ) = ( ( x e. ( A (,) B ) |-> ( G ` x ) ) limCC A ) ) |
191 |
8 190
|
eleqtrd |
|- ( ph -> E e. ( ( x e. ( A (,) B ) |-> ( G ` x ) ) limCC A ) ) |
192 |
|
simplrr |
|- ( ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) /\ -. ( G ` ( A + s ) ) = E ) -> ( A + s ) = A ) |
193 |
155
|
adantr |
|- ( ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) /\ -. ( G ` ( A + s ) ) = E ) -> -. ( A + s ) = A ) |
194 |
192 193
|
condan |
|- ( ( ph /\ ( s e. ( 0 (,) ( B - A ) ) /\ ( A + s ) = A ) ) -> ( G ` ( A + s ) ) = E ) |
195 |
140 189 149 191 108 194
|
limcco |
|- ( ph -> E e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) limCC 0 ) ) |
196 |
110
|
div1d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( G ` ( A + s ) ) / 1 ) = ( G ` ( A + s ) ) ) |
197 |
58 123
|
eqtrid |
|- ( ph -> ( RR _D N ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
198 |
197
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( RR _D N ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ) |
199 |
198
|
fveq1d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( RR _D N ) ` s ) = ( ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ` s ) ) |
200 |
|
fvmpt4 |
|- ( ( s e. ( 0 (,) ( B - A ) ) /\ ( G ` ( A + s ) ) e. RR ) -> ( ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ` s ) = ( G ` ( A + s ) ) ) |
201 |
34 75 200
|
syl2anc |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) ` s ) = ( G ` ( A + s ) ) ) |
202 |
199 201
|
eqtr2d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( G ` ( A + s ) ) = ( ( RR _D N ) ` s ) ) |
203 |
131
|
fveq1d |
|- ( ph -> ( ( RR _D D ) ` s ) = ( ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ` s ) ) |
204 |
203
|
adantr |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( RR _D D ) ` s ) = ( ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ` s ) ) |
205 |
|
fvmpt4 |
|- ( ( s e. ( 0 (,) ( B - A ) ) /\ 1 e. RR ) -> ( ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ` s ) = 1 ) |
206 |
34 76 205
|
syl2anc |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( s e. ( 0 (,) ( B - A ) ) |-> 1 ) ` s ) = 1 ) |
207 |
204 206
|
eqtr2d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> 1 = ( ( RR _D D ) ` s ) ) |
208 |
202 207
|
oveq12d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( G ` ( A + s ) ) / 1 ) = ( ( ( RR _D N ) ` s ) / ( ( RR _D D ) ` s ) ) ) |
209 |
196 208
|
eqtr3d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( G ` ( A + s ) ) = ( ( ( RR _D N ) ` s ) / ( ( RR _D D ) ` s ) ) ) |
210 |
209
|
mpteq2dva |
|- ( ph -> ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( ( RR _D N ) ` s ) / ( ( RR _D D ) ` s ) ) ) ) |
211 |
210
|
oveq1d |
|- ( ph -> ( ( s e. ( 0 (,) ( B - A ) ) |-> ( G ` ( A + s ) ) ) limCC 0 ) = ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( ( RR _D N ) ` s ) / ( ( RR _D D ) ` s ) ) ) limCC 0 ) ) |
212 |
195 211
|
eleqtrd |
|- ( ph -> E e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( ( RR _D N ) ` s ) / ( ( RR _D D ) ` s ) ) ) limCC 0 ) ) |
213 |
12 14 16 56 57 128 136 165 166 174 188 212
|
lhop1 |
|- ( ph -> E e. ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( N ` s ) / ( D ` s ) ) ) limCC 0 ) ) |
214 |
10
|
fvmpt2 |
|- ( ( s e. ( 0 (,) ( B - A ) ) /\ ( ( F ` ( A + s ) ) - Y ) e. RR ) -> ( N ` s ) = ( ( F ` ( A + s ) ) - Y ) ) |
215 |
34 55 214
|
syl2anc |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( N ` s ) = ( ( F ` ( A + s ) ) - Y ) ) |
216 |
11
|
fvmpt2 |
|- ( ( s e. ( 0 (,) ( B - A ) ) /\ s e. ( 0 (,) ( B - A ) ) ) -> ( D ` s ) = s ) |
217 |
34 34 216
|
syl2anc |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( D ` s ) = s ) |
218 |
215 217
|
oveq12d |
|- ( ( ph /\ s e. ( 0 (,) ( B - A ) ) ) -> ( ( N ` s ) / ( D ` s ) ) = ( ( ( F ` ( A + s ) ) - Y ) / s ) ) |
219 |
218
|
mpteq2dva |
|- ( ph -> ( s e. ( 0 (,) ( B - A ) ) |-> ( ( N ` s ) / ( D ` s ) ) ) = ( s e. ( 0 (,) ( B - A ) ) |-> ( ( ( F ` ( A + s ) ) - Y ) / s ) ) ) |
220 |
219 9
|
eqtr4di |
|- ( ph -> ( s e. ( 0 (,) ( B - A ) ) |-> ( ( N ` s ) / ( D ` s ) ) ) = H ) |
221 |
220
|
oveq1d |
|- ( ph -> ( ( s e. ( 0 (,) ( B - A ) ) |-> ( ( N ` s ) / ( D ` s ) ) ) limCC 0 ) = ( H limCC 0 ) ) |
222 |
213 221
|
eleqtrd |
|- ( ph -> E e. ( H limCC 0 ) ) |