Step |
Hyp |
Ref |
Expression |
1 |
|
itgperiod.a |
|- ( ph -> A e. RR ) |
2 |
|
itgperiod.b |
|- ( ph -> B e. RR ) |
3 |
|
itgperiod.aleb |
|- ( ph -> A <_ B ) |
4 |
|
itgperiod.t |
|- ( ph -> T e. RR+ ) |
5 |
|
itgperiod.f |
|- ( ph -> F : RR --> CC ) |
6 |
|
itgperiod.fper |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
7 |
|
itgperiod.fcn |
|- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) |
8 |
4
|
rpred |
|- ( ph -> T e. RR ) |
9 |
1 2 8 3
|
leadd1dd |
|- ( ph -> ( A + T ) <_ ( B + T ) ) |
10 |
9
|
ditgpos |
|- ( ph -> S_ [ ( A + T ) -> ( B + T ) ] ( F ` x ) _d x = S. ( ( A + T ) (,) ( B + T ) ) ( F ` x ) _d x ) |
11 |
1 8
|
readdcld |
|- ( ph -> ( A + T ) e. RR ) |
12 |
2 8
|
readdcld |
|- ( ph -> ( B + T ) e. RR ) |
13 |
5
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> F : RR --> CC ) |
14 |
11
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) e. RR ) |
15 |
12
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( B + T ) e. RR ) |
16 |
|
simpr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
17 |
|
eliccre |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
19 |
13 18
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( F ` x ) e. CC ) |
20 |
11 12 19
|
itgioo |
|- ( ph -> S. ( ( A + T ) (,) ( B + T ) ) ( F ` x ) _d x = S. ( ( A + T ) [,] ( B + T ) ) ( F ` x ) _d x ) |
21 |
10 20
|
eqtr2d |
|- ( ph -> S. ( ( A + T ) [,] ( B + T ) ) ( F ` x ) _d x = S_ [ ( A + T ) -> ( B + T ) ] ( F ` x ) _d x ) |
22 |
|
eqid |
|- ( y e. CC |-> ( y + T ) ) = ( y e. CC |-> ( y + T ) ) |
23 |
8
|
recnd |
|- ( ph -> T e. CC ) |
24 |
22
|
addccncf |
|- ( T e. CC -> ( y e. CC |-> ( y + T ) ) e. ( CC -cn-> CC ) ) |
25 |
23 24
|
syl |
|- ( ph -> ( y e. CC |-> ( y + T ) ) e. ( CC -cn-> CC ) ) |
26 |
1 2
|
iccssred |
|- ( ph -> ( A [,] B ) C_ RR ) |
27 |
|
ax-resscn |
|- RR C_ CC |
28 |
26 27
|
sstrdi |
|- ( ph -> ( A [,] B ) C_ CC ) |
29 |
11 12
|
iccssred |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) C_ RR ) |
30 |
29 27
|
sstrdi |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) C_ CC ) |
31 |
11
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( A + T ) e. RR ) |
32 |
12
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( B + T ) e. RR ) |
33 |
26
|
sselda |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y e. RR ) |
34 |
8
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> T e. RR ) |
35 |
33 34
|
readdcld |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) e. RR ) |
36 |
1
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> A e. RR ) |
37 |
|
simpr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y e. ( A [,] B ) ) |
38 |
2
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> B e. RR ) |
39 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
40 |
36 38 39
|
syl2anc |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
41 |
37 40
|
mpbid |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) ) |
42 |
41
|
simp2d |
|- ( ( ph /\ y e. ( A [,] B ) ) -> A <_ y ) |
43 |
36 33 34 42
|
leadd1dd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( A + T ) <_ ( y + T ) ) |
44 |
41
|
simp3d |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y <_ B ) |
45 |
33 38 34 44
|
leadd1dd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) <_ ( B + T ) ) |
46 |
31 32 35 43 45
|
eliccd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) e. ( ( A + T ) [,] ( B + T ) ) ) |
47 |
22 25 28 30 46
|
cncfmptssg |
|- ( ph -> ( y e. ( A [,] B ) |-> ( y + T ) ) e. ( ( A [,] B ) -cn-> ( ( A + T ) [,] ( B + T ) ) ) ) |
48 |
|
eqeq1 |
|- ( w = x -> ( w = ( z + T ) <-> x = ( z + T ) ) ) |
49 |
48
|
rexbidv |
|- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. z e. ( A [,] B ) x = ( z + T ) ) ) |
50 |
|
oveq1 |
|- ( z = y -> ( z + T ) = ( y + T ) ) |
51 |
50
|
eqeq2d |
|- ( z = y -> ( x = ( z + T ) <-> x = ( y + T ) ) ) |
52 |
51
|
cbvrexvw |
|- ( E. z e. ( A [,] B ) x = ( z + T ) <-> E. y e. ( A [,] B ) x = ( y + T ) ) |
53 |
49 52
|
bitrdi |
|- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. y e. ( A [,] B ) x = ( y + T ) ) ) |
54 |
53
|
cbvrabv |
|- { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } = { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |
55 |
5
|
ffdmd |
|- ( ph -> F : dom F --> CC ) |
56 |
|
simp3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ w = ( z + T ) ) -> w = ( z + T ) ) |
57 |
26
|
sselda |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z e. RR ) |
58 |
8
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> T e. RR ) |
59 |
57 58
|
readdcld |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) e. RR ) |
60 |
59
|
3adant3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ w = ( z + T ) ) -> ( z + T ) e. RR ) |
61 |
56 60
|
eqeltrd |
|- ( ( ph /\ z e. ( A [,] B ) /\ w = ( z + T ) ) -> w e. RR ) |
62 |
61
|
rexlimdv3a |
|- ( ph -> ( E. z e. ( A [,] B ) w = ( z + T ) -> w e. RR ) ) |
63 |
62
|
ralrimivw |
|- ( ph -> A. w e. CC ( E. z e. ( A [,] B ) w = ( z + T ) -> w e. RR ) ) |
64 |
|
rabss |
|- ( { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } C_ RR <-> A. w e. CC ( E. z e. ( A [,] B ) w = ( z + T ) -> w e. RR ) ) |
65 |
63 64
|
sylibr |
|- ( ph -> { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } C_ RR ) |
66 |
5
|
fdmd |
|- ( ph -> dom F = RR ) |
67 |
65 66
|
sseqtrrd |
|- ( ph -> { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } C_ dom F ) |
68 |
28 8 54 55 67 6 7
|
cncfperiod |
|- ( ph -> ( F |` { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) e. ( { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } -cn-> CC ) ) |
69 |
49
|
elrab |
|- ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) |
70 |
|
simprr |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
71 |
|
nfv |
|- F/ z ph |
72 |
|
nfv |
|- F/ z x e. CC |
73 |
|
nfre1 |
|- F/ z E. z e. ( A [,] B ) x = ( z + T ) |
74 |
72 73
|
nfan |
|- F/ z ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) |
75 |
71 74
|
nfan |
|- F/ z ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) |
76 |
|
nfv |
|- F/ z x e. ( ( A + T ) [,] ( B + T ) ) |
77 |
|
simp3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x = ( z + T ) ) |
78 |
1
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> A e. RR ) |
79 |
|
simpr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z e. ( A [,] B ) ) |
80 |
2
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> B e. RR ) |
81 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) ) |
82 |
78 80 81
|
syl2anc |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) ) |
83 |
79 82
|
mpbid |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. RR /\ A <_ z /\ z <_ B ) ) |
84 |
83
|
simp2d |
|- ( ( ph /\ z e. ( A [,] B ) ) -> A <_ z ) |
85 |
78 57 58 84
|
leadd1dd |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( A + T ) <_ ( z + T ) ) |
86 |
83
|
simp3d |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z <_ B ) |
87 |
57 80 58 86
|
leadd1dd |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) <_ ( B + T ) ) |
88 |
59 85 87
|
3jca |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) |
89 |
88
|
3adant3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) |
90 |
11
|
3ad2ant1 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( A + T ) e. RR ) |
91 |
12
|
3ad2ant1 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( B + T ) e. RR ) |
92 |
|
elicc2 |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) ) |
93 |
90 91 92
|
syl2anc |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) ) |
94 |
89 93
|
mpbird |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) ) |
95 |
77 94
|
eqeltrd |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
96 |
95
|
3exp |
|- ( ph -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) ) |
97 |
96
|
adantr |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) ) |
98 |
75 76 97
|
rexlimd |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( E. z e. ( A [,] B ) x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) |
99 |
70 98
|
mpd |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
100 |
69 99
|
sylan2b |
|- ( ( ph /\ x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
101 |
18
|
recnd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. CC ) |
102 |
1
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A e. RR ) |
103 |
2
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> B e. RR ) |
104 |
8
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. RR ) |
105 |
18 104
|
resubcld |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. RR ) |
106 |
1
|
recnd |
|- ( ph -> A e. CC ) |
107 |
106 23
|
pncand |
|- ( ph -> ( ( A + T ) - T ) = A ) |
108 |
107
|
eqcomd |
|- ( ph -> A = ( ( A + T ) - T ) ) |
109 |
108
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A = ( ( A + T ) - T ) ) |
110 |
|
elicc2 |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
111 |
14 15 110
|
syl2anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
112 |
16 111
|
mpbid |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) |
113 |
112
|
simp2d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) <_ x ) |
114 |
14 18 104 113
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( A + T ) - T ) <_ ( x - T ) ) |
115 |
109 114
|
eqbrtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A <_ ( x - T ) ) |
116 |
112
|
simp3d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x <_ ( B + T ) ) |
117 |
18 15 104 116
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ ( ( B + T ) - T ) ) |
118 |
2
|
recnd |
|- ( ph -> B e. CC ) |
119 |
118 23
|
pncand |
|- ( ph -> ( ( B + T ) - T ) = B ) |
120 |
119
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( B + T ) - T ) = B ) |
121 |
117 120
|
breqtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ B ) |
122 |
102 103 105 115 121
|
eliccd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. ( A [,] B ) ) |
123 |
23
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. CC ) |
124 |
101 123
|
npcand |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( x - T ) + T ) = x ) |
125 |
124
|
eqcomd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x = ( ( x - T ) + T ) ) |
126 |
|
oveq1 |
|- ( z = ( x - T ) -> ( z + T ) = ( ( x - T ) + T ) ) |
127 |
126
|
rspceeqv |
|- ( ( ( x - T ) e. ( A [,] B ) /\ x = ( ( x - T ) + T ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
128 |
122 125 127
|
syl2anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
129 |
101 128 69
|
sylanbrc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) |
130 |
100 129
|
impbida |
|- ( ph -> ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> x e. ( ( A + T ) [,] ( B + T ) ) ) ) |
131 |
130
|
eqrdv |
|- ( ph -> { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } = ( ( A + T ) [,] ( B + T ) ) ) |
132 |
131
|
reseq2d |
|- ( ph -> ( F |` { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) = ( F |` ( ( A + T ) [,] ( B + T ) ) ) ) |
133 |
131 67
|
eqsstrrd |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) C_ dom F ) |
134 |
55 133
|
feqresmpt |
|- ( ph -> ( F |` ( ( A + T ) [,] ( B + T ) ) ) = ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` x ) ) ) |
135 |
132 134
|
eqtr2d |
|- ( ph -> ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` x ) ) = ( F |` { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) ) |
136 |
1 2 8
|
iccshift |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) = { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) |
137 |
136
|
oveq1d |
|- ( ph -> ( ( ( A + T ) [,] ( B + T ) ) -cn-> CC ) = ( { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } -cn-> CC ) ) |
138 |
68 135 137
|
3eltr4d |
|- ( ph -> ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` x ) ) e. ( ( ( A + T ) [,] ( B + T ) ) -cn-> CC ) ) |
139 |
|
ioosscn |
|- ( A (,) B ) C_ CC |
140 |
139
|
a1i |
|- ( ph -> ( A (,) B ) C_ CC ) |
141 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
142 |
|
ssid |
|- CC C_ CC |
143 |
142
|
a1i |
|- ( ph -> CC C_ CC ) |
144 |
140 141 143
|
constcncfg |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC ) ) |
145 |
|
fconstmpt |
|- ( ( A (,) B ) X. { 1 } ) = ( y e. ( A (,) B ) |-> 1 ) |
146 |
|
ioombl |
|- ( A (,) B ) e. dom vol |
147 |
146
|
a1i |
|- ( ph -> ( A (,) B ) e. dom vol ) |
148 |
|
ioovolcl |
|- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A (,) B ) ) e. RR ) |
149 |
1 2 148
|
syl2anc |
|- ( ph -> ( vol ` ( A (,) B ) ) e. RR ) |
150 |
|
iblconst |
|- ( ( ( A (,) B ) e. dom vol /\ ( vol ` ( A (,) B ) ) e. RR /\ 1 e. CC ) -> ( ( A (,) B ) X. { 1 } ) e. L^1 ) |
151 |
147 149 141 150
|
syl3anc |
|- ( ph -> ( ( A (,) B ) X. { 1 } ) e. L^1 ) |
152 |
145 151
|
eqeltrrid |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. L^1 ) |
153 |
144 152
|
elind |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. ( ( ( A (,) B ) -cn-> CC ) i^i L^1 ) ) |
154 |
26
|
resmptd |
|- ( ph -> ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) = ( y e. ( A [,] B ) |-> ( y + T ) ) ) |
155 |
154
|
eqcomd |
|- ( ph -> ( y e. ( A [,] B ) |-> ( y + T ) ) = ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) |
156 |
155
|
oveq2d |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( RR _D ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) ) |
157 |
27
|
a1i |
|- ( ph -> RR C_ CC ) |
158 |
157
|
sselda |
|- ( ( ph /\ y e. RR ) -> y e. CC ) |
159 |
23
|
adantr |
|- ( ( ph /\ y e. RR ) -> T e. CC ) |
160 |
158 159
|
addcld |
|- ( ( ph /\ y e. RR ) -> ( y + T ) e. CC ) |
161 |
160
|
fmpttd |
|- ( ph -> ( y e. RR |-> ( y + T ) ) : RR --> CC ) |
162 |
|
ssid |
|- RR C_ RR |
163 |
162
|
a1i |
|- ( ph -> RR C_ RR ) |
164 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
165 |
164
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
166 |
164 165
|
dvres |
|- ( ( ( RR C_ CC /\ ( y e. RR |-> ( y + T ) ) : RR --> CC ) /\ ( RR C_ RR /\ ( A [,] B ) C_ RR ) ) -> ( RR _D ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
167 |
157 161 163 26 166
|
syl22anc |
|- ( ph -> ( RR _D ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
168 |
156 167
|
eqtrd |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
169 |
|
iccntr |
|- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
170 |
1 2 169
|
syl2anc |
|- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
171 |
170
|
reseq2d |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) ) |
172 |
|
reelprrecn |
|- RR e. { RR , CC } |
173 |
172
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
174 |
|
1cnd |
|- ( ( ph /\ y e. RR ) -> 1 e. CC ) |
175 |
173
|
dvmptid |
|- ( ph -> ( RR _D ( y e. RR |-> y ) ) = ( y e. RR |-> 1 ) ) |
176 |
|
0cnd |
|- ( ( ph /\ y e. RR ) -> 0 e. CC ) |
177 |
173 23
|
dvmptc |
|- ( ph -> ( RR _D ( y e. RR |-> T ) ) = ( y e. RR |-> 0 ) ) |
178 |
173 158 174 175 159 176 177
|
dvmptadd |
|- ( ph -> ( RR _D ( y e. RR |-> ( y + T ) ) ) = ( y e. RR |-> ( 1 + 0 ) ) ) |
179 |
178
|
reseq1d |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) = ( ( y e. RR |-> ( 1 + 0 ) ) |` ( A (,) B ) ) ) |
180 |
|
ioossre |
|- ( A (,) B ) C_ RR |
181 |
180
|
a1i |
|- ( ph -> ( A (,) B ) C_ RR ) |
182 |
181
|
resmptd |
|- ( ph -> ( ( y e. RR |-> ( 1 + 0 ) ) |` ( A (,) B ) ) = ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) ) |
183 |
|
1p0e1 |
|- ( 1 + 0 ) = 1 |
184 |
183
|
mpteq2i |
|- ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) = ( y e. ( A (,) B ) |-> 1 ) |
185 |
184
|
a1i |
|- ( ph -> ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
186 |
179 182 185
|
3eqtrd |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
187 |
168 171 186
|
3eqtrd |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
188 |
|
fveq2 |
|- ( x = ( y + T ) -> ( F ` x ) = ( F ` ( y + T ) ) ) |
189 |
|
oveq1 |
|- ( y = A -> ( y + T ) = ( A + T ) ) |
190 |
|
oveq1 |
|- ( y = B -> ( y + T ) = ( B + T ) ) |
191 |
1 2 3 47 138 153 187 188 189 190 11 12
|
itgsubsticc |
|- ( ph -> S_ [ ( A + T ) -> ( B + T ) ] ( F ` x ) _d x = S_ [ A -> B ] ( ( F ` ( y + T ) ) x. 1 ) _d y ) |
192 |
3
|
ditgpos |
|- ( ph -> S_ [ A -> B ] ( ( F ` ( y + T ) ) x. 1 ) _d y = S. ( A (,) B ) ( ( F ` ( y + T ) ) x. 1 ) _d y ) |
193 |
5
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> F : RR --> CC ) |
194 |
193 35
|
ffvelrnd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( F ` ( y + T ) ) e. CC ) |
195 |
|
1cnd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> 1 e. CC ) |
196 |
194 195
|
mulcld |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( ( F ` ( y + T ) ) x. 1 ) e. CC ) |
197 |
1 2 196
|
itgioo |
|- ( ph -> S. ( A (,) B ) ( ( F ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( ( F ` ( y + T ) ) x. 1 ) _d y ) |
198 |
|
fvoveq1 |
|- ( y = x -> ( F ` ( y + T ) ) = ( F ` ( x + T ) ) ) |
199 |
198
|
oveq1d |
|- ( y = x -> ( ( F ` ( y + T ) ) x. 1 ) = ( ( F ` ( x + T ) ) x. 1 ) ) |
200 |
199
|
cbvitgv |
|- S. ( A [,] B ) ( ( F ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( ( F ` ( x + T ) ) x. 1 ) _d x |
201 |
5
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC ) |
202 |
26
|
sselda |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR ) |
203 |
8
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> T e. RR ) |
204 |
202 203
|
readdcld |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( x + T ) e. RR ) |
205 |
201 204
|
ffvelrnd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` ( x + T ) ) e. CC ) |
206 |
205
|
mulid1d |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( F ` ( x + T ) ) x. 1 ) = ( F ` ( x + T ) ) ) |
207 |
206 6
|
eqtrd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( F ` ( x + T ) ) x. 1 ) = ( F ` x ) ) |
208 |
207
|
itgeq2dv |
|- ( ph -> S. ( A [,] B ) ( ( F ` ( x + T ) ) x. 1 ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |
209 |
200 208
|
eqtrid |
|- ( ph -> S. ( A [,] B ) ( ( F ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( F ` x ) _d x ) |
210 |
192 197 209
|
3eqtrd |
|- ( ph -> S_ [ A -> B ] ( ( F ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( F ` x ) _d x ) |
211 |
21 191 210
|
3eqtrd |
|- ( ph -> S. ( ( A + T ) [,] ( B + T ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |