Step |
Hyp |
Ref |
Expression |
1 |
|
itgiccshift.a |
|- ( ph -> A e. RR ) |
2 |
|
itgiccshift.b |
|- ( ph -> B e. RR ) |
3 |
|
itgiccshift.aleb |
|- ( ph -> A <_ B ) |
4 |
|
itgiccshift.f |
|- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) ) |
5 |
|
itgiccshift.t |
|- ( ph -> T e. RR+ ) |
6 |
|
itgiccshift.g |
|- G = ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( x - T ) ) ) |
7 |
5
|
rpred |
|- ( ph -> T e. RR ) |
8 |
1 2 7 3
|
leadd1dd |
|- ( ph -> ( A + T ) <_ ( B + T ) ) |
9 |
8
|
ditgpos |
|- ( ph -> S_ [ ( A + T ) -> ( B + T ) ] ( G ` x ) _d x = S. ( ( A + T ) (,) ( B + T ) ) ( G ` x ) _d x ) |
10 |
1 7
|
readdcld |
|- ( ph -> ( A + T ) e. RR ) |
11 |
2 7
|
readdcld |
|- ( ph -> ( B + T ) e. RR ) |
12 |
|
cncff |
|- ( F e. ( ( A [,] B ) -cn-> CC ) -> F : ( A [,] B ) --> CC ) |
13 |
4 12
|
syl |
|- ( ph -> F : ( A [,] B ) --> CC ) |
14 |
13
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> F : ( A [,] B ) --> CC ) |
15 |
1
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A e. RR ) |
16 |
2
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> B e. RR ) |
17 |
10
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) e. RR ) |
18 |
11
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( B + T ) e. RR ) |
19 |
|
simpr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
20 |
|
eliccre |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
21 |
17 18 19 20
|
syl3anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
22 |
7
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. RR ) |
23 |
21 22
|
resubcld |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. RR ) |
24 |
1
|
recnd |
|- ( ph -> A e. CC ) |
25 |
7
|
recnd |
|- ( ph -> T e. CC ) |
26 |
24 25
|
pncand |
|- ( ph -> ( ( A + T ) - T ) = A ) |
27 |
26
|
eqcomd |
|- ( ph -> A = ( ( A + T ) - T ) ) |
28 |
27
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A = ( ( A + T ) - T ) ) |
29 |
|
elicc2 |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
30 |
17 18 29
|
syl2anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
31 |
19 30
|
mpbid |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) |
32 |
31
|
simp2d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) <_ x ) |
33 |
17 21 22 32
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( A + T ) - T ) <_ ( x - T ) ) |
34 |
28 33
|
eqbrtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A <_ ( x - T ) ) |
35 |
31
|
simp3d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x <_ ( B + T ) ) |
36 |
21 18 22 35
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ ( ( B + T ) - T ) ) |
37 |
2
|
recnd |
|- ( ph -> B e. CC ) |
38 |
37 25
|
pncand |
|- ( ph -> ( ( B + T ) - T ) = B ) |
39 |
38
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( B + T ) - T ) = B ) |
40 |
36 39
|
breqtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ B ) |
41 |
15 16 23 34 40
|
eliccd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. ( A [,] B ) ) |
42 |
14 41
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( F ` ( x - T ) ) e. CC ) |
43 |
42 6
|
fmptd |
|- ( ph -> G : ( ( A + T ) [,] ( B + T ) ) --> CC ) |
44 |
43
|
ffvelrnda |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( G ` x ) e. CC ) |
45 |
10 11 44
|
itgioo |
|- ( ph -> S. ( ( A + T ) (,) ( B + T ) ) ( G ` x ) _d x = S. ( ( A + T ) [,] ( B + T ) ) ( G ` x ) _d x ) |
46 |
9 45
|
eqtr2d |
|- ( ph -> S. ( ( A + T ) [,] ( B + T ) ) ( G ` x ) _d x = S_ [ ( A + T ) -> ( B + T ) ] ( G ` x ) _d x ) |
47 |
|
eqid |
|- ( y e. CC |-> ( y + T ) ) = ( y e. CC |-> ( y + T ) ) |
48 |
47
|
addccncf |
|- ( T e. CC -> ( y e. CC |-> ( y + T ) ) e. ( CC -cn-> CC ) ) |
49 |
25 48
|
syl |
|- ( ph -> ( y e. CC |-> ( y + T ) ) e. ( CC -cn-> CC ) ) |
50 |
1 2
|
iccssred |
|- ( ph -> ( A [,] B ) C_ RR ) |
51 |
|
ax-resscn |
|- RR C_ CC |
52 |
50 51
|
sstrdi |
|- ( ph -> ( A [,] B ) C_ CC ) |
53 |
10 11
|
iccssred |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) C_ RR ) |
54 |
53 51
|
sstrdi |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) C_ CC ) |
55 |
10
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( A + T ) e. RR ) |
56 |
11
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( B + T ) e. RR ) |
57 |
50
|
sselda |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y e. RR ) |
58 |
7
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> T e. RR ) |
59 |
57 58
|
readdcld |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) e. RR ) |
60 |
1
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> A e. RR ) |
61 |
|
simpr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y e. ( A [,] B ) ) |
62 |
2
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> B e. RR ) |
63 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
64 |
60 62 63
|
syl2anc |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) ) |
65 |
61 64
|
mpbid |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y e. RR /\ A <_ y /\ y <_ B ) ) |
66 |
65
|
simp2d |
|- ( ( ph /\ y e. ( A [,] B ) ) -> A <_ y ) |
67 |
60 57 58 66
|
leadd1dd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( A + T ) <_ ( y + T ) ) |
68 |
65
|
simp3d |
|- ( ( ph /\ y e. ( A [,] B ) ) -> y <_ B ) |
69 |
57 62 58 68
|
leadd1dd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) <_ ( B + T ) ) |
70 |
55 56 59 67 69
|
eliccd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( y + T ) e. ( ( A + T ) [,] ( B + T ) ) ) |
71 |
47 49 52 54 70
|
cncfmptssg |
|- ( ph -> ( y e. ( A [,] B ) |-> ( y + T ) ) e. ( ( A [,] B ) -cn-> ( ( A + T ) [,] ( B + T ) ) ) ) |
72 |
|
fvoveq1 |
|- ( x = w -> ( F ` ( x - T ) ) = ( F ` ( w - T ) ) ) |
73 |
72
|
cbvmptv |
|- ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( x - T ) ) ) = ( w e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( w - T ) ) ) |
74 |
1 2 7
|
iccshift |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) = { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } ) |
75 |
74
|
mpteq1d |
|- ( ph -> ( w e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( w - T ) ) ) = ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) ) |
76 |
73 75
|
eqtrid |
|- ( ph -> ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( x - T ) ) ) = ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) ) |
77 |
6 76
|
eqtrid |
|- ( ph -> G = ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) ) |
78 |
|
eqeq1 |
|- ( w = x -> ( w = ( z + T ) <-> x = ( z + T ) ) ) |
79 |
78
|
rexbidv |
|- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. z e. ( A [,] B ) x = ( z + T ) ) ) |
80 |
|
oveq1 |
|- ( z = y -> ( z + T ) = ( y + T ) ) |
81 |
80
|
eqeq2d |
|- ( z = y -> ( x = ( z + T ) <-> x = ( y + T ) ) ) |
82 |
81
|
cbvrexvw |
|- ( E. z e. ( A [,] B ) x = ( z + T ) <-> E. y e. ( A [,] B ) x = ( y + T ) ) |
83 |
79 82
|
bitrdi |
|- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. y e. ( A [,] B ) x = ( y + T ) ) ) |
84 |
83
|
cbvrabv |
|- { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } = { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |
85 |
84
|
eqcomi |
|- { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } = { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } |
86 |
|
eqid |
|- ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) = ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) |
87 |
52 25 85 4 86
|
cncfshift |
|- ( ph -> ( w e. { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } |-> ( F ` ( w - T ) ) ) e. ( { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } -cn-> CC ) ) |
88 |
77 87
|
eqeltrd |
|- ( ph -> G e. ( { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } -cn-> CC ) ) |
89 |
43
|
feqmptd |
|- ( ph -> G = ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( G ` x ) ) ) |
90 |
74
|
eqcomd |
|- ( ph -> { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } = ( ( A + T ) [,] ( B + T ) ) ) |
91 |
90
|
oveq1d |
|- ( ph -> ( { x e. CC | E. y e. ( A [,] B ) x = ( y + T ) } -cn-> CC ) = ( ( ( A + T ) [,] ( B + T ) ) -cn-> CC ) ) |
92 |
88 89 91
|
3eltr3d |
|- ( ph -> ( x e. ( ( A + T ) [,] ( B + T ) ) |-> ( G ` x ) ) e. ( ( ( A + T ) [,] ( B + T ) ) -cn-> CC ) ) |
93 |
|
ioosscn |
|- ( A (,) B ) C_ CC |
94 |
93
|
a1i |
|- ( ph -> ( A (,) B ) C_ CC ) |
95 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
96 |
|
ssid |
|- CC C_ CC |
97 |
96
|
a1i |
|- ( ph -> CC C_ CC ) |
98 |
94 95 97
|
constcncfg |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC ) ) |
99 |
|
fconstmpt |
|- ( ( A (,) B ) X. { 1 } ) = ( y e. ( A (,) B ) |-> 1 ) |
100 |
|
ioombl |
|- ( A (,) B ) e. dom vol |
101 |
100
|
a1i |
|- ( ph -> ( A (,) B ) e. dom vol ) |
102 |
|
ioovolcl |
|- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A (,) B ) ) e. RR ) |
103 |
1 2 102
|
syl2anc |
|- ( ph -> ( vol ` ( A (,) B ) ) e. RR ) |
104 |
|
iblconst |
|- ( ( ( A (,) B ) e. dom vol /\ ( vol ` ( A (,) B ) ) e. RR /\ 1 e. CC ) -> ( ( A (,) B ) X. { 1 } ) e. L^1 ) |
105 |
101 103 95 104
|
syl3anc |
|- ( ph -> ( ( A (,) B ) X. { 1 } ) e. L^1 ) |
106 |
99 105
|
eqeltrrid |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. L^1 ) |
107 |
98 106
|
elind |
|- ( ph -> ( y e. ( A (,) B ) |-> 1 ) e. ( ( ( A (,) B ) -cn-> CC ) i^i L^1 ) ) |
108 |
50
|
resmptd |
|- ( ph -> ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) = ( y e. ( A [,] B ) |-> ( y + T ) ) ) |
109 |
108
|
eqcomd |
|- ( ph -> ( y e. ( A [,] B ) |-> ( y + T ) ) = ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) |
110 |
109
|
oveq2d |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( RR _D ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) ) |
111 |
51
|
a1i |
|- ( ph -> RR C_ CC ) |
112 |
111
|
sselda |
|- ( ( ph /\ y e. RR ) -> y e. CC ) |
113 |
25
|
adantr |
|- ( ( ph /\ y e. RR ) -> T e. CC ) |
114 |
112 113
|
addcld |
|- ( ( ph /\ y e. RR ) -> ( y + T ) e. CC ) |
115 |
114
|
fmpttd |
|- ( ph -> ( y e. RR |-> ( y + T ) ) : RR --> CC ) |
116 |
|
ssid |
|- RR C_ RR |
117 |
116
|
a1i |
|- ( ph -> RR C_ RR ) |
118 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
119 |
118
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
120 |
118 119
|
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 ) ) ) ) |
121 |
111 115 117 50 120
|
syl22anc |
|- ( ph -> ( RR _D ( ( y e. RR |-> ( y + T ) ) |` ( A [,] B ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
122 |
110 121
|
eqtrd |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
123 |
|
iccntr |
|- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
124 |
1 2 123
|
syl2anc |
|- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
125 |
124
|
reseq2d |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) = ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) ) |
126 |
|
reelprrecn |
|- RR e. { RR , CC } |
127 |
126
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
128 |
|
1cnd |
|- ( ( ph /\ y e. RR ) -> 1 e. CC ) |
129 |
127
|
dvmptid |
|- ( ph -> ( RR _D ( y e. RR |-> y ) ) = ( y e. RR |-> 1 ) ) |
130 |
|
0cnd |
|- ( ( ph /\ y e. RR ) -> 0 e. CC ) |
131 |
127 25
|
dvmptc |
|- ( ph -> ( RR _D ( y e. RR |-> T ) ) = ( y e. RR |-> 0 ) ) |
132 |
127 112 128 129 113 130 131
|
dvmptadd |
|- ( ph -> ( RR _D ( y e. RR |-> ( y + T ) ) ) = ( y e. RR |-> ( 1 + 0 ) ) ) |
133 |
132
|
reseq1d |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) = ( ( y e. RR |-> ( 1 + 0 ) ) |` ( A (,) B ) ) ) |
134 |
|
ioossre |
|- ( A (,) B ) C_ RR |
135 |
134
|
a1i |
|- ( ph -> ( A (,) B ) C_ RR ) |
136 |
135
|
resmptd |
|- ( ph -> ( ( y e. RR |-> ( 1 + 0 ) ) |` ( A (,) B ) ) = ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) ) |
137 |
|
1p0e1 |
|- ( 1 + 0 ) = 1 |
138 |
137
|
mpteq2i |
|- ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) = ( y e. ( A (,) B ) |-> 1 ) |
139 |
138
|
a1i |
|- ( ph -> ( y e. ( A (,) B ) |-> ( 1 + 0 ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
140 |
133 136 139
|
3eqtrd |
|- ( ph -> ( ( RR _D ( y e. RR |-> ( y + T ) ) ) |` ( A (,) B ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
141 |
122 125 140
|
3eqtrd |
|- ( ph -> ( RR _D ( y e. ( A [,] B ) |-> ( y + T ) ) ) = ( y e. ( A (,) B ) |-> 1 ) ) |
142 |
|
fveq2 |
|- ( x = ( y + T ) -> ( G ` x ) = ( G ` ( y + T ) ) ) |
143 |
|
oveq1 |
|- ( y = A -> ( y + T ) = ( A + T ) ) |
144 |
|
oveq1 |
|- ( y = B -> ( y + T ) = ( B + T ) ) |
145 |
1 2 3 71 92 107 141 142 143 144 10 11
|
itgsubsticc |
|- ( ph -> S_ [ ( A + T ) -> ( B + T ) ] ( G ` x ) _d x = S_ [ A -> B ] ( ( G ` ( y + T ) ) x. 1 ) _d y ) |
146 |
3
|
ditgpos |
|- ( ph -> S_ [ A -> B ] ( ( G ` ( y + T ) ) x. 1 ) _d y = S. ( A (,) B ) ( ( G ` ( y + T ) ) x. 1 ) _d y ) |
147 |
43
|
adantr |
|- ( ( ph /\ y e. ( A [,] B ) ) -> G : ( ( A + T ) [,] ( B + T ) ) --> CC ) |
148 |
147 70
|
ffvelrnd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( G ` ( y + T ) ) e. CC ) |
149 |
|
1cnd |
|- ( ( ph /\ y e. ( A [,] B ) ) -> 1 e. CC ) |
150 |
148 149
|
mulcld |
|- ( ( ph /\ y e. ( A [,] B ) ) -> ( ( G ` ( y + T ) ) x. 1 ) e. CC ) |
151 |
1 2 150
|
itgioo |
|- ( ph -> S. ( A (,) B ) ( ( G ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( ( G ` ( y + T ) ) x. 1 ) _d y ) |
152 |
|
fvoveq1 |
|- ( y = x -> ( G ` ( y + T ) ) = ( G ` ( x + T ) ) ) |
153 |
152
|
oveq1d |
|- ( y = x -> ( ( G ` ( y + T ) ) x. 1 ) = ( ( G ` ( x + T ) ) x. 1 ) ) |
154 |
153
|
cbvitgv |
|- S. ( A [,] B ) ( ( G ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( ( G ` ( x + T ) ) x. 1 ) _d x |
155 |
43
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> G : ( ( A + T ) [,] ( B + T ) ) --> CC ) |
156 |
10
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( A + T ) e. RR ) |
157 |
11
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( B + T ) e. RR ) |
158 |
50
|
sselda |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR ) |
159 |
7
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> T e. RR ) |
160 |
158 159
|
readdcld |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( x + T ) e. RR ) |
161 |
1
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR ) |
162 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
163 |
162
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR* ) |
164 |
2
|
rexrd |
|- ( ph -> B e. RR* ) |
165 |
164
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* ) |
166 |
|
simpr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) ) |
167 |
|
iccgelb |
|- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x ) |
168 |
163 165 166 167
|
syl3anc |
|- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x ) |
169 |
161 158 159 168
|
leadd1dd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( A + T ) <_ ( x + T ) ) |
170 |
2
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR ) |
171 |
|
iccleub |
|- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B ) |
172 |
163 165 166 171
|
syl3anc |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B ) |
173 |
158 170 159 172
|
leadd1dd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( x + T ) <_ ( B + T ) ) |
174 |
156 157 160 169 173
|
eliccd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( x + T ) e. ( ( A + T ) [,] ( B + T ) ) ) |
175 |
155 174
|
ffvelrnd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` ( x + T ) ) e. CC ) |
176 |
175
|
mulid1d |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( G ` ( x + T ) ) x. 1 ) = ( G ` ( x + T ) ) ) |
177 |
6 73
|
eqtri |
|- G = ( w e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( w - T ) ) ) |
178 |
177
|
a1i |
|- ( ( ph /\ x e. ( A [,] B ) ) -> G = ( w e. ( ( A + T ) [,] ( B + T ) ) |-> ( F ` ( w - T ) ) ) ) |
179 |
|
fvoveq1 |
|- ( w = ( x + T ) -> ( F ` ( w - T ) ) = ( F ` ( ( x + T ) - T ) ) ) |
180 |
158
|
recnd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC ) |
181 |
25
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> T e. CC ) |
182 |
180 181
|
pncand |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( x + T ) - T ) = x ) |
183 |
182
|
fveq2d |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` ( ( x + T ) - T ) ) = ( F ` x ) ) |
184 |
179 183
|
sylan9eqr |
|- ( ( ( ph /\ x e. ( A [,] B ) ) /\ w = ( x + T ) ) -> ( F ` ( w - T ) ) = ( F ` x ) ) |
185 |
13
|
ffvelrnda |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC ) |
186 |
178 184 174 185
|
fvmptd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` ( x + T ) ) = ( F ` x ) ) |
187 |
176 186
|
eqtrd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( G ` ( x + T ) ) x. 1 ) = ( F ` x ) ) |
188 |
187
|
itgeq2dv |
|- ( ph -> S. ( A [,] B ) ( ( G ` ( x + T ) ) x. 1 ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |
189 |
154 188
|
eqtrid |
|- ( ph -> S. ( A [,] B ) ( ( G ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( F ` x ) _d x ) |
190 |
146 151 189
|
3eqtrd |
|- ( ph -> S_ [ A -> B ] ( ( G ` ( y + T ) ) x. 1 ) _d y = S. ( A [,] B ) ( F ` x ) _d x ) |
191 |
46 145 190
|
3eqtrd |
|- ( ph -> S. ( ( A + T ) [,] ( B + T ) ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |