Step |
Hyp |
Ref |
Expression |
1 |
|
ftc2re.e |
|- E = ( C (,) D ) |
2 |
|
ftc2re.a |
|- ( ph -> A e. E ) |
3 |
|
ftc2re.b |
|- ( ph -> B e. E ) |
4 |
|
ftc2re.le |
|- ( ph -> A <_ B ) |
5 |
|
ftc2re.f |
|- ( ph -> F : E --> CC ) |
6 |
|
ftc2re.1 |
|- ( ph -> ( RR _D F ) e. ( E -cn-> CC ) ) |
7 |
|
ioossre |
|- ( C (,) D ) C_ RR |
8 |
1 7
|
eqsstri |
|- E C_ RR |
9 |
8
|
a1i |
|- ( ph -> E C_ RR ) |
10 |
9 2
|
sseldd |
|- ( ph -> A e. RR ) |
11 |
9 3
|
sseldd |
|- ( ph -> B e. RR ) |
12 |
|
ax-resscn |
|- RR C_ CC |
13 |
12
|
a1i |
|- ( ph -> RR C_ CC ) |
14 |
|
iccssre |
|- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR ) |
15 |
10 11 14
|
syl2anc |
|- ( ph -> ( A [,] B ) C_ RR ) |
16 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
17 |
16
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
18 |
16 17
|
dvres |
|- ( ( ( RR C_ CC /\ F : E --> CC ) /\ ( E C_ RR /\ ( A [,] B ) C_ RR ) ) -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
19 |
13 5 9 15 18
|
syl22anc |
|- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) ) |
20 |
|
iccntr |
|- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
21 |
10 11 20
|
syl2anc |
|- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) ) |
22 |
21
|
reseq2d |
|- ( ph -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( A (,) B ) ) ) |
23 |
19 22
|
eqtrd |
|- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( A (,) B ) ) ) |
24 |
|
ioossicc |
|- ( A (,) B ) C_ ( A [,] B ) |
25 |
24
|
a1i |
|- ( ph -> ( A (,) B ) C_ ( A [,] B ) ) |
26 |
1 2 3
|
fct2relem |
|- ( ph -> ( A [,] B ) C_ E ) |
27 |
25 26
|
sstrd |
|- ( ph -> ( A (,) B ) C_ E ) |
28 |
|
rescncf |
|- ( ( A (,) B ) C_ E -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) ) |
29 |
27 6 28
|
sylc |
|- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
30 |
23 29
|
eqeltrd |
|- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
31 |
|
ioombl |
|- ( A (,) B ) e. dom vol |
32 |
31
|
a1i |
|- ( ph -> ( A (,) B ) e. dom vol ) |
33 |
|
cnmbf |
|- ( ( ( A (,) B ) e. dom vol /\ ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn ) |
34 |
32 29 33
|
syl2anc |
|- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn ) |
35 |
|
dmres |
|- dom ( ( RR _D F ) |` ( A (,) B ) ) = ( ( A (,) B ) i^i dom ( RR _D F ) ) |
36 |
35
|
fveq2i |
|- ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) = ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) ) |
37 |
|
cncff |
|- ( ( RR _D F ) e. ( E -cn-> CC ) -> ( RR _D F ) : E --> CC ) |
38 |
6 37
|
syl |
|- ( ph -> ( RR _D F ) : E --> CC ) |
39 |
38
|
fdmd |
|- ( ph -> dom ( RR _D F ) = E ) |
40 |
39
|
ineq2d |
|- ( ph -> ( ( A (,) B ) i^i dom ( RR _D F ) ) = ( ( A (,) B ) i^i E ) ) |
41 |
|
df-ss |
|- ( ( A (,) B ) C_ E <-> ( ( A (,) B ) i^i E ) = ( A (,) B ) ) |
42 |
27 41
|
sylib |
|- ( ph -> ( ( A (,) B ) i^i E ) = ( A (,) B ) ) |
43 |
40 42
|
eqtrd |
|- ( ph -> ( ( A (,) B ) i^i dom ( RR _D F ) ) = ( A (,) B ) ) |
44 |
43
|
fveq2d |
|- ( ph -> ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) ) = ( vol ` ( A (,) B ) ) ) |
45 |
|
volioo |
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) ) |
46 |
10 11 4 45
|
syl3anc |
|- ( ph -> ( vol ` ( A (,) B ) ) = ( B - A ) ) |
47 |
11 10
|
resubcld |
|- ( ph -> ( B - A ) e. RR ) |
48 |
46 47
|
eqeltrd |
|- ( ph -> ( vol ` ( A (,) B ) ) e. RR ) |
49 |
44 48
|
eqeltrd |
|- ( ph -> ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) ) e. RR ) |
50 |
36 49
|
eqeltrid |
|- ( ph -> ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) e. RR ) |
51 |
|
rescncf |
|- ( ( A [,] B ) C_ E -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) ) |
52 |
26 51
|
syl |
|- ( ph -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) ) |
53 |
6 52
|
mpd |
|- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) |
54 |
|
cniccbdd |
|- ( ( A e. RR /\ B e. RR /\ ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) |
55 |
10 11 53 54
|
syl3anc |
|- ( ph -> E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) |
56 |
35 43
|
syl5eq |
|- ( ph -> dom ( ( RR _D F ) |` ( A (,) B ) ) = ( A (,) B ) ) |
57 |
56 25
|
eqsstrd |
|- ( ph -> dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) ) |
58 |
|
ssralv |
|- ( dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) ) |
59 |
57 58
|
syl |
|- ( ph -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) ) |
60 |
59
|
adantr |
|- ( ( ph /\ x e. RR ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) ) |
61 |
57
|
adantr |
|- ( ( ph /\ x e. RR ) -> dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) ) |
62 |
61
|
sselda |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. ( A [,] B ) ) |
63 |
|
fvres |
|- ( y e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( RR _D F ) ` y ) ) |
64 |
62 63
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( RR _D F ) ` y ) ) |
65 |
|
simpr |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) |
66 |
56
|
ad2antrr |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> dom ( ( RR _D F ) |` ( A (,) B ) ) = ( A (,) B ) ) |
67 |
65 66
|
eleqtrd |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. ( A (,) B ) ) |
68 |
|
fvres |
|- ( y e. ( A (,) B ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) = ( ( RR _D F ) ` y ) ) |
69 |
67 68
|
syl |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) = ( ( RR _D F ) ` y ) ) |
70 |
64 69
|
eqtr4d |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) |
71 |
70
|
fveq2d |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) = ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) ) |
72 |
71
|
breq1d |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x <-> ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) ) |
73 |
72
|
biimpd |
|- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) ) |
74 |
73
|
ralimdva |
|- ( ( ph /\ x e. RR ) -> ( A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) ) |
75 |
60 74
|
syld |
|- ( ( ph /\ x e. RR ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) ) |
76 |
75
|
reximdva |
|- ( ph -> ( E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) ) |
77 |
55 76
|
mpd |
|- ( ph -> E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) |
78 |
|
bddibl |
|- ( ( ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn /\ ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) e. RR /\ E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. L^1 ) |
79 |
34 50 77 78
|
syl3anc |
|- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. L^1 ) |
80 |
23 79
|
eqeltrd |
|- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) e. L^1 ) |
81 |
|
dvcn |
|- ( ( ( RR C_ CC /\ F : E --> CC /\ E C_ RR ) /\ dom ( RR _D F ) = E ) -> F e. ( E -cn-> CC ) ) |
82 |
13 5 9 39 81
|
syl31anc |
|- ( ph -> F e. ( E -cn-> CC ) ) |
83 |
|
rescncf |
|- ( ( A [,] B ) C_ E -> ( F e. ( E -cn-> CC ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) ) |
84 |
26 83
|
syl |
|- ( ph -> ( F e. ( E -cn-> CC ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) ) |
85 |
82 84
|
mpd |
|- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) |
86 |
10 11 4 30 80 85
|
ftc2 |
|- ( ph -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = ( ( ( F |` ( A [,] B ) ) ` B ) - ( ( F |` ( A [,] B ) ) ` A ) ) ) |
87 |
23
|
fveq1d |
|- ( ph -> ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( ( RR _D F ) |` ( A (,) B ) ) ` t ) ) |
88 |
|
fvres |
|- ( t e. ( A (,) B ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` t ) = ( ( RR _D F ) ` t ) ) |
89 |
87 88
|
sylan9eq |
|- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) ) |
90 |
89
|
ralrimiva |
|- ( ph -> A. t e. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) ) |
91 |
|
itgeq2 |
|- ( A. t e. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t ) |
92 |
90 91
|
syl |
|- ( ph -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t ) |
93 |
10
|
rexrd |
|- ( ph -> A e. RR* ) |
94 |
11
|
rexrd |
|- ( ph -> B e. RR* ) |
95 |
|
ubicc2 |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) ) |
96 |
93 94 4 95
|
syl3anc |
|- ( ph -> B e. ( A [,] B ) ) |
97 |
96
|
fvresd |
|- ( ph -> ( ( F |` ( A [,] B ) ) ` B ) = ( F ` B ) ) |
98 |
|
lbicc2 |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) ) |
99 |
93 94 4 98
|
syl3anc |
|- ( ph -> A e. ( A [,] B ) ) |
100 |
99
|
fvresd |
|- ( ph -> ( ( F |` ( A [,] B ) ) ` A ) = ( F ` A ) ) |
101 |
97 100
|
oveq12d |
|- ( ph -> ( ( ( F |` ( A [,] B ) ) ` B ) - ( ( F |` ( A [,] B ) ) ` A ) ) = ( ( F ` B ) - ( F ` A ) ) ) |
102 |
86 92 101
|
3eqtr3d |
|- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t = ( ( F ` B ) - ( F ` A ) ) ) |