Step |
Hyp |
Ref |
Expression |
1 |
|
itgsplitioo.1 |
|- ( ph -> A e. RR ) |
2 |
|
itgsplitioo.2 |
|- ( ph -> C e. RR ) |
3 |
|
itgsplitioo.3 |
|- ( ph -> B e. ( A [,] C ) ) |
4 |
|
itgsplitioo.4 |
|- ( ( ph /\ x e. ( A (,) C ) ) -> D e. CC ) |
5 |
|
itgsplitioo.5 |
|- ( ph -> ( x e. ( A (,) B ) |-> D ) e. L^1 ) |
6 |
|
itgsplitioo.6 |
|- ( ph -> ( x e. ( B (,) C ) |-> D ) e. L^1 ) |
7 |
|
elicc2 |
|- ( ( A e. RR /\ C e. RR ) -> ( B e. ( A [,] C ) <-> ( B e. RR /\ A <_ B /\ B <_ C ) ) ) |
8 |
1 2 7
|
syl2anc |
|- ( ph -> ( B e. ( A [,] C ) <-> ( B e. RR /\ A <_ B /\ B <_ C ) ) ) |
9 |
3 8
|
mpbid |
|- ( ph -> ( B e. RR /\ A <_ B /\ B <_ C ) ) |
10 |
9
|
simp2d |
|- ( ph -> A <_ B ) |
11 |
9
|
simp1d |
|- ( ph -> B e. RR ) |
12 |
1 11
|
leloed |
|- ( ph -> ( A <_ B <-> ( A < B \/ A = B ) ) ) |
13 |
10 12
|
mpbid |
|- ( ph -> ( A < B \/ A = B ) ) |
14 |
13
|
ord |
|- ( ph -> ( -. A < B -> A = B ) ) |
15 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
16 |
|
iooss1 |
|- ( ( A e. RR* /\ A <_ B ) -> ( B (,) C ) C_ ( A (,) C ) ) |
17 |
15 10 16
|
syl2anc |
|- ( ph -> ( B (,) C ) C_ ( A (,) C ) ) |
18 |
17
|
sselda |
|- ( ( ph /\ x e. ( B (,) C ) ) -> x e. ( A (,) C ) ) |
19 |
18 4
|
syldan |
|- ( ( ph /\ x e. ( B (,) C ) ) -> D e. CC ) |
20 |
19 6
|
itgcl |
|- ( ph -> S. ( B (,) C ) D _d x e. CC ) |
21 |
20
|
addid2d |
|- ( ph -> ( 0 + S. ( B (,) C ) D _d x ) = S. ( B (,) C ) D _d x ) |
22 |
21
|
eqcomd |
|- ( ph -> S. ( B (,) C ) D _d x = ( 0 + S. ( B (,) C ) D _d x ) ) |
23 |
|
oveq1 |
|- ( A = B -> ( A (,) C ) = ( B (,) C ) ) |
24 |
|
itgeq1 |
|- ( ( A (,) C ) = ( B (,) C ) -> S. ( A (,) C ) D _d x = S. ( B (,) C ) D _d x ) |
25 |
23 24
|
syl |
|- ( A = B -> S. ( A (,) C ) D _d x = S. ( B (,) C ) D _d x ) |
26 |
|
oveq1 |
|- ( A = B -> ( A (,) B ) = ( B (,) B ) ) |
27 |
|
iooid |
|- ( B (,) B ) = (/) |
28 |
26 27
|
eqtrdi |
|- ( A = B -> ( A (,) B ) = (/) ) |
29 |
|
itgeq1 |
|- ( ( A (,) B ) = (/) -> S. ( A (,) B ) D _d x = S. (/) D _d x ) |
30 |
28 29
|
syl |
|- ( A = B -> S. ( A (,) B ) D _d x = S. (/) D _d x ) |
31 |
|
itg0 |
|- S. (/) D _d x = 0 |
32 |
30 31
|
eqtrdi |
|- ( A = B -> S. ( A (,) B ) D _d x = 0 ) |
33 |
32
|
oveq1d |
|- ( A = B -> ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) = ( 0 + S. ( B (,) C ) D _d x ) ) |
34 |
25 33
|
eqeq12d |
|- ( A = B -> ( S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) <-> S. ( B (,) C ) D _d x = ( 0 + S. ( B (,) C ) D _d x ) ) ) |
35 |
22 34
|
syl5ibrcom |
|- ( ph -> ( A = B -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
36 |
14 35
|
syld |
|- ( ph -> ( -. A < B -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
37 |
9
|
simp3d |
|- ( ph -> B <_ C ) |
38 |
11 2
|
leloed |
|- ( ph -> ( B <_ C <-> ( B < C \/ B = C ) ) ) |
39 |
37 38
|
mpbid |
|- ( ph -> ( B < C \/ B = C ) ) |
40 |
39
|
ord |
|- ( ph -> ( -. B < C -> B = C ) ) |
41 |
2
|
rexrd |
|- ( ph -> C e. RR* ) |
42 |
|
iooss2 |
|- ( ( C e. RR* /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) ) |
43 |
41 37 42
|
syl2anc |
|- ( ph -> ( A (,) B ) C_ ( A (,) C ) ) |
44 |
43
|
sselda |
|- ( ( ph /\ x e. ( A (,) B ) ) -> x e. ( A (,) C ) ) |
45 |
44 4
|
syldan |
|- ( ( ph /\ x e. ( A (,) B ) ) -> D e. CC ) |
46 |
45 5
|
itgcl |
|- ( ph -> S. ( A (,) B ) D _d x e. CC ) |
47 |
46
|
addid1d |
|- ( ph -> ( S. ( A (,) B ) D _d x + 0 ) = S. ( A (,) B ) D _d x ) |
48 |
47
|
eqcomd |
|- ( ph -> S. ( A (,) B ) D _d x = ( S. ( A (,) B ) D _d x + 0 ) ) |
49 |
|
oveq2 |
|- ( B = C -> ( A (,) B ) = ( A (,) C ) ) |
50 |
|
itgeq1 |
|- ( ( A (,) B ) = ( A (,) C ) -> S. ( A (,) B ) D _d x = S. ( A (,) C ) D _d x ) |
51 |
49 50
|
syl |
|- ( B = C -> S. ( A (,) B ) D _d x = S. ( A (,) C ) D _d x ) |
52 |
|
oveq2 |
|- ( B = C -> ( B (,) B ) = ( B (,) C ) ) |
53 |
27 52
|
eqtr3id |
|- ( B = C -> (/) = ( B (,) C ) ) |
54 |
|
itgeq1 |
|- ( (/) = ( B (,) C ) -> S. (/) D _d x = S. ( B (,) C ) D _d x ) |
55 |
53 54
|
syl |
|- ( B = C -> S. (/) D _d x = S. ( B (,) C ) D _d x ) |
56 |
31 55
|
eqtr3id |
|- ( B = C -> 0 = S. ( B (,) C ) D _d x ) |
57 |
56
|
oveq2d |
|- ( B = C -> ( S. ( A (,) B ) D _d x + 0 ) = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) |
58 |
51 57
|
eqeq12d |
|- ( B = C -> ( S. ( A (,) B ) D _d x = ( S. ( A (,) B ) D _d x + 0 ) <-> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
59 |
48 58
|
syl5ibcom |
|- ( ph -> ( B = C -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
60 |
40 59
|
syld |
|- ( ph -> ( -. B < C -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
61 |
|
indir |
|- ( ( ( A (,) B ) u. { B } ) i^i ( B (,) C ) ) = ( ( ( A (,) B ) i^i ( B (,) C ) ) u. ( { B } i^i ( B (,) C ) ) ) |
62 |
11
|
rexrd |
|- ( ph -> B e. RR* ) |
63 |
15 62
|
jca |
|- ( ph -> ( A e. RR* /\ B e. RR* ) ) |
64 |
63
|
adantr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( A e. RR* /\ B e. RR* ) ) |
65 |
62 41
|
jca |
|- ( ph -> ( B e. RR* /\ C e. RR* ) ) |
66 |
65
|
adantr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( B e. RR* /\ C e. RR* ) ) |
67 |
11
|
adantr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> B e. RR ) |
68 |
67
|
leidd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> B <_ B ) |
69 |
|
ioodisj |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( B e. RR* /\ C e. RR* ) ) /\ B <_ B ) -> ( ( A (,) B ) i^i ( B (,) C ) ) = (/) ) |
70 |
64 66 68 69
|
syl21anc |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( A (,) B ) i^i ( B (,) C ) ) = (/) ) |
71 |
|
incom |
|- ( { B } i^i ( B (,) C ) ) = ( ( B (,) C ) i^i { B } ) |
72 |
67
|
ltnrd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> -. B < B ) |
73 |
|
eliooord |
|- ( B e. ( B (,) C ) -> ( B < B /\ B < C ) ) |
74 |
73
|
simpld |
|- ( B e. ( B (,) C ) -> B < B ) |
75 |
72 74
|
nsyl |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> -. B e. ( B (,) C ) ) |
76 |
|
disjsn |
|- ( ( ( B (,) C ) i^i { B } ) = (/) <-> -. B e. ( B (,) C ) ) |
77 |
75 76
|
sylibr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( B (,) C ) i^i { B } ) = (/) ) |
78 |
71 77
|
syl5eq |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( { B } i^i ( B (,) C ) ) = (/) ) |
79 |
70 78
|
uneq12d |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( ( A (,) B ) i^i ( B (,) C ) ) u. ( { B } i^i ( B (,) C ) ) ) = ( (/) u. (/) ) ) |
80 |
|
un0 |
|- ( (/) u. (/) ) = (/) |
81 |
79 80
|
eqtrdi |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( ( A (,) B ) i^i ( B (,) C ) ) u. ( { B } i^i ( B (,) C ) ) ) = (/) ) |
82 |
61 81
|
syl5eq |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( ( A (,) B ) u. { B } ) i^i ( B (,) C ) ) = (/) ) |
83 |
82
|
fveq2d |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( vol* ` ( ( ( A (,) B ) u. { B } ) i^i ( B (,) C ) ) ) = ( vol* ` (/) ) ) |
84 |
|
ovol0 |
|- ( vol* ` (/) ) = 0 |
85 |
83 84
|
eqtrdi |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( vol* ` ( ( ( A (,) B ) u. { B } ) i^i ( B (,) C ) ) ) = 0 ) |
86 |
15 62 41
|
3jca |
|- ( ph -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) ) |
87 |
|
ioojoin |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B < C ) ) -> ( ( ( A (,) B ) u. { B } ) u. ( B (,) C ) ) = ( A (,) C ) ) |
88 |
86 87
|
sylan |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( ( A (,) B ) u. { B } ) u. ( B (,) C ) ) = ( A (,) C ) ) |
89 |
88
|
eqcomd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( A (,) C ) = ( ( ( A (,) B ) u. { B } ) u. ( B (,) C ) ) ) |
90 |
4
|
adantlr |
|- ( ( ( ph /\ ( A < B /\ B < C ) ) /\ x e. ( A (,) C ) ) -> D e. CC ) |
91 |
5
|
adantr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( x e. ( A (,) B ) |-> D ) e. L^1 ) |
92 |
|
ssun1 |
|- ( A (,) B ) C_ ( ( A (,) B ) u. { B } ) |
93 |
92
|
a1i |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( A (,) B ) C_ ( ( A (,) B ) u. { B } ) ) |
94 |
|
ioossre |
|- ( A (,) B ) C_ RR |
95 |
94
|
a1i |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( A (,) B ) C_ RR ) |
96 |
67
|
snssd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> { B } C_ RR ) |
97 |
95 96
|
unssd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( A (,) B ) u. { B } ) C_ RR ) |
98 |
|
uncom |
|- ( ( A (,) B ) u. { B } ) = ( { B } u. ( A (,) B ) ) |
99 |
98
|
difeq1i |
|- ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) = ( ( { B } u. ( A (,) B ) ) \ ( A (,) B ) ) |
100 |
|
difun2 |
|- ( ( { B } u. ( A (,) B ) ) \ ( A (,) B ) ) = ( { B } \ ( A (,) B ) ) |
101 |
99 100
|
eqtri |
|- ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) = ( { B } \ ( A (,) B ) ) |
102 |
|
difss |
|- ( { B } \ ( A (,) B ) ) C_ { B } |
103 |
101 102
|
eqsstri |
|- ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) C_ { B } |
104 |
|
ovolsn |
|- ( B e. RR -> ( vol* ` { B } ) = 0 ) |
105 |
67 104
|
syl |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( vol* ` { B } ) = 0 ) |
106 |
|
ovolssnul |
|- ( ( ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) C_ { B } /\ { B } C_ RR /\ ( vol* ` { B } ) = 0 ) -> ( vol* ` ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) ) = 0 ) |
107 |
103 96 105 106
|
mp3an2i |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( vol* ` ( ( ( A (,) B ) u. { B } ) \ ( A (,) B ) ) ) = 0 ) |
108 |
|
ssun1 |
|- ( ( A (,) B ) u. { B } ) C_ ( ( ( A (,) B ) u. { B } ) u. ( B (,) C ) ) |
109 |
108 88
|
sseqtrid |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( A (,) B ) u. { B } ) C_ ( A (,) C ) ) |
110 |
109
|
sselda |
|- ( ( ( ph /\ ( A < B /\ B < C ) ) /\ x e. ( ( A (,) B ) u. { B } ) ) -> x e. ( A (,) C ) ) |
111 |
110 90
|
syldan |
|- ( ( ( ph /\ ( A < B /\ B < C ) ) /\ x e. ( ( A (,) B ) u. { B } ) ) -> D e. CC ) |
112 |
93 97 107 111
|
itgss3 |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( ( x e. ( A (,) B ) |-> D ) e. L^1 <-> ( x e. ( ( A (,) B ) u. { B } ) |-> D ) e. L^1 ) /\ S. ( A (,) B ) D _d x = S. ( ( A (,) B ) u. { B } ) D _d x ) ) |
113 |
112
|
simpld |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( ( x e. ( A (,) B ) |-> D ) e. L^1 <-> ( x e. ( ( A (,) B ) u. { B } ) |-> D ) e. L^1 ) ) |
114 |
91 113
|
mpbid |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( x e. ( ( A (,) B ) u. { B } ) |-> D ) e. L^1 ) |
115 |
6
|
adantr |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( x e. ( B (,) C ) |-> D ) e. L^1 ) |
116 |
85 89 90 114 115
|
itgsplit |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> S. ( A (,) C ) D _d x = ( S. ( ( A (,) B ) u. { B } ) D _d x + S. ( B (,) C ) D _d x ) ) |
117 |
112
|
simprd |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> S. ( A (,) B ) D _d x = S. ( ( A (,) B ) u. { B } ) D _d x ) |
118 |
117
|
oveq1d |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) = ( S. ( ( A (,) B ) u. { B } ) D _d x + S. ( B (,) C ) D _d x ) ) |
119 |
116 118
|
eqtr4d |
|- ( ( ph /\ ( A < B /\ B < C ) ) -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) |
120 |
119
|
ex |
|- ( ph -> ( ( A < B /\ B < C ) -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) ) |
121 |
36 60 120
|
ecased |
|- ( ph -> S. ( A (,) C ) D _d x = ( S. ( A (,) B ) D _d x + S. ( B (,) C ) D _d x ) ) |