Step |
Hyp |
Ref |
Expression |
1 |
|
itgsplitioo.1 |
|
2 |
|
itgsplitioo.2 |
|
3 |
|
itgsplitioo.3 |
|
4 |
|
itgsplitioo.4 |
|
5 |
|
itgsplitioo.5 |
|
6 |
|
itgsplitioo.6 |
|
7 |
|
elicc2 |
|
8 |
1 2 7
|
syl2anc |
|
9 |
3 8
|
mpbid |
|
10 |
9
|
simp2d |
|
11 |
9
|
simp1d |
|
12 |
1 11
|
leloed |
|
13 |
10 12
|
mpbid |
|
14 |
13
|
ord |
|
15 |
1
|
rexrd |
|
16 |
|
iooss1 |
|
17 |
15 10 16
|
syl2anc |
|
18 |
17
|
sselda |
|
19 |
18 4
|
syldan |
|
20 |
19 6
|
itgcl |
|
21 |
20
|
addid2d |
|
22 |
21
|
eqcomd |
|
23 |
|
oveq1 |
|
24 |
|
itgeq1 |
|
25 |
23 24
|
syl |
|
26 |
|
oveq1 |
|
27 |
|
iooid |
|
28 |
26 27
|
eqtrdi |
|
29 |
|
itgeq1 |
|
30 |
28 29
|
syl |
|
31 |
|
itg0 |
|
32 |
30 31
|
eqtrdi |
|
33 |
32
|
oveq1d |
|
34 |
25 33
|
eqeq12d |
|
35 |
22 34
|
syl5ibrcom |
|
36 |
14 35
|
syld |
|
37 |
9
|
simp3d |
|
38 |
11 2
|
leloed |
|
39 |
37 38
|
mpbid |
|
40 |
39
|
ord |
|
41 |
2
|
rexrd |
|
42 |
|
iooss2 |
|
43 |
41 37 42
|
syl2anc |
|
44 |
43
|
sselda |
|
45 |
44 4
|
syldan |
|
46 |
45 5
|
itgcl |
|
47 |
46
|
addid1d |
|
48 |
47
|
eqcomd |
|
49 |
|
oveq2 |
|
50 |
|
itgeq1 |
|
51 |
49 50
|
syl |
|
52 |
|
oveq2 |
|
53 |
27 52
|
syl5eqr |
|
54 |
|
itgeq1 |
|
55 |
53 54
|
syl |
|
56 |
31 55
|
syl5eqr |
|
57 |
56
|
oveq2d |
|
58 |
51 57
|
eqeq12d |
|
59 |
48 58
|
syl5ibcom |
|
60 |
40 59
|
syld |
|
61 |
|
indir |
|
62 |
11
|
rexrd |
|
63 |
15 62
|
jca |
|
64 |
63
|
adantr |
|
65 |
62 41
|
jca |
|
66 |
65
|
adantr |
|
67 |
11
|
adantr |
|
68 |
67
|
leidd |
|
69 |
|
ioodisj |
|
70 |
64 66 68 69
|
syl21anc |
|
71 |
|
incom |
|
72 |
67
|
ltnrd |
|
73 |
|
eliooord |
|
74 |
73
|
simpld |
|
75 |
72 74
|
nsyl |
|
76 |
|
disjsn |
|
77 |
75 76
|
sylibr |
|
78 |
71 77
|
syl5eq |
|
79 |
70 78
|
uneq12d |
|
80 |
|
un0 |
|
81 |
79 80
|
eqtrdi |
|
82 |
61 81
|
syl5eq |
|
83 |
82
|
fveq2d |
|
84 |
|
ovol0 |
|
85 |
83 84
|
eqtrdi |
|
86 |
15 62 41
|
3jca |
|
87 |
|
ioojoin |
|
88 |
86 87
|
sylan |
|
89 |
88
|
eqcomd |
|
90 |
4
|
adantlr |
|
91 |
5
|
adantr |
|
92 |
|
ssun1 |
|
93 |
92
|
a1i |
|
94 |
|
ioossre |
|
95 |
94
|
a1i |
|
96 |
67
|
snssd |
|
97 |
95 96
|
unssd |
|
98 |
|
uncom |
|
99 |
98
|
difeq1i |
|
100 |
|
difun2 |
|
101 |
99 100
|
eqtri |
|
102 |
|
difss |
|
103 |
101 102
|
eqsstri |
|
104 |
|
ovolsn |
|
105 |
67 104
|
syl |
|
106 |
|
ovolssnul |
|
107 |
103 96 105 106
|
mp3an2i |
|
108 |
|
ssun1 |
|
109 |
108 88
|
sseqtrid |
|
110 |
109
|
sselda |
|
111 |
110 90
|
syldan |
|
112 |
93 97 107 111
|
itgss3 |
|
113 |
112
|
simpld |
|
114 |
91 113
|
mpbid |
|
115 |
6
|
adantr |
|
116 |
85 89 90 114 115
|
itgsplit |
|
117 |
112
|
simprd |
|
118 |
117
|
oveq1d |
|
119 |
116 118
|
eqtr4d |
|
120 |
119
|
ex |
|
121 |
36 60 120
|
ecased |
|