Step |
Hyp |
Ref |
Expression |
1 |
|
ioombl |
|
2 |
|
mblvol |
|
3 |
1 2
|
ax-mp |
|
4 |
|
iccmbl |
|
5 |
|
mblvol |
|
6 |
4 5
|
syl |
|
7 |
6
|
3adant3 |
|
8 |
1
|
a1i |
|
9 |
|
prssi |
|
10 |
9
|
3adant3 |
|
11 |
|
prfi |
|
12 |
|
ovolfi |
|
13 |
11 10 12
|
sylancr |
|
14 |
|
nulmbl |
|
15 |
10 13 14
|
syl2anc |
|
16 |
|
df-pr |
|
17 |
16
|
ineq2i |
|
18 |
|
indi |
|
19 |
17 18
|
eqtri |
|
20 |
|
simp1 |
|
21 |
20
|
ltnrd |
|
22 |
|
eliooord |
|
23 |
22
|
simpld |
|
24 |
21 23
|
nsyl |
|
25 |
|
disjsn |
|
26 |
24 25
|
sylibr |
|
27 |
|
simp2 |
|
28 |
27
|
ltnrd |
|
29 |
|
eliooord |
|
30 |
29
|
simprd |
|
31 |
28 30
|
nsyl |
|
32 |
|
disjsn |
|
33 |
31 32
|
sylibr |
|
34 |
26 33
|
uneq12d |
|
35 |
|
un0 |
|
36 |
34 35
|
eqtrdi |
|
37 |
19 36
|
syl5eq |
|
38 |
|
ioossicc |
|
39 |
|
iccssre |
|
40 |
39
|
3adant3 |
|
41 |
|
ovolicc |
|
42 |
27 20
|
resubcld |
|
43 |
41 42
|
eqeltrd |
|
44 |
|
ovolsscl |
|
45 |
38 40 43 44
|
mp3an2i |
|
46 |
3 45
|
eqeltrid |
|
47 |
|
mblvol |
|
48 |
15 47
|
syl |
|
49 |
48 13
|
eqtrd |
|
50 |
|
0re |
|
51 |
49 50
|
eqeltrdi |
|
52 |
|
volun |
|
53 |
8 15 37 46 51 52
|
syl32anc |
|
54 |
|
rexr |
|
55 |
|
rexr |
|
56 |
|
id |
|
57 |
|
prunioo |
|
58 |
54 55 56 57
|
syl3an |
|
59 |
58
|
fveq2d |
|
60 |
49
|
oveq2d |
|
61 |
46
|
recnd |
|
62 |
61
|
addid1d |
|
63 |
60 62
|
eqtrd |
|
64 |
53 59 63
|
3eqtr3d |
|
65 |
7 64 41
|
3eqtr3d |
|
66 |
3 65
|
syl5eqr |
|