Step |
Hyp |
Ref |
Expression |
1 |
|
voliooico.1 |
|- ( ph -> A e. RR ) |
2 |
|
voliooico.2 |
|- ( ph -> B e. RR ) |
3 |
|
iftrue |
|- ( A < B -> if ( A < B , ( B - A ) , 0 ) = ( B - A ) ) |
4 |
3
|
adantl |
|- ( ( ( ph /\ A <_ B ) /\ A < B ) -> if ( A < B , ( B - A ) , 0 ) = ( B - A ) ) |
5 |
2
|
recnd |
|- ( ph -> B e. CC ) |
6 |
5
|
subidd |
|- ( ph -> ( B - B ) = 0 ) |
7 |
6
|
eqcomd |
|- ( ph -> 0 = ( B - B ) ) |
8 |
7
|
ad2antrr |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> 0 = ( B - B ) ) |
9 |
|
iffalse |
|- ( -. A < B -> if ( A < B , ( B - A ) , 0 ) = 0 ) |
10 |
9
|
adantl |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> if ( A < B , ( B - A ) , 0 ) = 0 ) |
11 |
|
simpll |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> ph ) |
12 |
11 1
|
syl |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> A e. RR ) |
13 |
11 2
|
syl |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> B e. RR ) |
14 |
|
simpr |
|- ( ( ph /\ A <_ B ) -> A <_ B ) |
15 |
14
|
adantr |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> A <_ B ) |
16 |
|
simpr |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> -. A < B ) |
17 |
12 13 15 16
|
lenlteq |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> A = B ) |
18 |
|
oveq2 |
|- ( A = B -> ( B - A ) = ( B - B ) ) |
19 |
18
|
adantl |
|- ( ( ph /\ A = B ) -> ( B - A ) = ( B - B ) ) |
20 |
11 17 19
|
syl2anc |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> ( B - A ) = ( B - B ) ) |
21 |
8 10 20
|
3eqtr4d |
|- ( ( ( ph /\ A <_ B ) /\ -. A < B ) -> if ( A < B , ( B - A ) , 0 ) = ( B - A ) ) |
22 |
4 21
|
pm2.61dan |
|- ( ( ph /\ A <_ B ) -> if ( A < B , ( B - A ) , 0 ) = ( B - A ) ) |
23 |
22
|
eqcomd |
|- ( ( ph /\ A <_ B ) -> ( B - A ) = if ( A < B , ( B - A ) , 0 ) ) |
24 |
1
|
adantr |
|- ( ( ph /\ A <_ B ) -> A e. RR ) |
25 |
2
|
adantr |
|- ( ( ph /\ A <_ B ) -> B e. RR ) |
26 |
|
volioo |
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) ) |
27 |
24 25 14 26
|
syl3anc |
|- ( ( ph /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) ) |
28 |
|
volico |
|- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A [,) B ) ) = if ( A < B , ( B - A ) , 0 ) ) |
29 |
1 2 28
|
syl2anc |
|- ( ph -> ( vol ` ( A [,) B ) ) = if ( A < B , ( B - A ) , 0 ) ) |
30 |
29
|
adantr |
|- ( ( ph /\ A <_ B ) -> ( vol ` ( A [,) B ) ) = if ( A < B , ( B - A ) , 0 ) ) |
31 |
23 27 30
|
3eqtr4d |
|- ( ( ph /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( vol ` ( A [,) B ) ) ) |
32 |
|
simpl |
|- ( ( ph /\ -. A <_ B ) -> ph ) |
33 |
|
simpr |
|- ( ( ph /\ -. A <_ B ) -> -. A <_ B ) |
34 |
32 2
|
syl |
|- ( ( ph /\ -. A <_ B ) -> B e. RR ) |
35 |
32 1
|
syl |
|- ( ( ph /\ -. A <_ B ) -> A e. RR ) |
36 |
34 35
|
ltnled |
|- ( ( ph /\ -. A <_ B ) -> ( B < A <-> -. A <_ B ) ) |
37 |
33 36
|
mpbird |
|- ( ( ph /\ -. A <_ B ) -> B < A ) |
38 |
2
|
adantr |
|- ( ( ph /\ B < A ) -> B e. RR ) |
39 |
1
|
adantr |
|- ( ( ph /\ B < A ) -> A e. RR ) |
40 |
|
simpr |
|- ( ( ph /\ B < A ) -> B < A ) |
41 |
38 39 40
|
ltled |
|- ( ( ph /\ B < A ) -> B <_ A ) |
42 |
39
|
rexrd |
|- ( ( ph /\ B < A ) -> A e. RR* ) |
43 |
38
|
rexrd |
|- ( ( ph /\ B < A ) -> B e. RR* ) |
44 |
|
ioo0 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) ) |
45 |
42 43 44
|
syl2anc |
|- ( ( ph /\ B < A ) -> ( ( A (,) B ) = (/) <-> B <_ A ) ) |
46 |
41 45
|
mpbird |
|- ( ( ph /\ B < A ) -> ( A (,) B ) = (/) ) |
47 |
|
ico0 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) ) |
48 |
42 43 47
|
syl2anc |
|- ( ( ph /\ B < A ) -> ( ( A [,) B ) = (/) <-> B <_ A ) ) |
49 |
41 48
|
mpbird |
|- ( ( ph /\ B < A ) -> ( A [,) B ) = (/) ) |
50 |
46 49
|
eqtr4d |
|- ( ( ph /\ B < A ) -> ( A (,) B ) = ( A [,) B ) ) |
51 |
50
|
fveq2d |
|- ( ( ph /\ B < A ) -> ( vol ` ( A (,) B ) ) = ( vol ` ( A [,) B ) ) ) |
52 |
32 37 51
|
syl2anc |
|- ( ( ph /\ -. A <_ B ) -> ( vol ` ( A (,) B ) ) = ( vol ` ( A [,) B ) ) ) |
53 |
31 52
|
pm2.61dan |
|- ( ph -> ( vol ` ( A (,) B ) ) = ( vol ` ( A [,) B ) ) ) |