Step |
Hyp |
Ref |
Expression |
1 |
|
incom |
|- ( ( A (,) B ) i^i ( B [,) C ) ) = ( ( B [,) C ) i^i ( A (,) B ) ) |
2 |
|
joiniooico |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( ( ( A (,) B ) i^i ( B [,) C ) ) = (/) /\ ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) ) ) |
3 |
2
|
anassrs |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( ( A (,) B ) i^i ( B [,) C ) ) = (/) /\ ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) ) ) |
4 |
3
|
simpld |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( B [,) C ) ) = (/) ) |
5 |
1 4
|
syl5eqr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) i^i ( A (,) B ) ) = (/) ) |
6 |
3
|
simprd |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) ) |
7 |
|
uncom |
|- ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) B ) u. ( B [,) C ) ) |
8 |
7
|
a1i |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) B ) u. ( B [,) C ) ) ) |
9 |
|
simpll1 |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> A e. RR* ) |
10 |
|
simpl3 |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> C e. RR* ) |
11 |
10
|
adantr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> C e. RR* ) |
12 |
9
|
xrleidd |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> A <_ A ) |
13 |
|
simpr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> B <_ C ) |
14 |
|
ioossioo |
|- ( ( ( A e. RR* /\ C e. RR* ) /\ ( A <_ A /\ B <_ C ) ) -> ( A (,) B ) C_ ( A (,) C ) ) |
15 |
9 11 12 13 14
|
syl22anc |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) ) |
16 |
|
ssequn2 |
|- ( ( A (,) B ) C_ ( A (,) C ) <-> ( ( A (,) C ) u. ( A (,) B ) ) = ( A (,) C ) ) |
17 |
15 16
|
sylib |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) C ) u. ( A (,) B ) ) = ( A (,) C ) ) |
18 |
6 8 17
|
3eqtr4d |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) C ) u. ( A (,) B ) ) ) |
19 |
|
difeq |
|- ( ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) <-> ( ( ( B [,) C ) i^i ( A (,) B ) ) = (/) /\ ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) C ) u. ( A (,) B ) ) ) ) |
20 |
5 18 19
|
sylanbrc |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) ) |
21 |
|
simpll1 |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> A e. RR* ) |
22 |
|
simpl2 |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> B e. RR* ) |
23 |
22
|
adantr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> B e. RR* ) |
24 |
21
|
xrleidd |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> A <_ A ) |
25 |
10
|
adantr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C e. RR* ) |
26 |
|
simpr |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C < B ) |
27 |
25 23 26
|
xrltled |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C <_ B ) |
28 |
|
ioossioo |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ A /\ C <_ B ) ) -> ( A (,) C ) C_ ( A (,) B ) ) |
29 |
21 23 24 27 28
|
syl22anc |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( A (,) C ) C_ ( A (,) B ) ) |
30 |
|
ssdif0 |
|- ( ( A (,) C ) C_ ( A (,) B ) <-> ( ( A (,) C ) \ ( A (,) B ) ) = (/) ) |
31 |
29 30
|
sylib |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = (/) ) |
32 |
|
ico0 |
|- ( ( B e. RR* /\ C e. RR* ) -> ( ( B [,) C ) = (/) <-> C <_ B ) ) |
33 |
32
|
biimpar |
|- ( ( ( B e. RR* /\ C e. RR* ) /\ C <_ B ) -> ( B [,) C ) = (/) ) |
34 |
23 25 27 33
|
syl21anc |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( B [,) C ) = (/) ) |
35 |
31 34
|
eqtr4d |
|- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) ) |
36 |
|
xrlelttric |
|- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C \/ C < B ) ) |
37 |
22 10 36
|
syl2anc |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( B <_ C \/ C < B ) ) |
38 |
20 35 37
|
mpjaodan |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) ) |