Step |
Hyp |
Ref |
Expression |
1 |
|
simpll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR ) |
2 |
|
simprl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR ) |
3 |
|
simplr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR ) |
4 |
|
ltsub1 |
|- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A - B ) < ( C - B ) ) ) |
5 |
1 2 3 4
|
syl3anc |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A - B ) < ( C - B ) ) ) |
6 |
|
simprr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR ) |
7 |
|
ltsub2 |
|- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D < B <-> ( C - B ) < ( C - D ) ) ) |
8 |
6 3 2 7
|
syl3anc |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( D < B <-> ( C - B ) < ( C - D ) ) ) |
9 |
5 8
|
anbi12d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ D < B ) <-> ( ( A - B ) < ( C - B ) /\ ( C - B ) < ( C - D ) ) ) ) |
10 |
|
resubcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR ) |
11 |
10
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR ) |
12 |
2 3
|
resubcld |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR ) |
13 |
|
resubcl |
|- ( ( C e. RR /\ D e. RR ) -> ( C - D ) e. RR ) |
14 |
13
|
adantl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - D ) e. RR ) |
15 |
|
lttr |
|- ( ( ( A - B ) e. RR /\ ( C - B ) e. RR /\ ( C - D ) e. RR ) -> ( ( ( A - B ) < ( C - B ) /\ ( C - B ) < ( C - D ) ) -> ( A - B ) < ( C - D ) ) ) |
16 |
11 12 14 15
|
syl3anc |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( A - B ) < ( C - B ) /\ ( C - B ) < ( C - D ) ) -> ( A - B ) < ( C - D ) ) ) |
17 |
9 16
|
sylbid |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ D < B ) -> ( A - B ) < ( C - D ) ) ) |