Step |
Hyp |
Ref |
Expression |
1 |
|
elicc1 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ A <_ C /\ C <_ B ) ) ) |
2 |
|
simp1 |
|- ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> C e. RR* ) |
3 |
2
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> C e. RR* ) ) |
4 |
|
xrletr |
|- ( ( A e. RR* /\ C e. RR* /\ B e. RR* ) -> ( ( A <_ C /\ C <_ B ) -> A <_ B ) ) |
5 |
4
|
exp5o |
|- ( A e. RR* -> ( C e. RR* -> ( B e. RR* -> ( A <_ C -> ( C <_ B -> A <_ B ) ) ) ) ) |
6 |
5
|
com23 |
|- ( A e. RR* -> ( B e. RR* -> ( C e. RR* -> ( A <_ C -> ( C <_ B -> A <_ B ) ) ) ) ) |
7 |
6
|
imp5q |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> A <_ B ) ) |
8 |
|
df-ne |
|- ( C =/= A <-> -. C = A ) |
9 |
|
xrleltne |
|- ( ( A e. RR* /\ C e. RR* /\ A <_ C ) -> ( A < C <-> C =/= A ) ) |
10 |
9
|
biimprd |
|- ( ( A e. RR* /\ C e. RR* /\ A <_ C ) -> ( C =/= A -> A < C ) ) |
11 |
8 10
|
syl5bir |
|- ( ( A e. RR* /\ C e. RR* /\ A <_ C ) -> ( -. C = A -> A < C ) ) |
12 |
11
|
3adant3r3 |
|- ( ( A e. RR* /\ ( C e. RR* /\ A <_ C /\ C <_ B ) ) -> ( -. C = A -> A < C ) ) |
13 |
12
|
adantlr |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C <_ B ) ) -> ( -. C = A -> A < C ) ) |
14 |
|
eqcom |
|- ( C = B <-> B = C ) |
15 |
14
|
necon3bbii |
|- ( -. C = B <-> B =/= C ) |
16 |
|
xrleltne |
|- ( ( C e. RR* /\ B e. RR* /\ C <_ B ) -> ( C < B <-> B =/= C ) ) |
17 |
16
|
biimprd |
|- ( ( C e. RR* /\ B e. RR* /\ C <_ B ) -> ( B =/= C -> C < B ) ) |
18 |
15 17
|
syl5bi |
|- ( ( C e. RR* /\ B e. RR* /\ C <_ B ) -> ( -. C = B -> C < B ) ) |
19 |
18
|
3exp |
|- ( C e. RR* -> ( B e. RR* -> ( C <_ B -> ( -. C = B -> C < B ) ) ) ) |
20 |
19
|
com12 |
|- ( B e. RR* -> ( C e. RR* -> ( C <_ B -> ( -. C = B -> C < B ) ) ) ) |
21 |
20
|
imp32 |
|- ( ( B e. RR* /\ ( C e. RR* /\ C <_ B ) ) -> ( -. C = B -> C < B ) ) |
22 |
21
|
3adantr2 |
|- ( ( B e. RR* /\ ( C e. RR* /\ A <_ C /\ C <_ B ) ) -> ( -. C = B -> C < B ) ) |
23 |
22
|
adantll |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C <_ B ) ) -> ( -. C = B -> C < B ) ) |
24 |
13 23
|
anim12d |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C <_ B ) ) -> ( ( -. C = A /\ -. C = B ) -> ( A < C /\ C < B ) ) ) |
25 |
24
|
ex |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> ( ( -. C = A /\ -. C = B ) -> ( A < C /\ C < B ) ) ) ) |
26 |
|
df-or |
|- ( ( C = A \/ ( ( A < C /\ C < B ) \/ C = B ) ) <-> ( -. C = A -> ( ( A < C /\ C < B ) \/ C = B ) ) ) |
27 |
|
3orass |
|- ( ( C = A \/ ( A < C /\ C < B ) \/ C = B ) <-> ( C = A \/ ( ( A < C /\ C < B ) \/ C = B ) ) ) |
28 |
|
pm5.6 |
|- ( ( ( -. C = A /\ -. C = B ) -> ( A < C /\ C < B ) ) <-> ( -. C = A -> ( C = B \/ ( A < C /\ C < B ) ) ) ) |
29 |
|
orcom |
|- ( ( C = B \/ ( A < C /\ C < B ) ) <-> ( ( A < C /\ C < B ) \/ C = B ) ) |
30 |
29
|
imbi2i |
|- ( ( -. C = A -> ( C = B \/ ( A < C /\ C < B ) ) ) <-> ( -. C = A -> ( ( A < C /\ C < B ) \/ C = B ) ) ) |
31 |
28 30
|
bitri |
|- ( ( ( -. C = A /\ -. C = B ) -> ( A < C /\ C < B ) ) <-> ( -. C = A -> ( ( A < C /\ C < B ) \/ C = B ) ) ) |
32 |
26 27 31
|
3bitr4ri |
|- ( ( ( -. C = A /\ -. C = B ) -> ( A < C /\ C < B ) ) <-> ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) |
33 |
25 32
|
syl6ib |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) ) |
34 |
3 7 33
|
3jcad |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) -> ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) ) ) |
35 |
|
simp1 |
|- ( ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) -> C e. RR* ) |
36 |
35
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) -> C e. RR* ) ) |
37 |
|
xrleid |
|- ( A e. RR* -> A <_ A ) |
38 |
37
|
ad3antrrr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> A <_ A ) |
39 |
|
breq2 |
|- ( C = A -> ( A <_ C <-> A <_ A ) ) |
40 |
38 39
|
syl5ibrcom |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( C = A -> A <_ C ) ) |
41 |
|
xrltle |
|- ( ( A e. RR* /\ C e. RR* ) -> ( A < C -> A <_ C ) ) |
42 |
41
|
adantr |
|- ( ( ( A e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( A < C -> A <_ C ) ) |
43 |
42
|
adantllr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( A < C -> A <_ C ) ) |
44 |
43
|
adantrd |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( ( A < C /\ C < B ) -> A <_ C ) ) |
45 |
|
simpr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> A <_ B ) |
46 |
|
breq2 |
|- ( C = B -> ( A <_ C <-> A <_ B ) ) |
47 |
45 46
|
syl5ibrcom |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( C = B -> A <_ C ) ) |
48 |
40 44 47
|
3jaod |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( ( C = A \/ ( A < C /\ C < B ) \/ C = B ) -> A <_ C ) ) |
49 |
48
|
exp31 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. RR* -> ( A <_ B -> ( ( C = A \/ ( A < C /\ C < B ) \/ C = B ) -> A <_ C ) ) ) ) |
50 |
49
|
3impd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) -> A <_ C ) ) |
51 |
|
breq1 |
|- ( C = A -> ( C <_ B <-> A <_ B ) ) |
52 |
45 51
|
syl5ibrcom |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( C = A -> C <_ B ) ) |
53 |
|
xrltle |
|- ( ( C e. RR* /\ B e. RR* ) -> ( C < B -> C <_ B ) ) |
54 |
53
|
ancoms |
|- ( ( B e. RR* /\ C e. RR* ) -> ( C < B -> C <_ B ) ) |
55 |
54
|
adantld |
|- ( ( B e. RR* /\ C e. RR* ) -> ( ( A < C /\ C < B ) -> C <_ B ) ) |
56 |
55
|
adantll |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) -> ( ( A < C /\ C < B ) -> C <_ B ) ) |
57 |
56
|
adantr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( ( A < C /\ C < B ) -> C <_ B ) ) |
58 |
|
xrleid |
|- ( B e. RR* -> B <_ B ) |
59 |
58
|
ad3antlr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> B <_ B ) |
60 |
|
breq1 |
|- ( C = B -> ( C <_ B <-> B <_ B ) ) |
61 |
59 60
|
syl5ibrcom |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( C = B -> C <_ B ) ) |
62 |
52 57 61
|
3jaod |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ A <_ B ) -> ( ( C = A \/ ( A < C /\ C < B ) \/ C = B ) -> C <_ B ) ) |
63 |
62
|
exp31 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. RR* -> ( A <_ B -> ( ( C = A \/ ( A < C /\ C < B ) \/ C = B ) -> C <_ B ) ) ) ) |
64 |
63
|
3impd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) -> C <_ B ) ) |
65 |
36 50 64
|
3jcad |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) -> ( C e. RR* /\ A <_ C /\ C <_ B ) ) ) |
66 |
34 65
|
impbid |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C <_ B ) <-> ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) ) ) |
67 |
1 66
|
bitrd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) ) ) |