Step |
Hyp |
Ref |
Expression |
1 |
|
icccntr.1 |
|- ( A / R ) = C |
2 |
|
icccntr.2 |
|- ( B / R ) = D |
3 |
|
simpl |
|- ( ( X e. RR /\ R e. RR+ ) -> X e. RR ) |
4 |
|
rerpdivcl |
|- ( ( X e. RR /\ R e. RR+ ) -> ( X / R ) e. RR ) |
5 |
3 4
|
2thd |
|- ( ( X e. RR /\ R e. RR+ ) -> ( X e. RR <-> ( X / R ) e. RR ) ) |
6 |
5
|
adantl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. RR <-> ( X / R ) e. RR ) ) |
7 |
|
elrp |
|- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) ) |
8 |
|
lediv1 |
|- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) ) |
9 |
7 8
|
syl3an3b |
|- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) ) |
10 |
9
|
3expb |
|- ( ( A e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) ) |
11 |
10
|
adantlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) ) |
12 |
1
|
breq1i |
|- ( ( A / R ) <_ ( X / R ) <-> C <_ ( X / R ) ) |
13 |
11 12
|
bitrdi |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> C <_ ( X / R ) ) ) |
14 |
|
lediv1 |
|- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) ) |
15 |
7 14
|
syl3an3b |
|- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) ) |
16 |
15
|
3expb |
|- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) ) |
17 |
16
|
an12s |
|- ( ( B e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) ) |
18 |
17
|
adantll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) ) |
19 |
2
|
breq2i |
|- ( ( X / R ) <_ ( B / R ) <-> ( X / R ) <_ D ) |
20 |
18 19
|
bitrdi |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ D ) ) |
21 |
6 13 20
|
3anbi123d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X e. RR /\ A <_ X /\ X <_ B ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) ) |
22 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) ) |
23 |
22
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) ) |
24 |
|
rerpdivcl |
|- ( ( A e. RR /\ R e. RR+ ) -> ( A / R ) e. RR ) |
25 |
1 24
|
eqeltrrid |
|- ( ( A e. RR /\ R e. RR+ ) -> C e. RR ) |
26 |
|
rerpdivcl |
|- ( ( B e. RR /\ R e. RR+ ) -> ( B / R ) e. RR ) |
27 |
2 26
|
eqeltrrid |
|- ( ( B e. RR /\ R e. RR+ ) -> D e. RR ) |
28 |
|
elicc2 |
|- ( ( C e. RR /\ D e. RR ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) ) |
29 |
25 27 28
|
syl2an |
|- ( ( ( A e. RR /\ R e. RR+ ) /\ ( B e. RR /\ R e. RR+ ) ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) ) |
30 |
29
|
anandirs |
|- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR+ ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) ) |
31 |
30
|
adantrl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) ) |
32 |
21 23 31
|
3bitr4d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) ) |