Step |
Hyp |
Ref |
Expression |
1 |
|
iccdil.1 |
|- ( A x. R ) = C |
2 |
|
iccdil.2 |
|- ( B x. R ) = D |
3 |
|
simpl |
|- ( ( X e. RR /\ R e. RR+ ) -> X e. RR ) |
4 |
|
rpre |
|- ( R e. RR+ -> R e. RR ) |
5 |
|
remulcl |
|- ( ( X e. RR /\ R e. RR ) -> ( X x. R ) e. RR ) |
6 |
4 5
|
sylan2 |
|- ( ( X e. RR /\ R e. RR+ ) -> ( X x. R ) e. RR ) |
7 |
3 6
|
2thd |
|- ( ( X e. RR /\ R e. RR+ ) -> ( X e. RR <-> ( X x. R ) e. RR ) ) |
8 |
7
|
adantl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. RR <-> ( X x. R ) e. RR ) ) |
9 |
|
elrp |
|- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) ) |
10 |
|
lemul1 |
|- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) ) |
11 |
9 10
|
syl3an3b |
|- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) ) |
12 |
11
|
3expb |
|- ( ( A e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) ) |
13 |
12
|
adantlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) ) |
14 |
1
|
breq1i |
|- ( ( A x. R ) <_ ( X x. R ) <-> C <_ ( X x. R ) ) |
15 |
13 14
|
bitrdi |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> C <_ ( X x. R ) ) ) |
16 |
|
lemul1 |
|- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) ) |
17 |
9 16
|
syl3an3b |
|- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) ) |
18 |
17
|
3expb |
|- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) ) |
19 |
18
|
an12s |
|- ( ( B e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) ) |
20 |
19
|
adantll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) ) |
21 |
2
|
breq2i |
|- ( ( X x. R ) <_ ( B x. R ) <-> ( X x. R ) <_ D ) |
22 |
20 21
|
bitrdi |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ D ) ) |
23 |
8 15 22
|
3anbi123d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X e. RR /\ A <_ X /\ X <_ B ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
24 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) ) |
25 |
24
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) ) |
26 |
|
remulcl |
|- ( ( A e. RR /\ R e. RR ) -> ( A x. R ) e. RR ) |
27 |
1 26
|
eqeltrrid |
|- ( ( A e. RR /\ R e. RR ) -> C e. RR ) |
28 |
|
remulcl |
|- ( ( B e. RR /\ R e. RR ) -> ( B x. R ) e. RR ) |
29 |
2 28
|
eqeltrrid |
|- ( ( B e. RR /\ R e. RR ) -> D e. RR ) |
30 |
|
elicc2 |
|- ( ( C e. RR /\ D e. RR ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
31 |
27 29 30
|
syl2an |
|- ( ( ( A e. RR /\ R e. RR ) /\ ( B e. RR /\ R e. RR ) ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
32 |
31
|
anandirs |
|- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
33 |
4 32
|
sylan2 |
|- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR+ ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
34 |
33
|
adantrl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) ) |
35 |
23 25 34
|
3bitr4d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) ) |