Metamath Proof Explorer

Theorem iccdil

Description: Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Hypotheses iccdil.1 
|- ( A x. R ) = C
iccdil.2 
|- ( B x. R ) = D
Assertion iccdil 
|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )

Proof

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 ) ) )