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Metamath Proof Explorer

Theorem icccntr

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

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

Proof

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 ) ) )