Metamath Proof Explorer

Theorem iccshftl

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

Ref Expression
Hypotheses iccshftl.1 
|- ( A - R ) = C
iccshftl.2 
|- ( B - R ) = D
Assertion iccshftl 
|- ( ( ( 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 iccshftl.1 
 |-  ( A - R ) = C
2 iccshftl.2 
 |-  ( B - R ) = D
3 simpl 
 |-  ( ( X e. RR /\ R e. RR ) -> X e. RR )
4 resubcl 
 |-  ( ( 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 lesub1 
 |-  ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
8 7 3expb 
 |-  ( ( A e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
9 8 adantlr 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
10 1 breq1i 
 |-  ( ( A - R ) <_ ( X - R ) <-> C <_ ( X - R ) )
11 9 10 bitrdi 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> C <_ ( X - R ) ) )
12 lesub1 
 |-  ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
13 12 3expb 
 |-  ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
14 13 an12s 
 |-  ( ( B e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
15 14 adantll 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
16 2 breq2i 
 |-  ( ( X - R ) <_ ( B - R ) <-> ( X - R ) <_ D )
17 15 16 bitrdi 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ D ) )
18 6 11 17 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 ) ) )
19 elicc2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
20 19 adantr 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
21 resubcl 
 |-  ( ( A e. RR /\ R e. RR ) -> ( A - R ) e. RR )
22 1 21 eqeltrrid 
 |-  ( ( A e. RR /\ R e. RR ) -> C e. RR )
23 resubcl 
 |-  ( ( B e. RR /\ R e. RR ) -> ( B - R ) e. RR )
24 2 23 eqeltrrid 
 |-  ( ( B e. RR /\ R e. RR ) -> D e. RR )
25 elicc2 
 |-  ( ( C e. RR /\ D e. RR ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
26 22 24 25 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 ) ) )
27 26 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 ) ) )
28 27 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 ) ) )
29 18 20 28 3bitr4d 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )