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Metamath Proof Explorer

Theorem eliooshift

Description: Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses eliooshift.a 
|- ( ph -> A e. RR )
eliooshift.b 
|- ( ph -> B e. RR )
eliooshift.c 
|- ( ph -> C e. RR )
eliooshift.d 
|- ( ph -> D e. RR )
Assertion eliooshift 
|- ( ph -> ( A e. ( B (,) C ) <-> ( A + D ) e. ( ( B + D ) (,) ( C + D ) ) ) )

Proof

Step Hyp Ref Expression
1 eliooshift.a 
 |-  ( ph -> A e. RR )
2 eliooshift.b 
 |-  ( ph -> B e. RR )
3 eliooshift.c 
 |-  ( ph -> C e. RR )
4 eliooshift.d 
 |-  ( ph -> D e. RR )
5 1 4 readdcld 
 |-  ( ph -> ( A + D ) e. RR )
6 5 1 2thd 
 |-  ( ph -> ( ( A + D ) e. RR <-> A e. RR ) )
7 2 1 4 ltadd1d 
 |-  ( ph -> ( B < A <-> ( B + D ) < ( A + D ) ) )
8 7 bicomd 
 |-  ( ph -> ( ( B + D ) < ( A + D ) <-> B < A ) )
9 1 3 4 ltadd1d 
 |-  ( ph -> ( A < C <-> ( A + D ) < ( C + D ) ) )
10 9 bicomd 
 |-  ( ph -> ( ( A + D ) < ( C + D ) <-> A < C ) )
11 6 8 10 3anbi123d 
 |-  ( ph -> ( ( ( A + D ) e. RR /\ ( B + D ) < ( A + D ) /\ ( A + D ) < ( C + D ) ) <-> ( A e. RR /\ B < A /\ A < C ) ) )
12 2 4 readdcld 
 |-  ( ph -> ( B + D ) e. RR )
13 12 rexrd 
 |-  ( ph -> ( B + D ) e. RR* )
14 3 4 readdcld 
 |-  ( ph -> ( C + D ) e. RR )
15 14 rexrd 
 |-  ( ph -> ( C + D ) e. RR* )
16 elioo2 
 |-  ( ( ( B + D ) e. RR* /\ ( C + D ) e. RR* ) -> ( ( A + D ) e. ( ( B + D ) (,) ( C + D ) ) <-> ( ( A + D ) e. RR /\ ( B + D ) < ( A + D ) /\ ( A + D ) < ( C + D ) ) ) )
17 13 15 16 syl2anc 
 |-  ( ph -> ( ( A + D ) e. ( ( B + D ) (,) ( C + D ) ) <-> ( ( A + D ) e. RR /\ ( B + D ) < ( A + D ) /\ ( A + D ) < ( C + D ) ) ) )
18 2 rexrd 
 |-  ( ph -> B e. RR* )
19 3 rexrd 
 |-  ( ph -> C e. RR* )
20 elioo2 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( A e. ( B (,) C ) <-> ( A e. RR /\ B < A /\ A < C ) ) )
21 18 19 20 syl2anc 
 |-  ( ph -> ( A e. ( B (,) C ) <-> ( A e. RR /\ B < A /\ A < C ) ) )
22 11 17 21 3bitr4rd 
 |-  ( ph -> ( A e. ( B (,) C ) <-> ( A + D ) e. ( ( B + D ) (,) ( C + D ) ) ) )