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Theorem iccshftl 9900
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccshftl.1  |-  ( A  -  R )  =  C
iccshftl.2  |-  ( B  -  R )  =  D
Assertion
Ref Expression
iccshftl  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  -  R
)  e.  ( C [,] D ) ) )

Proof of Theorem iccshftl
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 resubcl 8139 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  -  R
)  e.  RR )
31, 22thd 174 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
43adantl 275 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
5 lesub1 8331 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
653expb 1186 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
76adantlr 469 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R ) ) )
8 iccshftl.1 . . . . 5  |-  ( A  -  R )  =  C
98breq1i 3972 . . . 4  |-  ( ( A  -  R )  <_  ( X  -  R )  <->  C  <_  ( X  -  R ) )
107, 9bitrdi 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  -  R ) ) )
11 lesub1 8331 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
12113expb 1186 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1312an12s 555 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1413adantll 468 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R ) ) )
15 iccshftl.2 . . . . 5  |-  ( B  -  R )  =  D
1615breq2i 3973 . . . 4  |-  ( ( X  -  R )  <_  ( B  -  R )  <->  ( X  -  R )  <_  D
)
1714, 16bitrdi 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  -  R )  <_  D ) )
184, 10, 173anbi123d 1294 . 2  |-  ( ( ( 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 9842 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 274 . 2  |-  ( ( ( 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 8139 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  -  R
)  e.  RR )
228, 21eqeltrrid 2245 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 resubcl 8139 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  -  R
)  e.  RR )
2415, 23eqeltrrid 2245 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 9842 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( X  -  R )  e.  ( C [,] D )  <-> 
( ( X  -  R )  e.  RR  /\  C  <_  ( X  -  R )  /\  ( X  -  R )  <_  D ) ) )
2622, 24, 25syl2an 287 . . . 4  |-  ( ( ( 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 ) ) )
2726anandirs 583 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR )  ->  ( ( X  -  R )  e.  ( C [,] D
)  <->  ( ( X  -  R )  e.  RR  /\  C  <_ 
( X  -  R
)  /\  ( X  -  R )  <_  D
) ) )
2827adantrl 470 . 2  |-  ( ( ( 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 ) ) )
2918, 20, 283bitr4d 219 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  -  R
)  e.  ( C [,] D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731    <_ cle 7913    - cmin 8046   [,]cicc 9795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-icc 9799
This theorem is referenced by:  iccshftli  9901  iccf1o  9908
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