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Theorem iccshftr 8862
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccshftr.1  |-  ( A  +  R )  =  C
iccshftr.2  |-  ( B  +  R )  =  D
Assertion
Ref Expression
iccshftr  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  +  R
)  e.  ( C [,] D ) ) )

Proof of Theorem iccshftr
StepHypRef Expression
1 simpl 102 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 readdcl 7007 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  +  R
)  e.  RR )
31, 22thd 164 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
43adantl 262 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
5 leadd1 7425 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
653expb 1105 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
76adantlr 446 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R ) ) )
8 iccshftr.1 . . . . 5  |-  ( A  +  R )  =  C
98breq1i 3771 . . . 4  |-  ( ( A  +  R )  <_  ( X  +  R )  <->  C  <_  ( X  +  R ) )
107, 9syl6bb 185 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  +  R ) ) )
11 leadd1 7425 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
12113expb 1105 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1312an12s 499 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1413adantll 445 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R ) ) )
15 iccshftr.2 . . . . 5  |-  ( B  +  R )  =  D
1615breq2i 3772 . . . 4  |-  ( ( X  +  R )  <_  ( B  +  R )  <->  ( X  +  R )  <_  D
)
1714, 16syl6bb 185 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  +  R )  <_  D ) )
184, 10, 173anbi123d 1207 . 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 8807 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 261 . 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 readdcl 7007 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  +  R
)  e.  RR )
228, 21syl5eqelr 2125 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 readdcl 7007 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  +  R
)  e.  RR )
2415, 23syl5eqelr 2125 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 8807 . . . . 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 273 . . . 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 527 . . 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 447 . 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 209 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888    + caddc 6892    <_ cle 7061   [,]cicc 8760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-icc 8764
This theorem is referenced by:  iccshftri  8863  lincmb01cmp  8871
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