ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iccshftr Unicode version

Theorem iccshftr 9728
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 108 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 readdcl 7710 . . . . 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 273 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
5 leadd1 8156 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
653expb 1165 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
76adantlr 466 . . . 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 3904 . . . 4  |-  ( ( A  +  R )  <_  ( X  +  R )  <->  C  <_  ( X  +  R ) )
107, 9syl6bb 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  +  R ) ) )
11 leadd1 8156 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
12113expb 1165 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1312an12s 537 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1413adantll 465 . . . 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 3905 . . . 4  |-  ( ( X  +  R )  <_  ( B  +  R )  <->  ( X  +  R )  <_  D
)
1714, 16syl6bb 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  +  R )  <_  D ) )
184, 10, 173anbi123d 1273 . 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 9672 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 272 . 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 7710 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  +  R
)  e.  RR )
228, 21eqeltrrid 2203 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 readdcl 7710 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  +  R
)  e.  RR )
2415, 23eqeltrrid 2203 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 9672 . . . . 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 285 . . . 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 565 . . 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 467 . 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 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583    + caddc 7587    <_ cle 7765   [,]cicc 9625
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-icc 9629
This theorem is referenced by:  iccshftri  9729  lincmb01cmp  9737
  Copyright terms: Public domain W3C validator