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Theorem iccshftr 10986
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 444 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 readdcl 9029 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  +  R
)  e.  RR )
31, 22thd 232 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
43adantl 453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
5 leadd1 9452 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
653expb 1154 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
76adantlr 696 . . . 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 4179 . . . 4  |-  ( ( A  +  R )  <_  ( X  +  R )  <->  C  <_  ( X  +  R ) )
107, 9syl6bb 253 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  +  R ) ) )
11 leadd1 9452 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
12113expb 1154 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1312an12s 777 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1413adantll 695 . . . 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 4180 . . . 4  |-  ( ( X  +  R )  <_  ( B  +  R )  <->  ( X  +  R )  <_  D
)
1714, 16syl6bb 253 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  +  R )  <_  D ) )
184, 10, 173anbi123d 1254 . 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 10931 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 452 . 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 9029 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  +  R
)  e.  RR )
228, 21syl5eqelr 2489 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 readdcl 9029 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  +  R
)  e.  RR )
2415, 23syl5eqelr 2489 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 10931 . . . . 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 464 . . . 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 805 . . 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 697 . 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 277 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RRcr 8945    + caddc 8949    <_ cle 9077   [,]cicc 10875
This theorem is referenced by:  iccshftri  10987  lincmb01cmp  10994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-icc 10879
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