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Theorem iccshftr 10963
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 9007 . . . . 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 9429 . . . . . 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 4161 . . . 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 9429 . . . . . . 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 4162 . . . 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 10908 . . 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 9007 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  +  R
)  e.  RR )
228, 21syl5eqelr 2473 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 readdcl 9007 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  +  R
)  e.  RR )
2415, 23syl5eqelr 2473 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 10908 . . . . 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 1717   class class class wbr 4154  (class class class)co 6021   RRcr 8923    + caddc 8927    <_ cle 9055   [,]cicc 10852
This theorem is referenced by:  iccshftri  10964  lincmb01cmp  10971
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-icc 10856
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