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Theorem iccshftr 10785
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 443 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 readdcl 8836 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  +  R
)  e.  RR )
31, 22thd 231 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
43adantl 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  +  R )  e.  RR ) )
5 leadd1 9258 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
653expb 1152 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  +  R )  <_  ( X  +  R )
) )
76adantlr 695 . . . 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 4046 . . . 4  |-  ( ( A  +  R )  <_  ( X  +  R )  <->  C  <_  ( X  +  R ) )
107, 9syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  +  R ) ) )
11 leadd1 9258 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
12113expb 1152 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1312an12s 776 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  +  R )  <_  ( B  +  R )
) )
1413adantll 694 . . . 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 4047 . . . 4  |-  ( ( X  +  R )  <_  ( B  +  R )  <->  ( X  +  R )  <_  D
)
1714, 16syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  +  R )  <_  D ) )
184, 10, 173anbi123d 1252 . 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 10731 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 451 . 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 8836 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  +  R
)  e.  RR )
228, 21syl5eqelr 2381 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 readdcl 8836 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  +  R
)  e.  RR )
2415, 23syl5eqelr 2381 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 10731 . . . . 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 463 . . . 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 804 . . 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 696 . 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 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752    + caddc 8756    <_ cle 8884   [,]cicc 10675
This theorem is referenced by:  iccshftri  10786  lincmb01cmp  10793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679
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