MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icoshft Unicode version

Theorem icoshft 10944
Description: A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
Assertion
Ref Expression
icoshft  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C )
) ) )

Proof of Theorem icoshft
StepHypRef Expression
1 rexr 9056 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elico2 10899 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
31, 2sylan2 461 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
43biimpd 199 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  <  B
) ) )
543adant3 977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  < 
B ) ) )
6 3anass 940 . . 3  |-  ( ( X  e.  RR  /\  A  <_  X  /\  X  <  B )  <->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) )
75, 6syl6ib 218 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) ) )
8 leadd1 9421 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
983com12 1157 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
1093expib 1156 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) ) )
1110com12 29 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C ) ) ) )
12113adant2 976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) ) )
1312imp 419 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) )
14 ltadd1 9420 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) )
15143expib 1156 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( B  e.  RR  /\  C  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1615com12 29 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C ) ) ) )
17163adant1 975 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1817imp 419 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) )
1913, 18anbi12d 692 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( ( A  <_  X  /\  X  <  B )  <->  ( ( A  +  C )  <_  ( X  +  C
)  /\  ( X  +  C )  <  ( B  +  C )
) ) )
2019pm5.32da 623 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) )  <->  ( X  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) ) ) ) )
21 readdcl 8999 . . . . . . . 8  |-  ( ( X  e.  RR  /\  C  e.  RR )  ->  ( X  +  C
)  e.  RR )
2221expcom 425 . . . . . . 7  |-  ( C  e.  RR  ->  ( X  e.  RR  ->  ( X  +  C )  e.  RR ) )
2322anim1d 548 . . . . . 6  |-  ( C  e.  RR  ->  (
( X  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C
)  <  ( B  +  C ) ) )  ->  ( ( X  +  C )  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) ) ) ) )
24 3anass 940 . . . . . 6  |-  ( ( ( X  +  C
)  e.  RR  /\  ( A  +  C
)  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) )  <->  ( ( X  +  C )  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) ) ) )
2523, 24syl6ibr 219 . . . . 5  |-  ( C  e.  RR  ->  (
( X  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C
)  <  ( B  +  C ) ) )  ->  ( ( X  +  C )  e.  RR  /\  ( A  +  C )  <_ 
( X  +  C
)  /\  ( X  +  C )  <  ( B  +  C )
) ) )
26253ad2ant3 980 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( X  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C
)  <  ( B  +  C ) ) )  ->  ( ( X  +  C )  e.  RR  /\  ( A  +  C )  <_ 
( X  +  C
)  /\  ( X  +  C )  <  ( B  +  C )
) ) )
27 readdcl 8999 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
28273adant2 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
29 readdcl 8999 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
30293adant1 975 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
31 rexr 9056 . . . . . . 7  |-  ( ( B  +  C )  e.  RR  ->  ( B  +  C )  e.  RR* )
32 elico2 10899 . . . . . . 7  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR* )  ->  ( ( X  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) )  <-> 
( ( X  +  C )  e.  RR  /\  ( A  +  C
)  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) ) ) )
3331, 32sylan2 461 . . . . . 6  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C )
)  <->  ( ( X  +  C )  e.  RR  /\  ( A  +  C )  <_ 
( X  +  C
)  /\  ( X  +  C )  <  ( B  +  C )
) ) )
3433biimprd 215 . . . . 5  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( ( ( X  +  C )  e.  RR  /\  ( A  +  C )  <_  ( X  +  C
)  /\  ( X  +  C )  <  ( B  +  C )
)  ->  ( X  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) ) )
3528, 30, 34syl2anc 643 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( X  +  C )  e.  RR  /\  ( A  +  C
)  <_  ( X  +  C )  /\  ( X  +  C )  <  ( B  +  C
) )  ->  ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C )
) ) )
3626, 35syld 42 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( X  e.  RR  /\  ( ( A  +  C )  <_  ( X  +  C )  /\  ( X  +  C
)  <  ( B  +  C ) ) )  ->  ( X  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) ) )
3720, 36sylbid 207 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) )  ->  ( X  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) ) )
387, 37syld 42 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717   class class class wbr 4146  (class class class)co 6013   RRcr 8915    + caddc 8919   RR*cxr 9045    < clt 9046    <_ cle 9047   [,)cico 10843
This theorem is referenced by:  icoshftf1o  10945
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-ico 10847
  Copyright terms: Public domain W3C validator