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Theorem icoshft 9766
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 7804 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elico2 9713 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
31, 2sylan2 284 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
43biimpd 143 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  <  B
) ) )
543adant3 1001 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  < 
B ) ) )
6 3anass 966 . . 3  |-  ( ( X  e.  RR  /\  A  <_  X  /\  X  <  B )  <->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) )
75, 6syl6ib 160 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) ) )
8 leadd1 8185 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
983com12 1185 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
1093expib 1184 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) ) )
1110com12 30 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C ) ) ) )
12113adant2 1000 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) ) )
1312imp 123 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) )
14 ltadd1 8184 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) )
15143expib 1184 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( B  e.  RR  /\  C  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1615com12 30 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C ) ) ) )
17163adant1 999 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1817imp 123 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) )
1913, 18anbi12d 464 . . . 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 447 . . 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 7739 . . . . . . . 8  |-  ( ( X  e.  RR  /\  C  e.  RR )  ->  ( X  +  C
)  e.  RR )
2221expcom 115 . . . . . . 7  |-  ( C  e.  RR  ->  ( X  e.  RR  ->  ( X  +  C )  e.  RR ) )
2322anim1d 334 . . . . . 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 966 . . . . . 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 161 . . . . 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 1004 . . . 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 7739 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
28273adant2 1000 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
29 readdcl 7739 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
30293adant1 999 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
31 rexr 7804 . . . . . . 7  |-  ( ( B  +  C )  e.  RR  ->  ( B  +  C )  e.  RR* )
32 elico2 9713 . . . . . . 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 284 . . . . . 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 157 . . . . 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 408 . . . 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 45 . . 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 149 . 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 45 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    /\ wa 103    <-> wb 104    /\ w3a 962    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612    + caddc 7616   RR*cxr 7792    < clt 7793    <_ cle 7794   [,)cico 9666
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-ico 9670
This theorem is referenced by:  icoshftf1o  9767
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