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Theorem icoshft 9741
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 7779 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elico2 9688 . . . . . 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 986 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  < 
B ) ) )
6 3anass 951 . . 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 8160 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
983com12 1170 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
1093expib 1169 . . . . . . . 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 985 . . . . . 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 8159 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) )
15143expib 1169 . . . . . . . 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 984 . . . . . 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 7714 . . . . . . . 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 951 . . . . . 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 989 . . . 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 7714 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
28273adant2 985 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
29 readdcl 7714 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
30293adant1 984 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
31 rexr 7779 . . . . . . 7  |-  ( ( B  +  C )  e.  RR  ->  ( B  +  C )  e.  RR* )
32 elico2 9688 . . . . . . 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 947    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587    + caddc 7591   RR*cxr 7767    < clt 7768    <_ cle 7769   [,)cico 9641
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-ico 9645
This theorem is referenced by:  icoshftf1o  9742
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