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Theorem icoshft 8959
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 7130 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elico2 8907 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
31, 2sylan2 274 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
43biimpd 136 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  <  B
) ) )
543adant3 935 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  A  <_  X  /\  X  < 
B ) ) )
6 3anass 900 . . 3  |-  ( ( X  e.  RR  /\  A  <_  X  /\  X  <  B )  <->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) )
75, 6syl6ib 154 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) ) )
8 leadd1 7499 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
983com12 1119 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
1093expib 1118 . . . . . . . 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 934 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) ) )
1312imp 119 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) )
14 ltadd1 7498 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) )
15143expib 1118 . . . . . . . 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 933 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  RR  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1817imp 119 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) )
1913, 18anbi12d 450 . . . 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 433 . . 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 7065 . . . . . . . 8  |-  ( ( X  e.  RR  /\  C  e.  RR )  ->  ( X  +  C
)  e.  RR )
2221expcom 113 . . . . . . 7  |-  ( C  e.  RR  ->  ( X  e.  RR  ->  ( X  +  C )  e.  RR ) )
2322anim1d 323 . . . . . 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 900 . . . . . 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 155 . . . . 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 938 . . . 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 7065 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
28273adant2 934 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
29 readdcl 7065 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
30293adant1 933 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
31 rexr 7130 . . . . . . 7  |-  ( ( B  +  C )  e.  RR  ->  ( B  +  C )  e.  RR* )
32 elico2 8907 . . . . . . 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 274 . . . . . 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 151 . . . . 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 397 . . . 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 44 . . 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 143 . 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 44 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 101    <-> wb 102    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   RRcr 6946    + caddc 6950   RR*cxr 7118    < clt 7119    <_ cle 7120   [,)cico 8860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-po 4061  df-iso 4062  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-ico 8864
This theorem is referenced by:  icoshftf1o  8960
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