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Theorem icoshft 11394
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 9417 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
2 elico2 11347 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
31, 2sylan2 471 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,) B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <  B ) ) )
43biimpd 207 . . . 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 962 . . 3  |-  ( ( X  e.  RR  /\  A  <_  X  /\  X  <  B )  <->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) )
75, 6syl6ib 226 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  e.  RR  /\  ( A  <_  X  /\  X  <  B ) ) ) )
8 leadd1 9795 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
983com12 1184 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X  <->  ( A  +  C )  <_  ( X  +  C )
) )
1093expib 1183 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) ) )
1110com12 31 . . . . . . 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 429 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( A  <_  X 
<->  ( A  +  C
)  <_  ( X  +  C ) ) )
14 ltadd1 9794 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  <  B  <->  ( X  +  C )  <  ( B  +  C )
) )
15143expib 1183 . . . . . . . 8  |-  ( X  e.  RR  ->  (
( B  e.  RR  /\  C  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) ) )
1615com12 31 . . . . . . 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 429 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  X  e.  RR )  ->  ( X  < 
B  <->  ( X  +  C )  <  ( B  +  C )
) )
1913, 18anbi12d 703 . . . 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 634 . . 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 9353 . . . . . . . 8  |-  ( ( X  e.  RR  /\  C  e.  RR )  ->  ( X  +  C
)  e.  RR )
2221expcom 435 . . . . . . 7  |-  ( C  e.  RR  ->  ( X  e.  RR  ->  ( X  +  C )  e.  RR ) )
2322anim1d 559 . . . . . 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 962 . . . . . 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 227 . . . . 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 9353 . . . . . 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 9353 . . . . . 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 9417 . . . . . . 7  |-  ( ( B  +  C )  e.  RR  ->  ( B  +  C )  e.  RR* )
32 elico2 11347 . . . . . . 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 471 . . . . . 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 223 . . . . 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 654 . . . 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 215 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9269    + caddc 9273   RR*cxr 9405    < clt 9406    <_ cle 9407   [,)cico 11290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-ico 11294
This theorem is referenced by:  icoshftf1o  11395
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