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Theorem icoshft 9803
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 7835 . . . . . 6 |- ( B e. RR -> B e. RR* )
2 elico2 9750 . . . . . 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 1002 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
6 3anass 967 . . 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 8216 . . . . . . . . . 10 |- ( ( A e. RR /\ X e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
983com12 1186 . . . . . . . . 9 |- ( ( X e. RR /\ A e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
1093expib 1185 . . . . . . . 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 1001 . . . . . 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 8215 . . . . . . . . 9 |- ( ( X e. RR /\ B e. RR /\ C e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
15143expib 1185 . . . . . . . 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 1000 . . . . . 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 465 . . . 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 448 . . 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 7770 . . . . . . . 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 967 . . . . . 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 1005 . . . 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 7770 . . . . . 6 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
28273adant2 1001 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
29 readdcl 7770 . . . . . 6 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
30293adant1 1000 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
31 rexr 7835 . . . . . . 7 |- ( ( B + C ) e. RR -> ( B + C ) e. RR* )
32 elico2 9750 . . . . . . 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 409 . . . 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 963   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   RRcr 7643   + caddc 7647   RR*cxr 7823   < clt 7824   <_ cle 7825   [,)cico 9703
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-ico 9707
This theorem is referenced by:  icoshftf1o  9804
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