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Theorem icoshft 9992
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 8005 . . . . . 6 |- ( B e. RR -> B e. RR* )
2 elico2 9939 . . . . . 6 |- ( ( A e. RR /\ B e. RR* ) -> ( X e. ( A [,) B ) <-> ( X e. RR /\ A <_ X /\ X < B ) ) )
31, 2sylan2 286 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,) B ) <-> ( X e. RR /\ A <_ X /\ X < B ) ) )
43biimpd 144 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
543adant3 1017 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
6 3anass 982 . . 3 |- ( ( X e. RR /\ A <_ X /\ X < B ) <-> ( X e. RR /\ ( A <_ X /\ X < B ) ) )
75, 6imbitrdi 161 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ ( A <_ X /\ X < B ) ) ) )
8 leadd1 8389 . . . . . . . . . 10 |- ( ( A e. RR /\ X e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
983com12 1207 . . . . . . . . 9 |- ( ( X e. RR /\ A e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
1093expib 1206 . . . . . . . 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 1016 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. RR -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) ) )
1312imp 124 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ X e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
14 ltadd1 8388 . . . . . . . . 9 |- ( ( X e. RR /\ B e. RR /\ C e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
15143expib 1206 . . . . . . . 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 1015 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. RR -> ( X < B <-> ( X + C ) < ( B + C ) ) ) )
1817imp 124 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ X e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
1913, 18anbi12d 473 . . . 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 452 . . 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 7939 . . . . . . . 8 |- ( ( X e. RR /\ C e. RR ) -> ( X + C ) e. RR )
2221expcom 116 . . . . . . 7 |- ( C e. RR -> ( X e. RR -> ( X + C ) e. RR ) )
2322anim1d 336 . . . . . 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 982 . . . . . 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, 24imbitrrdi 162 . . . . 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 1020 . . . 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 7939 . . . . . 6 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
28273adant2 1016 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
29 readdcl 7939 . . . . . 6 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
30293adant1 1015 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
31 rexr 8005 . . . . . . 7 |- ( ( B + C ) e. RR -> ( B + C ) e. RR* )
32 elico2 9939 . . . . . . 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 286 . . . . . 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 158 . . . . 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 411 . . . 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 150 . 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 104   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   + caddc 7816   RR*cxr 7993   < clt 7994   <_ cle 7995   [,)cico 9892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-ico 9896
This theorem is referenced by:  icoshftf1o  9993
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