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Theorem icoshft 10059
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 8067 . . . . . 6 |- ( B e. RR -> B e. RR* )
2 elico2 10006 . . . . . 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 1019 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
6 3anass 984 . . 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 8451 . . . . . . . . . 10 |- ( ( A e. RR /\ X e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
983com12 1209 . . . . . . . . 9 |- ( ( X e. RR /\ A e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
1093expib 1208 . . . . . . . 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 1018 . . . . . 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 8450 . . . . . . . . 9 |- ( ( X e. RR /\ B e. RR /\ C e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
15143expib 1208 . . . . . . . 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 1017 . . . . . 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 8000 . . . . . . . 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 984 . . . . . 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 1022 . . . 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 8000 . . . . . 6 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
28273adant2 1018 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
29 readdcl 8000 . . . . . 6 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
30293adant1 1017 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
31 rexr 8067 . . . . . . 7 |- ( ( B + C ) e. RR -> ( B + C ) e. RR* )
32 elico2 10006 . . . . . . 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 980   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   + caddc 7877   RR*cxr 8055   < clt 8056   <_ cle 8057   [,)cico 9959
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-ico 9963
This theorem is referenced by:  icoshftf1o  10060
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