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Theorem icoshft 10269
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 8267 . . . . . 6 |- ( B e. RR -> B e. RR* )
2 elico2 10216 . . . . . 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 1044 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
6 3anass 1009 . . 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 8652 . . . . . . . . . 10 |- ( ( A e. RR /\ X e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
983com12 1234 . . . . . . . . 9 |- ( ( X e. RR /\ A e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
1093expib 1233 . . . . . . . 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 1043 . . . . . 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 8651 . . . . . . . . 9 |- ( ( X e. RR /\ B e. RR /\ C e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
15143expib 1233 . . . . . . . 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 1042 . . . . . 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 8201 . . . . . . . 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 1009 . . . . . 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 1047 . . . 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 8201 . . . . . 6 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
28273adant2 1043 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
29 readdcl 8201 . . . . . 6 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
30293adant1 1042 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
31 rexr 8267 . . . . . . 7 |- ( ( B + C ) e. RR -> ( B + C ) e. RR* )
32 elico2 10216 . . . . . . 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 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8074   + caddc 8078   RR*cxr 8255   < clt 8256   <_ cle 8257   [,)cico 10169
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-ico 10173
This theorem is referenced by:  icoshftf1o  10270
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