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Theorem icoshft 10409
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 8377 . . . . . 6 |- ( B e. RR -> B e. RR* )
2 elico2 10356 . . . . . 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 1048 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( X e. ( A [,) B ) -> ( X e. RR /\ A <_ X /\ X < B ) ) )
6 3anass 1013 . . 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 8765 . . . . . . . . . 10 |- ( ( A e. RR /\ X e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
983com12 1238 . . . . . . . . 9 |- ( ( X e. RR /\ A e. RR /\ C e. RR ) -> ( A <_ X <-> ( A + C ) <_ ( X + C ) ) )
1093expib 1237 . . . . . . . 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 1047 . . . . . 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 8764 . . . . . . . . 9 |- ( ( X e. RR /\ B e. RR /\ C e. RR ) -> ( X < B <-> ( X + C ) < ( B + C ) ) )
15143expib 1237 . . . . . . . 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 1046 . . . . . 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 477 . . . 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 456 . . 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 8311 . . . . . . . 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 1013 . . . . . 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 1051 . . . 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 8311 . . . . . 6 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
28273adant2 1047 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR )
29 readdcl 8311 . . . . . 6 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
30293adant1 1046 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
31 rexr 8377 . . . . . . 7 |- ( ( B + C ) e. RR -> ( B + C ) e. RR* )
32 elico2 10356 . . . . . . 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 415 . . . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   RRcr 8184   + caddc 8188   RR*cxr 8365   < clt 8366   <_ cle 8367   [,)cico 10309
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-ico 10313
This theorem is used by:  icoshftf1o  10410
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