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Theorem iooshf 10333
Description: Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
Assertion
Ref Expression
iooshf |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) )
Proof of Theorem iooshf
StepHypRef Expression
1 ltaddsub 8754 . . . . . 6 |- ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
213com13 1239 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
323expa 1234 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
43adantrr 483 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
5 ltsubadd 8750 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( ( A - B ) < D <-> A < ( D + B ) ) )
65bicomd 141 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
763expa 1234 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
87adantrl 482 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
94, 8anbi12d 477 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( C + B ) < A /\ A < ( D + B ) ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
10 readdcl 8295 . . . . . 6 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
1110rexrd 8365 . . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR* )
1211ad2ant2rl 515 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( C + B ) e. RR* )
13 readdcl 8295 . . . . . 6 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR )
1413rexrd 8365 . . . . 5 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR* )
1514ad2ant2l 512 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( D + B ) e. RR* )
16 rexr 8361 . . . . 5 |- ( A e. RR -> A e. RR* )
1716ad2antrl 494 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> A e. RR* )
18 elioo5 10314 . . . 4 |- ( ( ( C + B ) e. RR* /\ ( D + B ) e. RR* /\ A e. RR* ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
1912, 15, 17, 18syl3anc 1278 . . 3 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
2019ancoms 268 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) )
21 rexr 8361 . . . 4 |- ( C e. RR -> C e. RR* )
2221ad2antrl 494 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR* )
23 rexr 8361 . . . 4 |- ( D e. RR -> D e. RR* )
2423ad2antll 495 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR* )
25 resubcl 8580 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
2625rexrd 8365 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR* )
2726adantr 276 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR* )
28 elioo5 10314 . . 3 |- ( ( C e. RR* /\ D e. RR* /\ ( A - B ) e. RR* ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
2922, 24, 27, 28syl3anc 1278 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) )
309, 20, 293bitr4rd 221 1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   + caddc 8172   RR*cxr 8349   < clt 8350   - cmin 8487   (,)cioo 10269
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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltadd 8285
This theorem depends on definitions:  df-bi 117  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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-sub 8489  df-neg 8490  df-ioo 10273
This theorem is referenced by:  sinq34lt0t  15855
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