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Theorem iooshf 9909
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 8355 . . . . . 6 |- ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
213com13 1203 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
323expa 1198 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
43adantrr 476 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( C + B ) < A <-> C < ( A - B ) ) )
5 ltsubadd 8351 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( ( A - B ) < D <-> A < ( D + B ) ) )
65bicomd 140 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
763expa 1198 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
87adantrl 475 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < ( D + B ) <-> ( A - B ) < D ) )
94, 8anbi12d 470 . 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 7900 . . . . . 6 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
1110rexrd 7969 . . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR* )
1211ad2ant2rl 508 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( C + B ) e. RR* )
13 readdcl 7900 . . . . . 6 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR )
1413rexrd 7969 . . . . 5 |- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR* )
1514ad2ant2l 505 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( D + B ) e. RR* )
16 rexr 7965 . . . . 5 |- ( A e. RR -> A e. RR* )
1716ad2antrl 487 . . . 4 |- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> A e. RR* )
18 elioo5 9890 . . . 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 1233 . . 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 266 . 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 7965 . . . 4 |- ( C e. RR -> C e. RR* )
2221ad2antrl 487 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR* )
23 rexr 7965 . . . 4 |- ( D e. RR -> D e. RR* )
2423ad2antll 488 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR* )
25 resubcl 8183 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
2625rexrd 7969 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR* )
2726adantr 274 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR* )
28 elioo5 9890 . . 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 1233 . 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 220 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 103   <-> wb 104   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   + caddc 7777   RR*cxr 7953   < clt 7954   - cmin 8090   (,)cioo 9845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-sub 8092  df-neg 8093  df-ioo 9849
This theorem is referenced by:  sinq34lt0t  13546
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