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Theorem iooshf 8591
Description: Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
Assertion
Ref Expression
iooshf  RR  RR  C  RR  D  RR  -  C (,) D  C  +  (,) D  +

Proof of Theorem iooshf
StepHypRef Expression
1 ltaddsub 7226 . . . . . 6  C  RR  RR  RR  C  +  <  C  <  -
213com13 1108 . . . . 5  RR  RR  C  RR  C  +  <  C  <  -
323expa 1103 . . . 4  RR  RR  C  RR  C  +  <  C  <  -
43adantrr 448 . . 3  RR  RR  C  RR  D  RR  C  +  <  C  <  -
5 ltsubadd 7222 . . . . . 6  RR  RR  D  RR  -  <  D  <  D  +
65bicomd 129 . . . . 5  RR  RR  D  RR  <  D  +  -  <  D
763expa 1103 . . . 4  RR  RR  D  RR  <  D  +  -  <  D
87adantrl 447 . . 3  RR  RR  C  RR  D  RR  <  D  +  -  <  D
94, 8anbi12d 442 . 2  RR  RR  C  RR  D  RR  C  +  <  <  D  +  C  <  -  -  <  D
10 readdcl 6805 . . . . . 6  C  RR  RR  C  +  RR
1110rexrd 6872 . . . . 5  C  RR  RR  C  +  RR*
1211ad2ant2rl 480 . . . 4  C  RR  D  RR  RR  RR  C  + 
RR*
13 readdcl 6805 . . . . . 6  D  RR  RR  D  +  RR
1413rexrd 6872 . . . . 5  D  RR  RR  D  +  RR*
1514ad2ant2l 477 . . . 4  C  RR  D  RR  RR  RR  D  + 
RR*
16 rexr 6868 . . . . 5  RR  RR*
1716ad2antrl 459 . . . 4  C  RR  D  RR  RR  RR  RR*
18 elioo5 8572 . . . 4  C  +  RR*  D  +  RR*  RR*  C  +  (,) D  +  C  +  <  <  D  +
1912, 15, 17, 18syl3anc 1134 . . 3  C  RR  D  RR  RR  RR  C  +  (,) D  +  C  +  <  <  D  +
2019ancoms 255 . 2  RR  RR  C  RR  D  RR  C  +  (,) D  +  C  +  <  <  D  +
21 rexr 6868 . . . 4  C  RR  C  RR*
2221ad2antrl 459 . . 3  RR  RR  C  RR  D  RR  C 
RR*
23 rexr 6868 . . . 4  D  RR  D  RR*
2423ad2antll 460 . . 3  RR  RR  C  RR  D  RR  D 
RR*
25 resubcl 7071 . . . . 5  RR  RR  -  RR
2625rexrd 6872 . . . 4  RR  RR  -  RR*
2726adantr 261 . . 3  RR  RR  C  RR  D  RR  - 
RR*
28 elioo5 8572 . . 3  C  RR*  D  RR*  - 
RR*  -  C (,) D  C  <  -  -  < 
D
2922, 24, 27, 28syl3anc 1134 . 2  RR  RR  C  RR  D  RR  -  C (,) D  C  <  -  -  <  D
309, 20, 293bitr4rd 210 1  RR  RR  C  RR  D  RR  -  C (,) D  C  +  (,) D  +
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wcel 1390   class class class wbr 3755  (class class class)co 5455   RRcr 6710    + caddc 6714   RR*cxr 6856    < clt 6857    - cmin 6979   (,)cioo 8527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltadd 6799
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-sub 6981  df-neg 6982  df-ioo 8531
This theorem is referenced by: (None)
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