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Theorem iooshf 8922
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 7505 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
213com13 1120 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
323expa 1115 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( C  +  B )  < 
A  <->  C  <  ( A  -  B ) ) )
43adantrr 456 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  +  B )  <  A  <->  C  <  ( A  -  B ) ) )
5 ltsubadd 7501 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  (
( A  -  B
)  <  D  <->  A  <  ( D  +  B ) ) )
65bicomd 133 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  ( A  <  ( D  +  B )  <->  ( A  -  B )  <  D
) )
763expa 1115 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR )  ->  ( A  < 
( D  +  B
)  <->  ( A  -  B )  <  D
) )
87adantrl 455 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  ( D  +  B )  <->  ( A  -  B )  <  D ) )
94, 8anbi12d 450 . 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 7065 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1110rexrd 7134 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR* )
1211ad2ant2rl 488 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( C  +  B
)  e.  RR* )
13 readdcl 7065 . . . . . 6  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR )
1413rexrd 7134 . . . . 5  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR* )
1514ad2ant2l 485 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( D  +  B
)  e.  RR* )
16 rexr 7130 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
1716ad2antrl 467 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  A  e.  RR* )
18 elioo5 8903 . . . 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 1146 . . 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 259 . 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 7130 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
2221ad2antrl 467 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
23 rexr 7130 . . . 4  |-  ( D  e.  RR  ->  D  e.  RR* )
2423ad2antll 468 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
25 resubcl 7338 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
2625rexrd 7134 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR* )
2726adantr 265 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  -  B
)  e.  RR* )
28 elioo5 8903 . . 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 1146 . 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 214 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 101    <-> wb 102    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   RRcr 6946    + caddc 6950   RR*cxr 7118    < clt 7119    - cmin 7245   (,)cioo 8858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-sub 7247  df-neg 7248  df-ioo 8862
This theorem is referenced by: (None)
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