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Theorem iooshf 9742
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 8205 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
213com13 1186 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
323expa 1181 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( C  +  B )  < 
A  <->  C  <  ( A  -  B ) ) )
43adantrr 470 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  +  B )  <  A  <->  C  <  ( A  -  B ) ) )
5 ltsubadd 8201 . . . . . 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 1181 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR )  ->  ( A  < 
( D  +  B
)  <->  ( A  -  B )  <  D
) )
87adantrl 469 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  ( D  +  B )  <->  ( A  -  B )  <  D ) )
94, 8anbi12d 464 . 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 7753 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1110rexrd 7822 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR* )
1211ad2ant2rl 502 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( C  +  B
)  e.  RR* )
13 readdcl 7753 . . . . . 6  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR )
1413rexrd 7822 . . . . 5  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR* )
1514ad2ant2l 499 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( D  +  B
)  e.  RR* )
16 rexr 7818 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
1716ad2antrl 481 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  A  e.  RR* )
18 elioo5 9723 . . . 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 1216 . . 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 7818 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
2221ad2antrl 481 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
23 rexr 7818 . . . 4  |-  ( D  e.  RR  ->  D  e.  RR* )
2423ad2antll 482 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
25 resubcl 8033 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
2625rexrd 7822 . . . 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 9723 . . 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 1216 . 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 962    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7626    + caddc 7630   RR*cxr 7806    < clt 7807    - cmin 7940   (,)cioo 9678
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-sub 7942  df-neg 7943  df-ioo 9682
This theorem is referenced by:  sinq34lt0t  12922
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