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Theorem iooshf 11628
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 10047 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
213com13 1201 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
323expa 1196 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( C  +  B )  < 
A  <->  C  <  ( A  -  B ) ) )
43adantrr 716 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  +  B )  <  A  <->  C  <  ( A  -  B ) ) )
5 ltsubadd 10043 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  (
( A  -  B
)  <  D  <->  A  <  ( D  +  B ) ) )
65bicomd 201 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  ( A  <  ( D  +  B )  <->  ( A  -  B )  <  D
) )
763expa 1196 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR )  ->  ( A  < 
( D  +  B
)  <->  ( A  -  B )  <  D
) )
87adantrl 715 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  ( D  +  B )  <->  ( A  -  B )  <  D ) )
94, 8anbi12d 710 . 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 9592 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1110rexrd 9660 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR* )
1211ad2ant2rl 748 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( C  +  B
)  e.  RR* )
13 readdcl 9592 . . . . . 6  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR )
1413rexrd 9660 . . . . 5  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR* )
1514ad2ant2l 745 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( D  +  B
)  e.  RR* )
16 rexr 9656 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
1716ad2antrl 727 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  A  e.  RR* )
18 elioo5 11607 . . . 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 1228 . . 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 453 . 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 9656 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
2221ad2antrl 727 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
23 rexr 9656 . . . 4  |-  ( D  e.  RR  ->  D  e.  RR* )
2423ad2antll 728 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
25 resubcl 9902 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
2625rexrd 9660 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR* )
2726adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  -  B
)  e.  RR* )
28 elioo5 11607 . . 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 1228 . 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 286 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   RRcr 9508    + caddc 9512   RR*cxr 9644    < clt 9645    - cmin 9824   (,)cioo 11554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ioo 11558
This theorem is referenced by:  sinq34lt0t  23028
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