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Theorem iooshf 10981
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 9494 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
213com13 1158 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
323expa 1153 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( C  +  B )  < 
A  <->  C  <  ( A  -  B ) ) )
43adantrr 698 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  +  B )  <  A  <->  C  <  ( A  -  B ) ) )
5 ltsubadd 9490 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  (
( A  -  B
)  <  D  <->  A  <  ( D  +  B ) ) )
65bicomd 193 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  D  e.  RR )  ->  ( A  <  ( D  +  B )  <->  ( A  -  B )  <  D
) )
763expa 1153 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR )  ->  ( A  < 
( D  +  B
)  <->  ( A  -  B )  <  D
) )
87adantrl 697 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <  ( D  +  B )  <->  ( A  -  B )  <  D ) )
94, 8anbi12d 692 . 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 9065 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1110rexrd 9126 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR* )
1211ad2ant2rl 730 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( C  +  B
)  e.  RR* )
13 readdcl 9065 . . . . . 6  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR )
1413rexrd 9126 . . . . 5  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR* )
1514ad2ant2l 727 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( D  +  B
)  e.  RR* )
16 rexr 9122 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
1716ad2antrl 709 . . . 4  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  A  e.  RR* )
18 elioo5 10960 . . . 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 1184 . . 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 440 . 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 9122 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
2221ad2antrl 709 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
23 rexr 9122 . . . 4  |-  ( D  e.  RR  ->  D  e.  RR* )
2423ad2antll 710 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
25 resubcl 9357 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
2625rexrd 9126 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR* )
2726adantr 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  -  B
)  e.  RR* )
28 elioo5 10960 . . 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 1184 . 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 278 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    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981    + caddc 8985   RR*cxr 9111    < clt 9112    - cmin 9283   (,)cioo 10908
This theorem is referenced by:  sinq34lt0t  20409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-ioo 10912
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