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Theorem iooshf 11488
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 9927 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
213com13 1193 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  B
)  <  A  <->  C  <  ( A  -  B ) ) )
323expa 1188 . . . 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 9923 . . . . . 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 1188 . . . 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 9479 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
1110rexrd 9547 . . . . 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 9479 . . . . . 6  |-  ( ( D  e.  RR  /\  B  e.  RR )  ->  ( D  +  B
)  e.  RR )
1413rexrd 9547 . . . . 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 9543 . . . . 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 11467 . . . 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 1219 . . 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 9543 . . . 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 9543 . . . 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 9787 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
2625rexrd 9547 . . . 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 11467 . . 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 1219 . 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 965    e. wcel 1758   class class class wbr 4403  (class class class)co 6203   RRcr 9395    + caddc 9399   RR*cxr 9531    < clt 9532    - cmin 9709   (,)cioo 11414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-ioo 11418
This theorem is referenced by:  sinq34lt0t  22107
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