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Theorem difioo 27416
Description: The difference between two open intervals sharing the same lower bound (Contributed by Thierry Arnoux, 26-Sep-2017.)
Assertion
Ref Expression
difioo  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
( A (,) C
)  \  ( A (,) B ) )  =  ( B [,) C
) )

Proof of Theorem difioo
StepHypRef Expression
1 incom 3696 . . . 4  |-  ( ( A (,) B )  i^i  ( B [,) C ) )  =  ( ( B [,) C )  i^i  ( A (,) B ) )
2 joiniooico 27408 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( ( A (,) B )  i^i  ( B [,) C
) )  =  (/)  /\  ( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )
32anassrs 648 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( (
( A (,) B
)  i^i  ( B [,) C ) )  =  (/)  /\  ( ( A (,) B )  u.  ( B [,) C
) )  =  ( A (,) C ) ) )
43simpld 459 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( A (,) B )  i^i  ( B [,) C
) )  =  (/) )
51, 4syl5eqr 2522 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( B [,) C )  i^i  ( A (,) B
) )  =  (/) )
63simprd 463 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( A (,) B )  u.  ( B [,) C
) )  =  ( A (,) C ) )
7 uncom 3653 . . . . 5  |-  ( ( B [,) C )  u.  ( A (,) B ) )  =  ( ( A (,) B )  u.  ( B [,) C ) )
87a1i 11 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( B [,) C )  u.  ( A (,) B
) )  =  ( ( A (,) B
)  u.  ( B [,) C ) ) )
9 simpll1 1035 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  A  e.  RR* )
10 simpl3 1001 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  C  e.  RR* )
1110adantr 465 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  C  e.  RR* )
12 xrleid 11368 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
139, 12syl 16 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  A  <_  A )
14 simpr 461 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  B  <_  C )
15 ioossioo 11628 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  A  /\  B  <_  C ) )  ->  ( A (,) B )  C_  ( A (,) C ) )
169, 11, 13, 14, 15syl22anc 1229 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( A (,) B )  C_  ( A (,) C ) )
17 ssequn2 3682 . . . . 5  |-  ( ( A (,) B ) 
C_  ( A (,) C )  <->  ( ( A (,) C )  u.  ( A (,) B
) )  =  ( A (,) C ) )
1816, 17sylib 196 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( A (,) C )  u.  ( A (,) B
) )  =  ( A (,) C ) )
196, 8, 183eqtr4d 2518 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( B [,) C )  u.  ( A (,) B
) )  =  ( ( A (,) C
)  u.  ( A (,) B ) ) )
20 difeq 27239 . . 3  |-  ( ( ( A (,) C
)  \  ( A (,) B ) )  =  ( B [,) C
)  <->  ( ( ( B [,) C )  i^i  ( A (,) B ) )  =  (/)  /\  ( ( B [,) C )  u.  ( A (,) B
) )  =  ( ( A (,) C
)  u.  ( A (,) B ) ) ) )
215, 19, 20sylanbrc 664 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( ( A (,) C )  \ 
( A (,) B
) )  =  ( B [,) C ) )
22 simpll1 1035 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  A  e.  RR* )
23 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
2423adantr 465 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  B  e.  RR* )
2522, 12syl 16 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  A  <_  A )
2610adantr 465 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  C  e.  RR* )
27 simpr 461 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  C  <  B )
28 xrltle 11367 . . . . . . 7  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
2928imp 429 . . . . . 6  |-  ( ( ( C  e.  RR*  /\  B  e.  RR* )  /\  C  <  B )  ->  C  <_  B
)
3026, 24, 27, 29syl21anc 1227 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  C  <_  B )
31 ioossioo 11628 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  C  <_  B ) )  ->  ( A (,) C )  C_  ( A (,) B ) )
3222, 24, 25, 30, 31syl22anc 1229 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( A (,) C )  C_  ( A (,) B ) )
33 ssdif0 3890 . . . 4  |-  ( ( A (,) C ) 
C_  ( A (,) B )  <->  ( ( A (,) C )  \ 
( A (,) B
) )  =  (/) )
3432, 33sylib 196 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( ( A (,) C )  \ 
( A (,) B
) )  =  (/) )
35 ico0 11587 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( B [,) C
)  =  (/)  <->  C  <_  B ) )
3635biimpar 485 . . . 4  |-  ( ( ( B  e.  RR*  /\  C  e.  RR* )  /\  C  <_  B )  ->  ( B [,) C )  =  (/) )
3724, 26, 30, 36syl21anc 1227 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( B [,) C )  =  (/) )
3834, 37eqtr4d 2511 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( ( A (,) C )  \ 
( A (,) B
) )  =  ( B [,) C ) )
39 xrlelttric 27395 . . 3  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  \/  C  <  B ) )
4023, 10, 39syl2anc 661 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  ( B  <_  C  \/  C  <  B ) )
4121, 38, 40mpjaodan 784 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
( A (,) C
)  \  ( A (,) B ) )  =  ( B [,) C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   class class class wbr 4453  (class class class)co 6295   RR*cxr 9639    < clt 9640    <_ cle 9641   (,)cioo 11541   [,)cico 11543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-ioo 11545  df-ico 11547
This theorem is referenced by:  dya2iocbrsiga  28071  dya2icobrsiga  28072
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