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Theorem difioo 26077
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 3548 . . . 4  |-  ( ( A (,) B )  i^i  ( B [,) C ) )  =  ( ( B [,) C )  i^i  ( A (,) B ) )
2 joiniooico 26069 . . . . . 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 2489 . . 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 3505 . . . . 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 1027 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  A  e.  RR* )
10 simpl3 993 . . . . . . 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 11132 . . . . . . 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 26068 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  A  /\  B  <_  C ) )  ->  ( A (,) B )  C_  ( A (,) C ) )
169, 11, 13, 14, 15syl22anc 1219 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  B  <_  C )  ->  ( A (,) B )  C_  ( A (,) C ) )
17 ssequn2 3534 . . . . 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 2485 . . 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 25904 . . 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 1027 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  A  e.  RR* )
23 simpl2 992 . . . . . 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 11131 . . . . . . 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 1217 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  C  <_  B )
31 ioossioo 26068 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  C  <_  B ) )  ->  ( A (,) C )  C_  ( A (,) B ) )
3222, 24, 25, 30, 31syl22anc 1219 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( A (,) C )  C_  ( A (,) B ) )
33 ssdif0 3742 . . . 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 11351 . . . . 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 1217 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( B [,) C )  =  (/) )
3834, 37eqtr4d 2478 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  <  B
)  /\  C  <  B )  ->  ( ( A (,) C )  \ 
( A (,) B
) )  =  ( B [,) C ) )
39 xrlelttric 26050 . . 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 965    = wceq 1369    e. wcel 1756    \ cdif 3330    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   class class class wbr 4297  (class class class)co 6096   RR*cxr 9422    < clt 9423    <_ cle 9424   (,)cioo 11305   [,)cico 11307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-ioo 11309  df-ico 11311
This theorem is referenced by:  dya2iocbrsiga  26695  dya2icobrsiga  26696
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