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Theorem ioojoin 10919
Description: Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
ioojoin  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( ( A (,) B )  u. 
{ B } )  u.  ( B (,) C ) )  =  ( A (,) C
) )

Proof of Theorem ioojoin
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unass 3420 . 2  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )  =  ( ( A (,) B )  u.  ( { B }  u.  ( B (,) C
) ) )
2 snunioo 10915 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  B  < 
C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
323expa 1152 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C
) )
433adantl1 1112 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
54adantrl 696 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( { B }  u.  ( B (,) C
) )  =  ( B [,) C ) )
65uneq2d 3417 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  ( { B }  u.  ( B (,) C ) ) )  =  ( ( A (,) B )  u.  ( B [,) C ) ) )
7 df-ioo 10813 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
8 df-ico 10815 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
9 xrlenlt 9037 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
10 xrlttr 10626 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  <  B  /\  B  <  C )  ->  w  <  C
) )
11 xrltletr 10640 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
127, 8, 9, 7, 10, 11ixxun 10825 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
136, 12eqtrd 2398 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  ( { B }  u.  ( B (,) C ) ) )  =  ( A (,) C ) )
141, 13syl5eq 2410 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( ( A (,) B )  u. 
{ B } )  u.  ( B (,) C ) )  =  ( A (,) C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    u. cun 3236   {csn 3729   class class class wbr 4125  (class class class)co 5981   RR*cxr 9013    < clt 9014    <_ cle 9015   (,)cioo 10809   [,)cico 10811
This theorem is referenced by:  reconnlem1  18545  itgsplitioo  19407  lhop  19578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-ioo 10813  df-ico 10815  df-icc 10816
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