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Theorem ioojoin 10983
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 3464 . 2  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )  =  ( ( A (,) B )  u.  ( { B }  u.  ( B (,) C
) ) )
2 snunioo 10979 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  B  < 
C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
323expa 1153 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C
) )
433adantl1 1113 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
54adantrl 697 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( { B }  u.  ( B (,) C
) )  =  ( B [,) C ) )
65uneq2d 3461 . . 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 10876 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
8 df-ico 10878 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
9 xrlenlt 9099 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
10 xrlttr 10689 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  <  B  /\  B  <  C )  ->  w  <  C
) )
11 xrltletr 10703 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
127, 8, 9, 7, 10, 11ixxun 10888 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
136, 12eqtrd 2436 . 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 2448 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    u. cun 3278   {csn 3774   class class class wbr 4172  (class class class)co 6040   RR*cxr 9075    < clt 9076    <_ cle 9077   (,)cioo 10872   [,)cico 10874
This theorem is referenced by:  reconnlem1  18810  itgsplitioo  19682  lhop  19853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-ioo 10876  df-ico 10878  df-icc 10879
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