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Theorem ioojoin 11663
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 3666 . 2  |-  ( ( ( A (,) B
)  u.  { B } )  u.  ( B (,) C ) )  =  ( ( A (,) B )  u.  ( { B }  u.  ( B (,) C
) ) )
2 snunioo 11658 . . . . . . 7  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  B  < 
C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
323expa 1196 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C
) )
433adantl1 1152 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <  C )  ->  ( { B }  u.  ( B (,) C ) )  =  ( B [,) C ) )
54adantrl 715 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( { B }  u.  ( B (,) C
) )  =  ( B [,) C ) )
65uneq2d 3663 . . 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 11545 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
8 df-ico 11547 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
9 xrlenlt 9664 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
10 xrlttr 11358 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  <  B  /\  B  <  C )  ->  w  <  C
) )
11 xrltletr 11372 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
127, 8, 9, 7, 10, 11ixxun 11557 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  -> 
( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
136, 12eqtrd 2508 . 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 2520 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3479   {csn 4033   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-pre-lttri 9578  ax-pre-lttrn 9579
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-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  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-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-ioo 11545  df-ico 11547  df-icc 11548
This theorem is referenced by:  reconnlem1  21199  itgsplitioo  22112  lhop  22285  iocunico  31107
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