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Theorem ixxun 10924
Description: Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxun.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
ixxun.3  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B T w  <->  -.  w S B ) )
ixxun.4  |-  Q  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z U y ) } )
ixxun.5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w S B  /\  B X C )  ->  w U C ) )
ixxun.6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
Assertion
Ref Expression
ixxun  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( ( A O B )  u.  ( B P C ) )  =  ( A Q C ) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, O   
w, Q    w, B, x, y, z    w, P   
x, R, y, z   
x, S, y, z   
x, T, y, z   
x, U, y, z   
w, W    w, X
Allowed substitution hints:    P( x, y, z)    Q( x, y, z)    R( w)    S( w)    T( w)    U( w)    O( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem ixxun
StepHypRef Expression
1 elun 3480 . . 3  |-  ( w  e.  ( ( A O B )  u.  ( B P C ) )  <->  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )
2 simpl1 960 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  A  e.  RR* )
3 simpl2 961 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  B  e.  RR* )
4 ixx.1 . . . . . . . . . . 11  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
54elixx1 10917 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
62, 3, 5syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( A O B )  <-> 
( w  e.  RR*  /\  A R w  /\  w S B ) ) )
76biimpa 471 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  -> 
( w  e.  RR*  /\  A R w  /\  w S B ) )
87simp1d 969 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
97simp2d 970 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  A R w )
107simp3d 971 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  w S B )
11 simplrr 738 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  B X C )
123adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
13 simpl3 962 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  C  e.  RR* )
1413adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  C  e.  RR* )
15 ixxun.5 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w S B  /\  B X C )  ->  w U C ) )
168, 12, 14, 15syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  -> 
( ( w S B  /\  B X C )  ->  w U C ) )
1710, 11, 16mp2and 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  ->  w U C )
188, 9, 173jca 1134 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A O B ) )  -> 
( w  e.  RR*  /\  A R w  /\  w U C ) )
19 ixxun.2 . . . . . . . . . . 11  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
2019elixx1 10917 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
213, 13, 20syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( B P C )  <-> 
( w  e.  RR*  /\  B T w  /\  w U C ) ) )
2221biimpa 471 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  -> 
( w  e.  RR*  /\  B T w  /\  w U C ) )
2322simp1d 969 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  w  e.  RR* )
24 simplrl 737 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  A W B )
2522simp2d 970 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  B T w )
262adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  A  e.  RR* )
273adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
28 ixxun.6 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
2926, 27, 23, 28syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  -> 
( ( A W B  /\  B T w )  ->  A R w ) )
3024, 25, 29mp2and 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  A R w )
3122simp3d 971 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  ->  w U C )
3223, 30, 313jca 1134 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( B P C ) )  -> 
( w  e.  RR*  /\  A R w  /\  w U C ) )
3318, 32jaodan 761 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )  ->  ( w  e.  RR*  /\  A R w  /\  w U C ) )
34 ixxun.4 . . . . . . . 8  |-  Q  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z U y ) } )
3534elixx1 10917 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A Q C )  <->  ( w  e.  RR*  /\  A R w  /\  w U C ) ) )
362, 13, 35syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( A Q C )  <-> 
( w  e.  RR*  /\  A R w  /\  w U C ) ) )
3736biimpar 472 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  ( w  e.  RR*  /\  A R w  /\  w U C ) )  ->  w  e.  ( A Q C ) )
3833, 37syldan 457 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )  ->  w  e.  ( A Q C ) )
39 exmid 405 . . . . 5  |-  ( w S B  \/  -.  w S B )
40 df-3an 938 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  <->  ( (
w  e.  RR*  /\  A R w )  /\  w S B ) )
416, 40syl6bb 253 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( A O B )  <-> 
( ( w  e. 
RR*  /\  A R w )  /\  w S B ) ) )
4241adantr 452 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  ( A O B )  <-> 
( ( w  e. 
RR*  /\  A R w )  /\  w S B ) ) )
4336biimpa 471 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  RR*  /\  A R w  /\  w U C ) )
4443simp1d 969 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  ->  w  e.  RR* )
4543simp2d 970 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  ->  A R w )
4644, 45jca 519 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  RR*  /\  A R w ) )
4746biantrurd 495 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w S B  <-> 
( ( w  e. 
RR*  /\  A R w )  /\  w S B ) ) )
4842, 47bitr4d 248 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  ( A O B )  <-> 
w S B ) )
49 3anan12 949 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  B T w  /\  w U C )  <->  ( B T w  /\  (
w  e.  RR*  /\  w U C ) ) )
5021, 49syl6bb 253 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( B P C )  <-> 
( B T w  /\  ( w  e. 
RR*  /\  w U C ) ) ) )
5150adantr 452 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  ( B P C )  <-> 
( B T w  /\  ( w  e. 
RR*  /\  w U C ) ) ) )
5243simp3d 971 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  ->  w U C )
5344, 52jca 519 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  RR*  /\  w U C ) )
5453biantrud 494 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( B T w  <-> 
( B T w  /\  ( w  e. 
RR*  /\  w U C ) ) ) )
553adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  ->  B  e.  RR* )
56 ixxun.3 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B T w  <->  -.  w S B ) )
5755, 44, 56syl2anc 643 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( B T w  <->  -.  w S B ) )
5851, 54, 573bitr2d 273 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  ( B P C )  <->  -.  w S B ) )
5948, 58orbi12d 691 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( ( w  e.  ( A O B )  \/  w  e.  ( B P C ) )  <->  ( w S B  \/  -.  w S B ) ) )
6039, 59mpbiri 225 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  /\  w  e.  ( A Q C ) )  -> 
( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )
6138, 60impbida 806 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( ( w  e.  ( A O B )  \/  w  e.  ( B P C ) )  <->  w  e.  ( A Q C ) ) )
621, 61syl5bb 249 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( w  e.  ( ( A O B )  u.  ( B P C ) )  <-> 
w  e.  ( A Q C ) ) )
6362eqrdv 2433 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  -> 
( ( A O B )  u.  ( B P C ) )  =  ( A Q C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    u. cun 3310   class class class wbr 4204  (class class class)co 6073    e. cmpt2 6075   RR*cxr 9111
This theorem is referenced by:  icoun  11013  snunioo  11015  snunico  11016  ioojoin  11019  leordtval2  17268  lecldbas  17275  icopnfcld  18794  iocmnfcld  18795  ioombl  19451  ismbf3d  19538  joiniooico  24127  snunioc  24129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-xr 9116
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