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Theorem ixxun 10688
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 3329 . . 3  |-  ( w  e.  ( ( A O B )  u.  ( B P C ) )  <->  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )
2 simpl1 958 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  A  e.  RR* )
3 simpl2 959 . . . . . . . . . 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 10681 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
62, 3, 5syl2anc 642 . . . . . . . . 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 470 . . . . . . . 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 967 . . . . . . 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 968 . . . . . . 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 969 . . . . . . . 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 737 . . . . . . . 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 451 . . . . . . . . 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 960 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  C  e.  RR* )
1413adantr 451 . . . . . . . . 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 1182 . . . . . . . 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 660 . . . . . . 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 1132 . . . . . 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 10681 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
213, 13, 20syl2anc 642 . . . . . . . . 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 470 . . . . . . . 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 967 . . . . . . 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 736 . . . . . . . 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 968 . . . . . . . 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 451 . . . . . . . . 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 451 . . . . . . . . 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 1182 . . . . . . . 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 660 . . . . . . 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 969 . . . . . . 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 1132 . . . . . 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 760 . . . . 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 10681 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A Q C )  <->  ( w  e.  RR*  /\  A R w  /\  w U C ) ) )
362, 13, 35syl2anc 642 . . . . . 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 471 . . . . 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 456 . . . 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 404 . . . . 5  |-  ( w S B  \/  -.  w S B )
40 df-3an 936 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  <->  ( (
w  e.  RR*  /\  A R w )  /\  w S B ) )
416, 40syl6bb 252 . . . . . . . 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 451 . . . . . . 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 470 . . . . . . . . . 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 967 . . . . . . . . 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 968 . . . . . . . . 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 518 . . . . . . . 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 494 . . . . . . 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 247 . . . . . 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 947 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  B T w  /\  w U C )  <->  ( B T w  /\  (
w  e.  RR*  /\  w U C ) ) )
5021, 49syl6bb 252 . . . . . . . 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 451 . . . . . . 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 969 . . . . . . . . 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 518 . . . . . . . 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 493 . . . . . . 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 451 . . . . . . . 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 642 . . . . . . 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 272 . . . . . 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 690 . . . . 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 224 . . . 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 805 . . 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 248 . 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 2294 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    u. cun 3163   class class class wbr 4039  (class class class)co 5874    e. cmpt2 5876   RR*cxr 8882
This theorem is referenced by:  icoun  10776  snunioo  10778  snunico  10779  ioojoin  10782  leordtval2  16958  lecldbas  16965  icopnfcld  18293  iocmnfcld  18294  ioombl  18938  ismbf3d  19025  joiniooico  23280  snunioc  23282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-xr 8887
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