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Theorem ixxun 11321
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 3502 . . 3  |-  ( w  e.  ( ( A O B )  u.  ( B P C ) )  <->  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )
2 simpl1 991 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  A  e.  RR* )
3 simpl2 992 . . . . . . . . . 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 11314 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
62, 3, 5syl2anc 661 . . . . . . . . 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 484 . . . . . . . 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 1000 . . . . . . 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 1001 . . . . . . 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 1002 . . . . . . . 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 760 . . . . . . . 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 465 . . . . . . . . 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 993 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  C  e.  RR* )
1413adantr 465 . . . . . . . . 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 1218 . . . . . . . 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 679 . . . . . . 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 1168 . . . . . 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 11314 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
213, 13, 20syl2anc 661 . . . . . . . . 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 484 . . . . . . . 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 1000 . . . . . . 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 759 . . . . . . . 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 1001 . . . . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . 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 1218 . . . . . . . 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 679 . . . . . . 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 1002 . . . . . . 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 1168 . . . . . 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 783 . . . . 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 11314 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A Q C )  <->  ( w  e.  RR*  /\  A R w  /\  w U C ) ) )
362, 13, 35syl2anc 661 . . . . . 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 485 . . . . 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 470 . . . 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 415 . . . . 5  |-  ( w S B  \/  -.  w S B )
40 df-3an 967 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  <->  ( (
w  e.  RR*  /\  A R w )  /\  w S B ) )
416, 40syl6bb 261 . . . . . . . 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 465 . . . . . . 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 484 . . . . . . . . . 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 1000 . . . . . . . . 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 1001 . . . . . . . . 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 532 . . . . . . . 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 508 . . . . . . 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 256 . . . . . 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 978 . . . . . . . . 9  |-  ( ( w  e.  RR*  /\  B T w  /\  w U C )  <->  ( B T w  /\  (
w  e.  RR*  /\  w U C ) ) )
5021, 49syl6bb 261 . . . . . . . 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 465 . . . . . . 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 1002 . . . . . . . . 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 532 . . . . . . . 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 507 . . . . . . 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 465 . . . . . . . 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 661 . . . . . . 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 281 . . . . . 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 709 . . . . 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 233 . . . 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 828 . . 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 257 . 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 2441 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2724    u. cun 3331   class class class wbr 4297  (class class class)co 6096    e. cmpt2 6098   RR*cxr 9422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-xr 9427
This theorem is referenced by:  icoun  11414  snunioo  11416  snunico  11417  snunioc  11418  ioojoin  11421  leordtval2  18821  lecldbas  18828  icopnfcld  20352  iocmnfcld  20353  ioombl  21051  ismbf3d  21137  joiniooico  26069  asindmre  28484  ioounsn  29590
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