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Theorem ixxun 11545
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 3645 . . 3  |-  ( w  e.  ( ( A O B )  u.  ( B P C ) )  <->  ( w  e.  ( A O B )  \/  w  e.  ( B P C ) ) )
2 simpl1 999 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A W B  /\  B X C ) )  ->  A  e.  RR* )
3 simpl2 1000 . . . . . . . . . 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 11538 . . . . . . . . . 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 1008 . . . . . . 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 1009 . . . . . . 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 1010 . . . . . . . 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 1001 . . . . . . . . . 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 1228 . . . . . . . 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 1176 . . . . . 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 11538 . . . . . . . . . 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 1008 . . . . . . 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 1009 . . . . . . . 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 1228 . . . . . . . 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 1010 . . . . . . 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 1176 . . . . . 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 11538 . . . . . . 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 975 . . . . . . . . 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 1008 . . . . . . . . 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 1009 . . . . . . . . 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 986 . . . . . . . . 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 1010 . . . . . . . . 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 830 . . 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 2464 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 973    = wceq 1379    e. wcel 1767   {crab 2818    u. cun 3474   class class class wbr 4447  (class class class)co 6284    |-> cmpt2 6286   RR*cxr 9627
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 4568  ax-nul 4576  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-xr 9632
This theorem is referenced by:  icoun  11644  snunioo  11646  snunico  11647  snunioc  11648  ioojoin  11651  leordtval2  19507  lecldbas  19514  icopnfcld  21038  iocmnfcld  21039  ioombl  21738  ismbf3d  21824  joiniooico  27281  asindmre  29707  ioounsn  30810  snunioo2  31136  snunioo1  31144
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