MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supxrun Unicode version

Theorem supxrun 10850
Description: The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
supxrun  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( ( A  u.  B ) , 
RR* ,  <  )  =  sup ( B ,  RR* ,  <  ) )

Proof of Theorem supxrun
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unss 3481 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  <->  ( A  u.  B )  C_  RR* )
21biimpi 187 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( A  u.  B )  C_ 
RR* )
323adant3 977 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( A  u.  B )  C_  RR* )
4 supxrcl 10849 . . 3  |-  ( B 
C_  RR*  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
543ad2ant2 979 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( B ,  RR* ,  <  )  e. 
RR* )
6 elun 3448 . . . 4  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
7 xrltso 10690 . . . . . . . . 9  |-  <  Or  RR*
87a1i 11 . . . . . . . 8  |-  ( A 
C_  RR*  ->  <  Or  RR* )
9 xrsupss 10843 . . . . . . . 8  |-  ( A 
C_  RR*  ->  E. y  e.  RR*  ( A. z  e.  A  -.  y  <  z  /\  A. z  e.  RR*  ( z  < 
y  ->  E. w  e.  A  z  <  w ) ) )
108, 9supub 7420 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
11103ad2ant1 978 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
12 supxrcl 10849 . . . . . . . . . . . . 13  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
1312ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
144ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
15 ssel2 3303 . . . . . . . . . . . . 13  |-  ( ( A  C_  RR*  /\  x  e.  A )  ->  x  e.  RR* )
1615adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  x  e.  RR* )
17 xrlelttr 10702 . . . . . . . . . . . 12  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( B ,  RR* ,  <  )  e.  RR*  /\  x  e.  RR* )  ->  (
( sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  )  /\  sup ( B ,  RR* ,  <  )  <  x
)  ->  sup ( A ,  RR* ,  <  )  <  x ) )
1813, 14, 16, 17syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  (
( sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  )  /\  sup ( B ,  RR* ,  <  )  <  x
)  ->  sup ( A ,  RR* ,  <  )  <  x ) )
1918expdimp 427 . . . . . . . . . 10  |-  ( ( ( ( A  C_  RR* 
/\  B  C_  RR* )  /\  x  e.  A
)  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( sup ( B ,  RR* ,  <  )  <  x  ->  sup ( A ,  RR* ,  <  )  <  x ) )
2019con3d 127 . . . . . . . . 9  |-  ( ( ( ( A  C_  RR* 
/\  B  C_  RR* )  /\  x  e.  A
)  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( -.  sup ( A ,  RR* ,  <  )  <  x  ->  -.  sup ( B ,  RR* ,  <  )  <  x
) )
2120exp41 594 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( B  C_ 
RR*  ->  ( x  e.  A  ->  ( sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  )  -> 
( -.  sup ( A ,  RR* ,  <  )  <  x  ->  -.  sup ( B ,  RR* ,  <  )  <  x
) ) ) ) )
2221com34 79 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( B  C_ 
RR*  ->  ( sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  )  -> 
( x  e.  A  ->  ( -.  sup ( A ,  RR* ,  <  )  <  x  ->  -.  sup ( B ,  RR* ,  <  )  <  x
) ) ) ) )
23223imp 1147 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  ( -.  sup ( A ,  RR* ,  <  )  <  x  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) ) )
2411, 23mpdd 38 . . . . 5  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
257a1i 11 . . . . . . 7  |-  ( B 
C_  RR*  ->  <  Or  RR* )
26 xrsupss 10843 . . . . . . 7  |-  ( B 
C_  RR*  ->  E. y  e.  RR*  ( A. z  e.  B  -.  y  <  z  /\  A. z  e.  RR*  ( z  < 
y  ->  E. w  e.  B  z  <  w ) ) )
2725, 26supub 7420 . . . . . 6  |-  ( B 
C_  RR*  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
28273ad2ant2 979 . . . . 5  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
2924, 28jaod 370 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( ( x  e.  A  \/  x  e.  B )  ->  -.  sup ( B ,  RR* ,  <  )  <  x
) )
306, 29syl5bi 209 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  ( A  u.  B
)  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
3130ralrimiv 2748 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  A. x  e.  ( A  u.  B )  -.  sup ( B ,  RR* ,  <  )  <  x )
32 rexr 9086 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
33 xrsupss 10843 . . . . . . . 8  |-  ( B 
C_  RR*  ->  E. x  e.  RR*  ( A. z  e.  B  -.  x  <  z  /\  A. z  e.  RR*  ( z  < 
x  ->  E. y  e.  B  z  <  y ) ) )
3425, 33suplub 7421 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR*  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
3532, 34sylani 636 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
36 elun2 3475 . . . . . . . 8  |-  ( y  e.  B  ->  y  e.  ( A  u.  B
) )
3736anim1i 552 . . . . . . 7  |-  ( ( y  e.  B  /\  x  <  y )  -> 
( y  e.  ( A  u.  B )  /\  x  <  y
) )
3837reximi2 2772 . . . . . 6  |-  ( E. y  e.  B  x  <  y  ->  E. y  e.  ( A  u.  B
) x  <  y
)
3935, 38syl6 31 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  ( A  u.  B ) x  <  y ) )
4039exp3a 426 . . . 4  |-  ( B 
C_  RR*  ->  ( x  e.  RR  ->  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B )
x  <  y )
) )
4140ralrimiv 2748 . . 3  |-  ( B 
C_  RR*  ->  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B
) x  <  y
) )
42413ad2ant2 979 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B ) x  <  y ) )
43 supxr 10847 . 2  |-  ( ( ( ( A  u.  B )  C_  RR*  /\  sup ( B ,  RR* ,  <  )  e.  RR* )  /\  ( A. x  e.  ( A  u.  B )  -.  sup ( B ,  RR* ,  <  )  < 
x  /\  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B
) x  <  y
) ) )  ->  sup ( ( A  u.  B ) ,  RR* ,  <  )  =  sup ( B ,  RR* ,  <  ) )
443, 5, 31, 42, 43syl22anc 1185 1  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( ( A  u.  B ) , 
RR* ,  <  )  =  sup ( B ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    u. cun 3278    C_ wss 3280   class class class wbr 4172    Or wor 4462   supcsup 7403   RRcr 8945   RR*cxr 9075    < clt 9076    <_ cle 9077
This theorem is referenced by:  supxrmnf  10852  xpsdsval  18364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250
  Copyright terms: Public domain W3C validator