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Theorem supxrun 11270
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 3525 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  <->  ( A  u.  B )  C_  RR* )
21biimpi 194 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( A  u.  B )  C_ 
RR* )
323adant3 1008 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( A  u.  B )  C_  RR* )
4 supxrcl 11269 . . 3  |-  ( B 
C_  RR*  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
543ad2ant2 1010 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( B ,  RR* ,  <  )  e. 
RR* )
6 elun 3492 . . . 4  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
7 xrltso 11110 . . . . . . . . 9  |-  <  Or  RR*
87a1i 11 . . . . . . . 8  |-  ( A 
C_  RR*  ->  <  Or  RR* )
9 xrsupss 11263 . . . . . . . 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 7701 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
11103ad2ant1 1009 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
12 supxrcl 11269 . . . . . . . . . . . . 13  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
1312ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
144ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
15 ssel2 3346 . . . . . . . . . . . . 13  |-  ( ( A  C_  RR*  /\  x  e.  A )  ->  x  e.  RR* )
1615adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  x  e.  RR* )
17 xrlelttr 11122 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 437 . . . . . . . . . 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 133 . . . . . . . . 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 610 . . . . . . . 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 83 . . . . . . 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 1181 . . . . . 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 40 . . . . 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 11263 . . . . . . 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 7701 . . . . . 6  |-  ( B 
C_  RR*  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
28273ad2ant2 1010 . . . . 5  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
2924, 28jaod 380 . . . 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 217 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  ( A  u.  B
)  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
3130ralrimiv 2793 . 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 9421 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
33 xrsupss 11263 . . . . . . . 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 7702 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR*  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
3532, 34sylani 654 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
36 elun2 3519 . . . . . . . 8  |-  ( y  e.  B  ->  y  e.  ( A  u.  B
) )
3736anim1i 568 . . . . . . 7  |-  ( ( y  e.  B  /\  x  <  y )  -> 
( y  e.  ( A  u.  B )  /\  x  <  y
) )
3837reximi2 2817 . . . . . 6  |-  ( E. y  e.  B  x  <  y  ->  E. y  e.  ( A  u.  B
) x  <  y
)
3935, 38syl6 33 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  ( A  u.  B ) x  <  y ) )
4039expd 436 . . . 4  |-  ( B 
C_  RR*  ->  ( x  e.  RR  ->  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B )
x  <  y )
) )
4140ralrimiv 2793 . . 3  |-  ( B 
C_  RR*  ->  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B
) x  <  y
) )
42413ad2ant2 1010 . 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 11267 . 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 1219 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    u. cun 3321    C_ wss 3323   class class class wbr 4287    Or wor 4635   supcsup 7682   RRcr 9273   RR*cxr 9409    < clt 9410    <_ cle 9411
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590
This theorem is referenced by:  supxrmnf  11272  xpsdsval  19931
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