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Theorem supxrun 11382
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 3631 . . . 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 11381 . . 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 3598 . . . 4  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
7 xrltso 11222 . . . . . . . . 9  |-  <  Or  RR*
87a1i 11 . . . . . . . 8  |-  ( A 
C_  RR*  ->  <  Or  RR* )
9 xrsupss 11375 . . . . . . . 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 7813 . . . . . . 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 11381 . . . . . . . . . . . . 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 3452 . . . . . . . . . . . . 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 11234 . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 1182 . . . . . 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 11375 . . . . . . 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 7813 . . . . . 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 2823 . 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 9533 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
33 xrsupss 11375 . . . . . . . 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 7814 . . . . . . 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 3625 . . . . . . . 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 2921 . . . . . 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 2823 . . 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 11379 . 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 1220 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    u. cun 3427    C_ wss 3429   class class class wbr 4393    Or wor 4741   supcsup 7794   RRcr 9385   RR*cxr 9521    < clt 9522    <_ cle 9523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702
This theorem is referenced by:  supxrmnf  11384  xpsdsval  20081
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