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Theorem supxrun 11503
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 3678 . . . 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 1016 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( A  u.  B )  C_  RR* )
4 supxrcl 11502 . . 3  |-  ( B 
C_  RR*  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
543ad2ant2 1018 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( B ,  RR* ,  <  )  e. 
RR* )
6 elun 3645 . . . 4  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
7 xrltso 11343 . . . . . . . . 9  |-  <  Or  RR*
87a1i 11 . . . . . . . 8  |-  ( A 
C_  RR*  ->  <  Or  RR* )
9 xrsupss 11496 . . . . . . . 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 7915 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
11103ad2ant1 1017 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
12 supxrcl 11502 . . . . . . . . . . . . 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 3499 . . . . . . . . . . . . 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 11355 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 1190 . . . . . 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 11496 . . . . . . 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 7915 . . . . . 6  |-  ( B 
C_  RR*  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
28273ad2ant2 1018 . . . . 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 2876 . 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 9635 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
33 xrsupss 11496 . . . . . . . 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 7916 . . . . . . 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 3672 . . . . . . . 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 2931 . . . . . 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 2876 . . 3  |-  ( B 
C_  RR*  ->  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B
) x  <  y
) )
42413ad2ant2 1018 . 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 11500 . 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 1229 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    u. cun 3474    C_ wss 3476   class class class wbr 4447    Or wor 4799   supcsup 7896   RRcr 9487   RR*cxr 9623    < clt 9624    <_ cle 9625
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-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804
This theorem is referenced by:  supxrmnf  11505  xpsdsval  20619
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