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Theorem supxrun 10723
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 3425 . . . 4  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  <->  ( A  u.  B )  C_  RR* )
21biimpi 186 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR* )  ->  ( A  u.  B )  C_ 
RR* )
323adant3 975 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( A  u.  B )  C_  RR* )
4 supxrcl 10722 . . 3  |-  ( B 
C_  RR*  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
543ad2ant2 977 . 2  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  sup ( B ,  RR* ,  <  )  e. 
RR* )
6 elun 3392 . . . 4  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
7 xrltso 10564 . . . . . . . . 9  |-  <  Or  RR*
87a1i 10 . . . . . . . 8  |-  ( A 
C_  RR*  ->  <  Or  RR* )
9 xrsupss 10716 . . . . . . . 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 7297 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
11103ad2ant1 976 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( A ,  RR* ,  <  )  <  x ) )
12 supxrcl 10722 . . . . . . . . . . . . 13  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
1312ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
144ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  sup ( B ,  RR* ,  <  )  e.  RR* )
15 ssel2 3251 . . . . . . . . . . . . 13  |-  ( ( A  C_  RR*  /\  x  e.  A )  ->  x  e.  RR* )
1615adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  B  C_ 
RR* )  /\  x  e.  A )  ->  x  e.  RR* )
17 xrlelttr 10576 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 426 . . . . . . . . . 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 125 . . . . . . . . 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 593 . . . . . . . 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 77 . . . . . . 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 1145 . . . . . 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 36 . . . . 5  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  A  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
257a1i 10 . . . . . . 7  |-  ( B 
C_  RR*  ->  <  Or  RR* )
26 xrsupss 10716 . . . . . . 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 7297 . . . . . 6  |-  ( B 
C_  RR*  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
28273ad2ant2 977 . . . . 5  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  B  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
2924, 28jaod 369 . . . 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 208 . . 3  |-  ( ( A  C_  RR*  /\  B  C_ 
RR*  /\  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR* ,  <  ) )  ->  ( x  e.  ( A  u.  B
)  ->  -.  sup ( B ,  RR* ,  <  )  <  x ) )
3130ralrimiv 2701 . 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 8964 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
33 xrsupss 10716 . . . . . . . 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 7298 . . . . . . 7  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR*  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
3532, 34sylani 635 . . . . . 6  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  B  x  <  y ) )
36 elun2 3419 . . . . . . . 8  |-  ( y  e.  B  ->  y  e.  ( A  u.  B
) )
3736anim1i 551 . . . . . . 7  |-  ( ( y  e.  B  /\  x  <  y )  -> 
( y  e.  ( A  u.  B )  /\  x  <  y
) )
3837reximi2 2725 . . . . . 6  |-  ( E. y  e.  B  x  <  y  ->  E. y  e.  ( A  u.  B
) x  <  y
)
3935, 38syl6 29 . . . . 5  |-  ( B 
C_  RR*  ->  ( (
x  e.  RR  /\  x  <  sup ( B ,  RR* ,  <  ) )  ->  E. y  e.  ( A  u.  B ) x  <  y ) )
4039exp3a 425 . . . 4  |-  ( B 
C_  RR*  ->  ( x  e.  RR  ->  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B )
x  <  y )
) )
4140ralrimiv 2701 . . 3  |-  ( B 
C_  RR*  ->  A. x  e.  RR  ( x  <  sup ( B ,  RR* ,  <  )  ->  E. y  e.  ( A  u.  B
) x  <  y
) )
42413ad2ant2 977 . 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 10720 . 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 1183 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 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    u. cun 3226    C_ wss 3228   class class class wbr 4102    Or wor 4392   supcsup 7280   RRcr 8823   RR*cxr 8953    < clt 8954    <_ cle 8955
This theorem is referenced by:  supxrmnf  10725  xpsdsval  18041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127
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