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Theorem xrre 10494
Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
Assertion
Ref Expression
xrre  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )

Proof of Theorem xrre
StepHypRef Expression
1 simprl 732 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  -oo  <  A )
2 ltpnf 10459 . . . . . 6  |-  ( B  e.  RR  ->  B  <  +oo )
32adantl 452 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  <  +oo )
4 rexr 8873 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 10451 . . . . . . 7  |-  +oo  e.  RR*
6 xrlelttr 10483 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo ) )
75, 6mp3an3 1266 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo )
)
84, 7sylan2 460 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  <  +oo )  ->  A  <  +oo )
)
93, 8mpan2d 655 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  <  +oo ) )
109imp 418 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  <  +oo )
1110adantrl 696 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  A  <  +oo )
12 xrrebnd 10493 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
1312ad2antrr 706 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
141, 11, 13mpbir2and 888 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685   class class class wbr 4024   RRcr 8732    +oocpnf 8860    -oocmnf 8861   RR*cxr 8862    < clt 8863    <_ cle 8864
This theorem is referenced by:  xrrege0  10499  supxrre  10642  infmxrre  10650  caucvgrlem  12141  pcgcd1  12925  tgioo  18298  ovolunlem1a  18851  ovoliunlem1  18857  ioombl1lem2  18912  itg2monolem2  19102  dvferm1lem  19327  radcnvle  19792  psercnlem1  19797  nmobndi  21347  ubthlem3  21445  nmophmi  22607  bdophsi  22672  bdopcoi  22674  areacirclem6  24340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-pre-lttri 8807  ax-pre-lttrn 8808
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869
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