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Theorem xrre 11371
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 755 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  -> -oo  <  A )
2 ltpnf 11332 . . . . . 6  |-  ( B  e.  RR  ->  B  < +oo )
32adantl 466 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  B  < +oo )
4 rexr 9640 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
5 pnfxr 11322 . . . . . . 7  |- +oo  e.  RR*
6 xrlelttr 11360 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A  <_  B  /\  B  < +oo )  ->  A  < +oo ) )
75, 6mp3an3 1313 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
84, 7sylan2 474 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A  <_  B  /\  B  < +oo )  ->  A  < +oo )
)
93, 8mpan2d 674 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  ->  A  < +oo ) )
109imp 429 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  < +oo )
1110adantrl 715 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  < +oo )
12 xrrebnd 11370 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
1312ad2antrr 725 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  ( A  e.  RR  <->  ( -oo  <  A  /\  A  < +oo ) ) )
141, 11, 13mpbir2and 920 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( -oo  <  A  /\  A  <_  B
) )  ->  A  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   class class class wbr 4447   RRcr 9492   +oocpnf 9626   -oocmnf 9627   RR*cxr 9628    < clt 9629    <_ cle 9630
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 6577  ax-cnex 9549  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635
This theorem is referenced by:  xrrege0  11376  supxrre  11520  infmxrre  11528  caucvgrlem  13461  pcgcd1  14262  tgioo  21128  ovolunlem1a  21734  ovoliunlem1  21740  ioombl1lem2  21796  itg2monolem2  21985  dvferm1lem  22212  radcnvle  22641  psercnlem1  22646  nmobndi  25463  ubthlem3  25561  nmophmi  26723  bdophsi  26788  bdopcoi  26790  orvclteel  28162  itg2addnclem  29919  itg2gt0cn  29923  areacirclem5  29964  eliocre  31338  fourierdlem87  31721
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