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Theorem xrltletr 9820
Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
xrltletr  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <  C
) )

Proof of Theorem xrltletr
StepHypRef Expression
1 simprr 531 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 1002 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR* )
3 simpl3 1003 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR* )
4 xrlenlt 8035 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
52, 3, 4syl2anc 411 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( B  <_  C  <->  -.  C  <  B ) )
61, 5mpbid 147 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  -.  C  <  B )
7 simprl 529 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  B )
8 xrltso 9809 . . . . . 6  |-  <  Or  RR*
9 sowlin 4332 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )  -> 
( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
108, 9mpan 424 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
1110adantr 276 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
127, 11mpd 13 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( A  <  C  \/  C  <  B ) )
136, 12ecased 1359 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1413ex 115 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    /\ w3a 979    e. wcel 2158   class class class wbr 4015    Or wor 4307   RR*cxr 8004    < clt 8005    <_ cle 8006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-po 4308  df-iso 4309  df-xp 4644  df-cnv 4646  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011
This theorem is referenced by:  xrltletrd  9824  xrre2  9834  xrre3  9835  ge0gtmnf  9836  iooss2  9930  iccssioo  9955  icossico  9956  icossioo  9977  ioossioo  9978  ioc0  10276
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