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Theorem xrltletr 9431
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 502 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 953 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR* )
3 simpl3 954 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR* )
4 xrlenlt 7701 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
52, 3, 4syl2anc 406 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( B  <_  C  <->  -.  C  <  B ) )
61, 5mpbid 146 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  -.  C  <  B )
7 simprl 501 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  B )
8 xrltso 9423 . . . . . 6  |-  <  Or  RR*
9 sowlin 4180 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* ) )  -> 
( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
108, 9mpan 418 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
1110adantr 272 . . . 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 1295 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1413ex 114 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 103    <-> wb 104    \/ wo 670    /\ w3a 930    e. wcel 1448   class class class wbr 3875    Or wor 4155   RR*cxr 7671    < clt 7672    <_ cle 7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-po 4156  df-iso 4157  df-xp 4483  df-cnv 4485  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678
This theorem is referenced by:  xrltletrd  9435  xrre2  9445  xrre3  9446  ge0gtmnf  9447  iooss2  9541  iccssioo  9566  icossico  9567  icossioo  9588  ioossioo  9589  ioc0  9881
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