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Theorem xrletr 9810
Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrletr  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  C )  ->  A  <_  C
) )

Proof of Theorem xrletr
StepHypRef Expression
1 xrltso 9798 . . . . . 6  |-  <  Or  RR*
2 sowlin 4322 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( C  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* ) )  -> 
( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
31, 2mpan 424 . . . . 5  |-  ( ( C  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  ( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
433coml 1210 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
5 orcom 728 . . . 4  |-  ( ( C  <  B  \/  B  <  A )  <->  ( B  <  A  \/  C  < 
B ) )
64, 5imbitrdi 161 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  <  A  ->  ( B  <  A  \/  C  <  B ) ) )
76con3d 631 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  ( B  <  A  \/  C  <  B )  ->  -.  C  <  A ) )
8 xrlenlt 8024 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
983adant3 1017 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
10 xrlenlt 8024 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
11103adant1 1015 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
129, 11anbi12d 473 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  C )  <-> 
( -.  B  < 
A  /\  -.  C  <  B ) ) )
13 ioran 752 . . 3  |-  ( -.  ( B  <  A  \/  C  <  B )  <-> 
( -.  B  < 
A  /\  -.  C  <  B ) )
1412, 13bitr4di 198 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  C )  <->  -.  ( B  <  A  \/  C  <  B ) ) )
15 xrlenlt 8024 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <_  C  <->  -.  C  <  A ) )
16153adant2 1016 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <_  C  <->  -.  C  <  A ) )
177, 14, 163imtr4d 203 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 708    /\ w3a 978    e. wcel 2148   class class class wbr 4005    Or wor 4297   RR*cxr 7993    < clt 7994    <_ cle 7995
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-po 4298  df-iso 4299  df-xp 4634  df-cnv 4636  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000
This theorem is referenced by:  xrletrd  9814  xle2add  9881  icc0r  9928  iccss  9943  icossico  9945  iccss2  9946  iccssico  9947  bdxmet  14086
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