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Theorem xrletr 9273
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 9266 . . . . . 6  |-  <  Or  RR*
2 sowlin 4147 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( C  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* ) )  -> 
( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
31, 2mpan 415 . . . . 5  |-  ( ( C  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  ( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
433coml 1150 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  <  A  ->  ( C  <  B  \/  B  <  A ) ) )
5 orcom 682 . . . 4  |-  ( ( C  <  B  \/  B  <  A )  <->  ( B  <  A  \/  C  < 
B ) )
64, 5syl6ib 159 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  <  A  ->  ( B  <  A  \/  C  <  B ) ) )
76con3d 596 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  ( B  <  A  \/  C  <  B )  ->  -.  C  <  A ) )
8 xrlenlt 7551 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
983adant3 963 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
10 xrlenlt 7551 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
11103adant1 961 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  <_  C  <->  -.  C  <  B ) )
129, 11anbi12d 457 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  C )  <-> 
( -.  B  < 
A  /\  -.  C  <  B ) ) )
13 ioran 704 . . 3  |-  ( -.  ( B  <  A  \/  C  <  B )  <-> 
( -.  B  < 
A  /\  -.  C  <  B ) )
1412, 13syl6bbr 196 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  C )  <->  -.  ( B  <  A  \/  C  <  B ) ) )
15 xrlenlt 7551 . . 3  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <_  C  <->  -.  C  <  A ) )
16153adant2 962 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <_  C  <->  -.  C  <  A ) )
177, 14, 163imtr4d 201 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    e. wcel 1438   class class class wbr 3845    Or wor 4122   RR*cxr 7521    < clt 7522    <_ cle 7523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-po 4123  df-iso 4124  df-xp 4444  df-cnv 4446  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528
This theorem is referenced by:  xrletrd  9277  icc0r  9344  iccss  9359  icossico  9361  iccss2  9362  iccssico  9363
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