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Theorem ltletr 7877
Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
Assertion
Ref Expression
ltletr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <  C
) )

Proof of Theorem ltletr
StepHypRef Expression
1 simprr 522 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 986 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR )
3 simpl3 987 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR )
4 lenlt 7864 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
52, 3, 4syl2anc 409 . . . 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 521 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  B )
8 axltwlin 7856 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
98adantr 274 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( A  <  B  ->  ( A  <  C  \/  C  < 
B ) ) )
107, 9mpd 13 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( A  <  C  \/  C  < 
B ) )
116, 10ecased 1328 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1211ex 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 698    /\ w3a 963    e. wcel 1481   class class class wbr 3937   RRcr 7643    < clt 7824    <_ cle 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltwlin 7757
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830
This theorem is referenced by:  ltletri  7894  ltletrd  8209  ltleadd  8232  nngt0  8769  nnrecgt0  8782  elnnnn0c  9046  elnnz1  9101  zltp1le  9132  uz3m2nn  9395  ledivge1le  9543  addlelt  9585  zltaddlt1le  9820  elfz1b  9901  elfzodifsumelfzo  10009  ssfzo12bi  10033  cos01gt0  11505  oddge22np1  11614  nn0seqcvgd  11758  coprm  11858  logdivlti  13010
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