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Theorem ltletr 8024
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 531 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 1001 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR )
3 simpl3 1002 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR )
4 lenlt 8010 . . . . 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 axltwlin 8002 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
98adantr 276 . . . 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 1349 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1211ex 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 708    /\ w3a 978    e. wcel 2148   class class class wbr 4000   RRcr 7788    < clt 7969    <_ cle 7970
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 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-pre-ltwlin 7902
This theorem depends on definitions:  df-bi 117  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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-cnv 4630  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975
This theorem is referenced by:  ltletri  8041  ltletrd  8357  ltleadd  8380  nngt0  8920  nnrecgt0  8933  elnnnn0c  9197  elnnz1  9252  zltp1le  9283  uz3m2nn  9549  ledivge1le  9700  addlelt  9742  zltaddlt1le  9981  elfz1b  10063  elfzodifsumelfzo  10174  ssfzo12bi  10198  cos01gt0  11741  oddge22np1  11856  nn0seqcvgd  12011  coprm  12114  logdivlti  13935
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