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Theorem ltletr 7107
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 484 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 908 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR )
3 simpl3 909 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR )
4 lenlt 7094 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
52, 3, 4syl2anc 391 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( B  <_  C  <->  -.  C  <  B ) )
61, 5mpbid 135 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  -.  C  <  B )
7 simprl 483 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  B )
8 axltwlin 7087 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
98adantr 261 . . . 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 1239 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1211ex 108 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 97    <-> wb 98    \/ wo 629    /\ w3a 885    e. wcel 1393   class class class wbr 3764   RRcr 6888    < clt 7060    <_ cle 7061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltwlin 6997
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066
This theorem is referenced by:  ltletri  7124  ltletrd  7420  ltleadd  7441  nngt0  7939  nnrecgt0  7951  elnnnn0c  8227  elnnz1  8268  zltp1le  8298  uz3m2nn  8515  ledivge1le  8652  elfz1b  8952  elfzodifsumelfzo  9057  ssfzo12bi  9081  nn0seqcvgd  9880
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