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Theorem lelttr 7165
Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
Assertion
Ref Expression
lelttr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  B  /\  B  <  C )  ->  A  <  C
) )

Proof of Theorem lelttr
StepHypRef Expression
1 simprl 491 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 918 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR )
3 simpl2 919 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR )
4 lenlt 7153 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 397 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 139 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 559 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( B  <  A  ->  A  <  C ) )
8 idd 21 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( A  <  C  ->  A  <  C ) )
9 simprr 492 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 920 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR )
11 axltwlin 7146 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  A  e.  RR )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
123, 10, 2, 11syl3anc 1146 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( B  <  C  ->  ( B  <  A  \/  A  < 
C ) ) )
139, 12mpd 13 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( B  <  A  \/  A  < 
C ) )
147, 8, 13mpjaod 648 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1514ex 112 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   class class class wbr 3792   RRcr 6946    < clt 7119    <_ cle 7120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltwlin 7055
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125
This theorem is referenced by:  lelttri  7182  lelttrd  7200  letrp1  7889  ltmul12a  7901  bndndx  8238  uzind  8408  fnn0ind  8413  elfzo0z  9142  fzofzim  9146  elfzodifsumelfzo  9159  flqge  9232  modfzo0difsn  9345  expnlbnd2  9542  caubnd2  9944  mulcn2  10064  cn1lem  10065  climsqz  10086  climsqz2  10087  climcvg1nlem  10099  ltoddhalfle  10205  algcvgblem  10271
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