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Theorem lelttr 7552
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 498 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 946 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR )
3 simpl2 947 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR )
4 lenlt 7540 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 403 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 145 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 584 . . 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 499 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 948 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR )
11 axltwlin 7533 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  A  e.  RR )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
123, 10, 2, 11syl3anc 1174 . . . 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 673 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1514ex 113 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 3837   RRcr 7328    < clt 7501    <_ cle 7502
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 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-pre-ltwlin 7437
This theorem depends on definitions:  df-bi 115  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 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507
This theorem is referenced by:  lelttri  7569  lelttrd  7587  letrp1  8281  ltmul12a  8293  bndndx  8642  uzind  8827  fnn0ind  8832  elfzo0z  9560  fzofzim  9564  elfzodifsumelfzo  9577  flqge  9654  modfzo0difsn  9767  expnlbnd2  10044  caubnd2  10515  mulcn2  10665  cn1lem  10666  climsqz  10687  climsqz2  10688  climcvg1nlem  10702  ltoddhalfle  10986  algcvgblem  11124
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