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Theorem lelttr 7816
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 503 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 967 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR )
3 simpl2 968 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR )
4 lenlt 7804 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 406 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  ( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 146 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 591 . . 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 504 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 969 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR )
11 axltwlin 7796 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  A  e.  RR )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
123, 10, 2, 11syl3anc 1199 . . . 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 690 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1514ex 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 680    /\ w3a 945    e. wcel 1463   class class class wbr 3897   RRcr 7583    < clt 7764    <_ cle 7765
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltwlin 7697
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770
This theorem is referenced by:  lelttri  7833  lelttrd  7851  letrp1  8563  ltmul12a  8575  bndndx  8927  uzind  9113  fnn0ind  9118  elfzo0z  9901  fzofzim  9905  elfzodifsumelfzo  9918  flqge  9995  modfzo0difsn  10108  expnlbnd2  10357  caubnd2  10829  mulcn2  11021  cn1lem  11023  climsqz  11044  climsqz2  11045  climcvg1nlem  11058  ltoddhalfle  11486  algcvgblem  11626  metss2lem  12561
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