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Theorem lelttr 8110
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 529 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2 simpl1 1002 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A e. RR )
3 simpl2 1003 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B e. RR )
4 lenlt 8097 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
52, 3, 4syl2anc 411 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> ( A <_ B <-> -. B < A ) )
61, 5mpbid 147 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> -. B < A )
76pm2.21d 620 . . 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 531 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10 simpl3 1004 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> C e. RR )
11 axltwlin 8089 . . . . 5 |- ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( B < C -> ( B < A \/ A < C ) ) )
123, 10, 2, 11syl3anc 1249 . . . 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 719 . 2 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A < C )
1514ex 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 709   /\ w3a 980   e. wcel 2164   class class class wbr 4030   RRcr 7873   < clt 8056   <_ cle 8057
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltwlin 7987
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062
This theorem is referenced by:  lelttri  8127  lelttrd  8146  letrp1  8869  ltmul12a  8881  bndndx  9242  uzind  9431  fnn0ind  9436  elfzo0z  10254  fzofzim  10258  elfzodifsumelfzo  10271  flqge  10354  modfzo0difsn  10469  expnlbnd2  10739  caubnd2  11264  mulcn2  11458  cn1lem  11460  climsqz  11481  climsqz2  11482  climcvg1nlem  11495  ltoddhalfle  12037  algcvgblem  12190  pclemub  12428  metss2lem  14676  logdivlti  15057  gausslemma2dlem2  15219
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