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Theorem lelttr 7959
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 521 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2 simpl1 985 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A e. RR )
3 simpl2 986 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B e. RR )
4 lenlt 7947 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
52, 3, 4syl2anc 409 . . . . 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 609 . . 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 522 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10 simpl3 987 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> C e. RR )
11 axltwlin 7939 . . . . 5 |- ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( B < C -> ( B < A \/ A < C ) ) )
123, 10, 2, 11syl3anc 1220 . . . 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 708 . 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 698   /\ w3a 963   e. wcel 2128   class class class wbr 3965   RRcr 7725   < clt 7906   <_ cle 7907
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-pre-ltwlin 7839
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-cnv 4593  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912
This theorem is referenced by:  lelttri  7976  lelttrd  7994  letrp1  8713  ltmul12a  8725  bndndx  9083  uzind  9269  fnn0ind  9274  elfzo0z  10076  fzofzim  10080  elfzodifsumelfzo  10093  flqge  10174  modfzo0difsn  10287  expnlbnd2  10536  caubnd2  11010  mulcn2  11202  cn1lem  11204  climsqz  11225  climsqz2  11226  climcvg1nlem  11239  ltoddhalfle  11776  algcvgblem  11917  metss2lem  12868  logdivlti  13173
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