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Theorem lelttr 8008
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 526 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A <_ B )
2 simpl1 995 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> A e. RR )
3 simpl2 996 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B e. RR )
4 lenlt 7995 . . . . . 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 614 . . 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 527 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> B < C )
10 simpl3 997 . . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ B /\ B < C ) ) -> C e. RR )
11 axltwlin 7987 . . . . 5 |- ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( B < C -> ( B < A \/ A < C ) ) )
123, 10, 2, 11syl3anc 1233 . . . 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 713 . 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 703   /\ w3a 973   e. wcel 2141   class class class wbr 3989   RRcr 7773   < clt 7954   <_ cle 7955
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltwlin 7887
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960
This theorem is referenced by:  lelttri  8025  lelttrd  8044  letrp1  8764  ltmul12a  8776  bndndx  9134  uzind  9323  fnn0ind  9328  elfzo0z  10140  fzofzim  10144  elfzodifsumelfzo  10157  flqge  10238  modfzo0difsn  10351  expnlbnd2  10601  caubnd2  11081  mulcn2  11275  cn1lem  11277  climsqz  11298  climsqz2  11299  climcvg1nlem  11312  ltoddhalfle  11852  algcvgblem  12003  pclemub  12241  metss2lem  13291  logdivlti  13596
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