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Theorem lelttr 8132
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 8119 . . . . . 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 8111 . . . . 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 2167   class class class wbr 4034   RRcr 7895   < clt 8078   <_ cle 8079
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-ltwlin 8009
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084
This theorem is referenced by:  lelttri  8149  lelttrd  8168  letrp1  8892  ltmul12a  8904  bndndx  9265  uzind  9454  fnn0ind  9459  elfzo0z  10277  fzofzim  10281  elfzodifsumelfzo  10294  flqge  10389  modfzo0difsn  10504  expnlbnd2  10774  caubnd2  11299  mulcn2  11494  cn1lem  11496  climsqz  11517  climsqz2  11518  climcvg1nlem  11531  ltoddhalfle  12075  algcvgblem  12242  pclemub  12481  metss2lem  14817  logdivlti  15201  gausslemma2dlem2  15387
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