Theorem ltletr 8162

Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)

Assertion

Ref	Expression
ltletr	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A < C ) )

Proof of Theorem ltletr

Step	Hyp	Ref	Expression
1		simprr 531	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> B <_ C )
2		simpl2 1004	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> B e. RR )
3		simpl3 1005	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> C e. RR )
4		lenlt 8148	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
5	2, 3, 4	syl2anc 411	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> ( B <_ C <-> -. C < B ) )
6	1, 5	mpbid 147	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> -. C < B )
7		simprl 529	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> A < B )
8		axltwlin 8140	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )
9	8	adantr 276	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
10	7, 9	mpd 13	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> ( A < C \/ C < B ) )
11	6, 10	ecased 1362	. 2 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> A < C )
12	11	ex 115	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   e. wcel 2176   class class class wbr 4044   RRcr 7924   < clt 8107   <_ cle 8108

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltwlin 8038

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113

This theorem is referenced by:  ltletri  8179  ltletrd  8496  ltleadd  8519  nngt0  9061  nnrecgt0  9074  elnnnn0c  9340  elnnz1  9395  zltp1le  9427  uz3m2nn  9694  ledivge1le  9848  addlelt  9890  zltaddlt1le  10129  elfz1b  10212  elfzodifsumelfzo  10330  ssfzo12bi  10354  cos01gt0  12074  oddge22np1  12192  nn0seqcvgd  12363  coprm  12466  logdivlti  15353  gausslemma2dlem1a  15535