Theorem ltletr 8247

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 1025	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> B e. RR )
3		simpl3 1026	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < B /\ B <_ C ) ) -> C e. RR )
4		lenlt 8233	. . . . 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 8225	. . . . 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 1383	. 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 713   /\ w3a 1002   e. wcel 2200   class class class wbr 4083   RRcr 8009   < clt 8192   <_ cle 8193

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-pre-ltwlin 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198

This theorem is referenced by:  ltletri  8264  ltletrd  8581  ltleadd  8604  nngt0  9146  nnrecgt0  9159  elnnnn0c  9425  elnnz1  9480  zltp1le  9512  uz3m2nn  9780  ledivge1le  9934  addlelt  9976  zltaddlt1le  10215  elfz1b  10298  elfzodifsumelfzo  10419  ssfzo12bi  10443  swrdswrd  11253  swrdccatin1  11273  cos01gt0  12290  oddge22np1  12408  nn0seqcvgd  12579  coprm  12682  logdivlti  15571  gausslemma2dlem1a  15753