Theorem ltletr 8164

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 8150	. . . . 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 8142	. . . . 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 4045   RRcr 7926   < clt 8109   <_ cle 8110

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 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-pre-ltwlin 8040

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 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115

This theorem is referenced by:  ltletri  8181  ltletrd  8498  ltleadd  8521  nngt0  9063  nnrecgt0  9076  elnnnn0c  9342  elnnz1  9397  zltp1le  9429  uz3m2nn  9696  ledivge1le  9850  addlelt  9892  zltaddlt1le  10131  elfz1b  10214  elfzodifsumelfzo  10332  ssfzo12bi  10356  cos01gt0  12107  oddge22np1  12225  nn0seqcvgd  12396  coprm  12499  logdivlti  15386  gausslemma2dlem1a  15568