Theorem xrltletr 9791

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Assertion

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

Proof of Theorem xrltletr

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 1001	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> B e. RR* )
3		simpl3 1002	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> C e. RR* )
4		xrlenlt 8009	. . . . 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		xrltso 9780	. . . . . 6 |- < Or RR*
9		sowlin 4317	. . . . . 6 |- ( ( < Or RR* /\ ( A e. RR* /\ B e. RR* /\ C e. RR* ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
10	8, 9	mpan 424	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( A < C \/ C < B ) ) )
11	10	adantr 276	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
12	7, 11	mpd 13	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( A < C \/ C < B ) )
13	6, 12	ecased 1349	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> A < C )
14	13	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 708   /\ w3a 978   e. wcel 2148   class class class wbr 4000   Or wor 4292   RR*cxr 7978   < clt 7979   <_ cle 7980

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-po 4293  df-iso 4294  df-xp 4629  df-cnv 4631  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985

This theorem is referenced by:  xrltletrd  9795  xrre2  9805  xrre3  9806  ge0gtmnf  9807  iooss2  9901  iccssioo  9926  icossico  9927  icossioo  9948  ioossioo  9949  ioc0  10246