Theorem xrltletr 9743

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 522	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> B <_ C )
2		simpl2 991	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> B e. RR* )
3		simpl3 992	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> C e. RR* )
4		xrlenlt 7963	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
5	2, 3, 4	syl2anc 409	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( B <_ C <-> -. C < B ) )
6	1, 5	mpbid 146	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> -. C < B )
7		simprl 521	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> A < B )
8		xrltso 9732	. . . . . 6 |- < Or RR*
9		sowlin 4298	. . . . . 6 |- ( ( < Or RR* /\ ( A e. RR* /\ B e. RR* /\ C e. RR* ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
10	8, 9	mpan 421	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( A < C \/ C < B ) ) )
11	10	adantr 274	. . . 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 1339	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> A < C )
14	13	ex 114	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B <_ C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   e. wcel 2136   class class class wbr 3982   Or wor 4273   RR*cxr 7932   < clt 7933   <_ cle 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-po 4274  df-iso 4275  df-xp 4610  df-cnv 4612  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by:  xrltletrd  9747  xrre2  9757  xrre3  9758  ge0gtmnf  9759  iooss2  9853  iccssioo  9878  icossico  9879  icossioo  9900  ioossioo  9901  ioc0  10198