Theorem xrltletr 10015

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 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		xrlenlt 8222	. . . . 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 10004	. . . . . 6 |- < Or RR*
9		sowlin 4411	. . . . . 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 1383	. 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 713   /\ w3a 1002   e. wcel 2200   class class class wbr 4083   Or wor 4386   RR*cxr 8191   < 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-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124

This theorem depends on definitions:  df-bi 117  df-3or 1003  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-po 4387  df-iso 4388  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:  xrltletrd  10019  xrre2  10029  xrre3  10030  ge0gtmnf  10031  iooss2  10125  iccssioo  10150  icossico  10151  icossioo  10172  ioossioo  10173  ioc0  10494