Theorem xrletr 10087

Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Assertion

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

Proof of Theorem xrletr

Step	Hyp	Ref	Expression
1		xrltso 10075	. . . . . 6 |- < Or RR*
2		sowlin 4423	. . . . . 6 |- ( ( < Or RR* /\ ( C e. RR* /\ A e. RR* /\ B e. RR* ) ) -> ( C < A -> ( C < B \/ B < A ) ) )
3	1, 2	mpan 424	. . . . 5 |- ( ( C e. RR* /\ A e. RR* /\ B e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
4	3	3coml 1237	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
5		orcom 736	. . . 4 |- ( ( C < B \/ B < A ) <-> ( B < A \/ C < B ) )
6	4, 5	imbitrdi 161	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( B < A \/ C < B ) ) )
7	6	con3d 636	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. ( B < A \/ C < B ) -> -. C < A ) )
8		xrlenlt 8286	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
9	8	3adant3 1044	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B <-> -. B < A ) )
10		xrlenlt 8286	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
11	10	3adant1 1042	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
12	9, 11	anbi12d 473	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) <-> ( -. B < A /\ -. C < B ) ) )
13		ioran 760	. . 3 |- ( -. ( B < A \/ C < B ) <-> ( -. B < A /\ -. C < B ) )
14	12, 13	bitr4di 198	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) <-> -. ( B < A \/ C < B ) ) )
15		xrlenlt 8286	. . 3 |- ( ( A e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
16	15	3adant2 1043	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
17	7, 14, 16	3imtr4d 203	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 716   /\ w3a 1005   e. wcel 2202   class class class wbr 4093   Or wor 4398   RR*cxr 8255   < clt 8256   <_ cle 8257

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-po 4399  df-iso 4400  df-xp 4737  df-cnv 4739  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262

This theorem is referenced by:  xrletrd  10091  xle2add  10158  icc0r  10205  iccss  10220  icossico  10222  iccss2  10223  iccssico  10224  bdxmet  15295