Theorem xrletr 10189

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 10177	. . . . . 6 |- < Or RR*
2		sowlin 4460	. . . . . 6 |- ( ( < Or RR* /\ ( C e. RR* /\ A e. RR* /\ B e. RR* ) ) -> ( C < A -> ( C < B \/ B < A ) ) )
3	1, 2	mpan 428	. . . . 5 |- ( ( C e. RR* /\ A e. RR* /\ B e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
4	3	3coml 1241	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
5		orcom 740	. . . 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 640	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. ( B < A \/ C < B ) -> -. C < A ) )
8		xrlenlt 8380	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
9	8	3adant3 1048	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B <-> -. B < A ) )
10		xrlenlt 8380	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
11	10	3adant1 1046	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
12	9, 11	anbi12d 477	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) <-> ( -. B < A /\ -. C < B ) ) )
13		ioran 764	. . 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 8380	. . 3 |- ( ( A e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
16	15	3adant2 1047	. 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 720   /\ w3a 1009   e. wcel 2209   class class class wbr 4125   Or wor 4435   RR*cxr 8349   < clt 8350   <_ cle 8351

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-po 4436  df-iso 4437  df-xp 4775  df-cnv 4777  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356

This theorem is referenced by:  xrletrd  10193  xle2add  10260  icc0r  10307  iccss  10322  icossico  10324  iccss2  10325  iccssico  10326  bdxmet  15525