Theorem xrletr 10000

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 9988	. . . . . 6 |- < Or RR*
2		sowlin 4410	. . . . . 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 1234	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < A -> ( C < B \/ B < A ) ) )
5		orcom 733	. . . 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 634	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. ( B < A \/ C < B ) -> -. C < A ) )
8		xrlenlt 8207	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
9	8	3adant3 1041	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B <-> -. B < A ) )
10		xrlenlt 8207	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
11	10	3adant1 1039	. . . 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 757	. . 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 8207	. . 3 |- ( ( A e. RR* /\ C e. RR* ) -> ( A <_ C <-> -. C < A ) )
16	15	3adant2 1040	. 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 713   /\ w3a 1002   e. wcel 2200   class class class wbr 4082   Or wor 4385   RR*cxr 8176   < clt 8177   <_ cle 8178

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109

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 3888  df-br 4083  df-opab 4145  df-po 4386  df-iso 4387  df-xp 4724  df-cnv 4726  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183

This theorem is referenced by:  xrletrd  10004  xle2add  10071  icc0r  10118  iccss  10133  icossico  10135  iccss2  10136  iccssico  10137  bdxmet  15169