MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xrlttr Structured version   Unicode version

Theorem xrlttr 11371
Description: Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrlttr  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )

Proof of Theorem xrlttr
StepHypRef Expression
1 elxr 11350 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11350 . . 3  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
3 elxr 11350 . . . . . . . . 9  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4 lttr 9678 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
543expa 1196 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
65an32s 804 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7 rexr 9656 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  RR* )
8 pnfnlt 11362 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR*  ->  -. +oo  <  C )
97, 8syl 16 . . . . . . . . . . . . . . 15  |-  ( C  e.  RR  ->  -. +oo 
<  C )
109adantr 465 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -. +oo  <  C
)
11 breq1 4459 . . . . . . . . . . . . . . 15  |-  ( B  = +oo  ->  ( B  <  C  <-> +oo  <  C
) )
1211adantl 466 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  <-> +oo 
<  C ) )
1310, 12mtbird 301 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -.  B  <  C
)
1413pm2.21d 106 . . . . . . . . . . . 12  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  ->  A  <  C ) )
1514adantll 713 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( B  < 
C  ->  A  <  C ) )
1615adantld 467 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
17 rexr 9656 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  A  e.  RR* )
18 nltmnf 11363 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR*  ->  -.  A  < -oo )
1917, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  -.  A  < -oo )
2019adantr 465 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
21 breq2 4460 . . . . . . . . . . . . . . 15  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2221adantl 466 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
2320, 22mtbird 301 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
2423pm2.21d 106 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  A  <  C ) )
2524adantlr 714 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  ->  A  <  C ) )
2625adantrd 468 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
276, 16, 263jaodan 1294 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
283, 27sylan2b 475 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR* )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
2928an32s 804 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
30 ltpnf 11356 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  < +oo )
3130adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  < +oo )
32 breq2 4460 . . . . . . . . . . 11  |-  ( C  = +oo  ->  ( A  <  C  <->  A  < +oo ) )
3332adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  ( A  <  C  <->  A  < +oo ) )
3431, 33mpbird 232 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  <  C )
3534adantlr 714 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  A  <  C
)
3635a1d 25 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
37 nltmnf 11363 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -.  B  < -oo )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  < -oo )
39 breq2 4460 . . . . . . . . . . . 12  |-  ( C  = -oo  ->  ( B  <  C  <->  B  < -oo ) )
4039adantl 466 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  <->  B  < -oo ) )
4138, 40mtbird 301 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  <  C )
4241pm2.21d 106 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  ->  A  <  C ) )
4342adantld 467 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4443adantll 713 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
4529, 36, 443jaodan 1294 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4645anasss 647 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
47 pnfnlt 11362 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -. +oo  <  B )
4847adantl 466 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
49 breq1 4459 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
5049adantr 465 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
5148, 50mtbird 301 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
5251pm2.21d 106 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <  C ) )
5352adantrd 468 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
5453adantrr 716 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
55 mnflt 11358 . . . . . . . . . . 11  |-  ( C  e.  RR  -> -oo  <  C )
5655adantl 466 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  -> -oo  <  C )
57 breq1 4459 . . . . . . . . . . 11  |-  ( A  = -oo  ->  ( A  <  C  <-> -oo  <  C
) )
5857adantr 465 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( A  <  C  <-> -oo 
<  C ) )
5956, 58mpbird 232 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  A  <  C )
6059a1d 25 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6160adantlr 714 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
62 mnfltpnf 11360 . . . . . . . . . 10  |- -oo  < +oo
63 breq12 4461 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( A  <  C  <-> -oo 
< +oo ) )
6462, 63mpbiri 233 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  = +oo )  ->  A  <  C )
6564a1d 25 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6665adantlr 714 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6743adantll 713 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6861, 66, 673jaodan 1294 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
6968anasss 647 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7046, 54, 693jaoian 1293 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e. 
RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
71703impb 1192 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
722, 71syl3an3b 1266 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
731, 72syl3an1b 1264 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   RRcr 9508   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650
This theorem is referenced by:  xrltso  11372  xrlelttr  11384  xrltletr  11385  xrlttrd  11387  xrub  11528  ioo0  11579  ioojoin  11676  leordtval2  19840  icopnfcld  21401  iocmnfcld  21402  ismbf3d  22187  tanord1  23050  tan2h  30252  asindmre  30307
  Copyright terms: Public domain W3C validator