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Theorem xrlttr 9521
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 9503 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9503 . . 3  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
3 elxr 9503 . . . . . . . . 9  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4 lttr 7802 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
543expa 1164 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
65an32s 540 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7 rexr 7775 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  RR* )
8 pnfnlt 9513 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR*  ->  -. +oo  <  C )
97, 8syl 14 . . . . . . . . . . . . . . 15  |-  ( C  e.  RR  ->  -. +oo 
<  C )
109adantr 272 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -. +oo  <  C
)
11 breq1 3900 . . . . . . . . . . . . . . 15  |-  ( B  = +oo  ->  ( B  <  C  <-> +oo  <  C
) )
1211adantl 273 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  <-> +oo 
<  C ) )
1310, 12mtbird 645 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -.  B  <  C
)
1413pm2.21d 591 . . . . . . . . . . . 12  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  ->  A  <  C ) )
1514adantll 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( B  < 
C  ->  A  <  C ) )
1615adantld 274 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
17 rexr 7775 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  A  e.  RR* )
18 nltmnf 9514 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR*  ->  -.  A  < -oo )
1917, 18syl 14 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  -.  A  < -oo )
2019adantr 272 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
21 breq2 3901 . . . . . . . . . . . . . . 15  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2221adantl 273 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
2320, 22mtbird 645 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
2423pm2.21d 591 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  A  <  C ) )
2524adantlr 466 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  ->  A  <  C ) )
2625adantrd 275 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
276, 16, 263jaodan 1267 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
283, 27sylan2b 283 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR* )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
2928an32s 540 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
30 ltpnf 9507 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  < +oo )
3130adantr 272 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  < +oo )
32 breq2 3901 . . . . . . . . . . 11  |-  ( C  = +oo  ->  ( A  <  C  <->  A  < +oo ) )
3332adantl 273 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  ( A  <  C  <->  A  < +oo ) )
3431, 33mpbird 166 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  <  C )
3534adantlr 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  A  <  C
)
3635a1d 22 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
37 nltmnf 9514 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -.  B  < -oo )
3837adantr 272 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  < -oo )
39 breq2 3901 . . . . . . . . . . . 12  |-  ( C  = -oo  ->  ( B  <  C  <->  B  < -oo ) )
4039adantl 273 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  <->  B  < -oo ) )
4138, 40mtbird 645 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  <  C )
4241pm2.21d 591 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  ->  A  <  C ) )
4342adantld 274 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4443adantll 465 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
4529, 36, 443jaodan 1267 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4645anasss 394 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
47 pnfnlt 9513 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -. +oo  <  B )
4847adantl 273 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
49 breq1 3900 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
5049adantr 272 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
5148, 50mtbird 645 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
5251pm2.21d 591 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <  C ) )
5352adantrd 275 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
5453adantrr 468 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
55 mnflt 9509 . . . . . . . . . . 11  |-  ( C  e.  RR  -> -oo  <  C )
5655adantl 273 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  -> -oo  <  C )
57 breq1 3900 . . . . . . . . . . 11  |-  ( A  = -oo  ->  ( A  <  C  <-> -oo  <  C
) )
5857adantr 272 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( A  <  C  <-> -oo 
<  C ) )
5956, 58mpbird 166 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  A  <  C )
6059a1d 22 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6160adantlr 466 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
62 mnfltpnf 9511 . . . . . . . . . 10  |- -oo  < +oo
63 breq12 3902 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( A  <  C  <-> -oo 
< +oo ) )
6462, 63mpbiri 167 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  = +oo )  ->  A  <  C )
6564a1d 22 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6665adantlr 466 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6743adantll 465 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6861, 66, 673jaodan 1267 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
6968anasss 394 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7046, 54, 693jaoian 1266 . . . 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 1160 . . 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 1237 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
731, 72syl3an1b 1235 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 944    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897   RRcr 7583   +oocpnf 7761   -oocmnf 7762   RR*cxr 7763    < clt 7764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-lttrn 7698
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769
This theorem is referenced by:  xrltso  9522  xrlttrd  9532  ioo0  9977
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