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Theorem xrlttr 8486
Description: Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrlttr  RR*  RR*  C 
RR*  <  <  C  <  C

Proof of Theorem xrlttr
StepHypRef Expression
1 elxr 8466 . 2  RR*  RR +oo -oo
2 elxr 8466 . . 3  C  RR*  C  RR  C +oo  C -oo
3 elxr 8466 . . . . . . . . 9  RR*  RR +oo -oo
4 lttr 6889 . . . . . . . . . . . 12  RR  RR  C  RR  <  <  C  <  C
543expa 1103 . . . . . . . . . . 11  RR  RR  C  RR  <  <  C  <  C
65an32s 502 . . . . . . . . . 10  RR  C  RR  RR  <  <  C  <  C
7 rexr 6868 . . . . . . . . . . . . . . . 16  C  RR  C  RR*
8 pnfnlt 8478 . . . . . . . . . . . . . . . 16  C  RR* +oo  <  C
97, 8syl 14 . . . . . . . . . . . . . . 15  C  RR +oo 
<  C
109adantr 261 . . . . . . . . . . . . . 14  C  RR +oo +oo  <  C
11 breq1 3758 . . . . . . . . . . . . . . 15 +oo  <  C +oo  <  C
1211adantl 262 . . . . . . . . . . . . . 14  C  RR +oo  <  C +oo  <  C
1310, 12mtbird 597 . . . . . . . . . . . . 13  C  RR +oo  <  C
1413pm2.21d 549 . . . . . . . . . . . 12  C  RR +oo  <  C  <  C
1514adantll 445 . . . . . . . . . . 11  RR  C  RR +oo  <  C  <  C
1615adantld 263 . . . . . . . . . 10  RR  C  RR +oo  <  <  C  <  C
17 rexr 6868 . . . . . . . . . . . . . . . 16  RR  RR*
18 nltmnf 8479 . . . . . . . . . . . . . . . 16  RR*  < -oo
1917, 18syl 14 . . . . . . . . . . . . . . 15  RR  < -oo
2019adantr 261 . . . . . . . . . . . . . 14  RR -oo  < -oo
21 breq2 3759 . . . . . . . . . . . . . . 15 -oo  <  < -oo
2221adantl 262 . . . . . . . . . . . . . 14  RR -oo  <  < -oo
2320, 22mtbird 597 . . . . . . . . . . . . 13  RR -oo  <
2423pm2.21d 549 . . . . . . . . . . . 12  RR -oo  <  <  C
2524adantlr 446 . . . . . . . . . . 11  RR  C  RR -oo  <  <  C
2625adantrd 264 . . . . . . . . . 10  RR  C  RR -oo  <  <  C  <  C
276, 16, 263jaodan 1200 . . . . . . . . 9  RR  C  RR  RR +oo -oo  <  <  C  <  C
283, 27sylan2b 271 . . . . . . . 8  RR  C  RR  RR*  <  <  C  <  C
2928an32s 502 . . . . . . 7  RR  RR*  C  RR  <  <  C  <  C
30 ltpnf 8472 . . . . . . . . . . 11  RR  < +oo
3130adantr 261 . . . . . . . . . 10  RR  C +oo  < +oo
32 breq2 3759 . . . . . . . . . . 11  C +oo  <  C  < +oo
3332adantl 262 . . . . . . . . . 10  RR  C +oo  <  C  < +oo
3431, 33mpbird 156 . . . . . . . . 9  RR  C +oo  <  C
3534adantlr 446 . . . . . . . 8  RR  RR*  C +oo  <  C
3635a1d 22 . . . . . . 7  RR  RR*  C +oo  <  <  C  <  C
37 nltmnf 8479 . . . . . . . . . . . 12  RR*  < -oo
3837adantr 261 . . . . . . . . . . 11  RR*  C -oo  < -oo
39 breq2 3759 . . . . . . . . . . . 12  C -oo  <  C  < -oo
4039adantl 262 . . . . . . . . . . 11  RR*  C -oo  <  C  < -oo
4138, 40mtbird 597 . . . . . . . . . 10  RR*  C -oo  <  C
4241pm2.21d 549 . . . . . . . . 9  RR*  C -oo  <  C  <  C
4342adantld 263 . . . . . . . 8  RR*  C -oo  <  <  C  <  C
4443adantll 445 . . . . . . 7  RR  RR*  C -oo  <  <  C  <  C
4529, 36, 443jaodan 1200 . . . . . 6  RR  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
4645anasss 379 . . . . 5  RR  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
47 pnfnlt 8478 . . . . . . . . . 10  RR* +oo  <
4847adantl 262 . . . . . . . . 9 +oo  RR* +oo  <
49 breq1 3758 . . . . . . . . . 10 +oo  < +oo  <
5049adantr 261 . . . . . . . . 9 +oo  RR*  < +oo  <
5148, 50mtbird 597 . . . . . . . 8 +oo  RR*  <
5251pm2.21d 549 . . . . . . 7 +oo  RR*  <  <  C
5352adantrd 264 . . . . . 6 +oo  RR*  <  <  C  <  C
5453adantrr 448 . . . . 5 +oo  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
55 mnflt 8474 . . . . . . . . . . 11  C  RR -oo  <  C
5655adantl 262 . . . . . . . . . 10 -oo  C  RR -oo  <  C
57 breq1 3758 . . . . . . . . . . 11 -oo  <  C -oo  <  C
5857adantr 261 . . . . . . . . . 10 -oo  C  RR  <  C -oo  <  C
5956, 58mpbird 156 . . . . . . . . 9 -oo  C  RR  <  C
6059a1d 22 . . . . . . . 8 -oo  C  RR  <  <  C  <  C
6160adantlr 446 . . . . . . 7 -oo  RR*  C  RR  <  <  C  <  C
62 mnfltpnf 8476 . . . . . . . . . 10 -oo  < +oo
63 breq12 3760 . . . . . . . . . 10 -oo  C +oo  <  C -oo  < +oo
6462, 63mpbiri 157 . . . . . . . . 9 -oo  C +oo  <  C
6564a1d 22 . . . . . . . 8 -oo  C +oo  <  <  C  <  C
6665adantlr 446 . . . . . . 7 -oo  RR*  C +oo  <  <  C  <  C
6743adantll 445 . . . . . . 7 -oo  RR*  C -oo  <  <  C  <  C
6861, 66, 673jaodan 1200 . . . . . 6 -oo  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
6968anasss 379 . . . . 5 -oo  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
7046, 54, 693jaoian 1199 . . . 4  RR +oo -oo  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
71703impb 1099 . . 3  RR +oo -oo  RR*  C  RR  C +oo  C -oo  <  <  C  <  C
722, 71syl3an3b 1172 . 2  RR +oo -oo  RR*  C  RR*  <  <  C  <  C
731, 72syl3an1b 1170 1  RR*  RR*  C 
RR*  <  <  C  <  C
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3o 883   w3a 884   wceq 1242   wcel 1390   class class class wbr 3755   RRcr 6710   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856    < clt 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-lttrn 6797
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  xrltso  8487  xrlttrd  8495
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