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Theorem xrlttr 11117
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 11096 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11096 . . 3  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
3 elxr 11096 . . . . . . . . 9  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4 lttr 9451 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
543expa 1187 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
65an32s 802 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7 rexr 9429 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  RR* )
8 pnfnlt 11108 . . . . . . . . . . . . . . . 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 4295 . . . . . . . . . . . . . . 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 9429 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  A  e.  RR* )
18 nltmnf 11109 . . . . . . . . . . . . . . . 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 4296 . . . . . . . . . . . . . . 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 1284 . . . . . . . . 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 802 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
30 ltpnf 11102 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  < +oo )
3130adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  < +oo )
32 breq2 4296 . . . . . . . . . . 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 11109 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -.  B  < -oo )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  < -oo )
39 breq2 4296 . . . . . . . . . . . 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 1284 . . . . . 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 11108 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -. +oo  <  B )
4847adantl 466 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
49 breq1 4295 . . . . . . . . . 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 11104 . . . . . . . . . . 11  |-  ( C  e.  RR  -> -oo  <  C )
5655adantl 466 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  -> -oo  <  C )
57 breq1 4295 . . . . . . . . . . 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 11106 . . . . . . . . . 10  |- -oo  < +oo
63 breq12 4297 . . . . . . . . . 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 1284 . . . . . 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 1283 . . . 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 1183 . . 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 1256 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
731, 72syl3an1b 1254 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 964    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   RRcr 9281   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423
This theorem is referenced by:  xrltso  11118  xrlelttr  11130  xrltletr  11131  xrlttrd  11133  xrub  11274  ioo0  11325  ioojoin  11416  leordtval2  18816  icopnfcld  20347  iocmnfcld  20348  ismbf3d  21132  tanord1  21993  tan2h  28424  asindmre  28479
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