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Theorem xrlttr 11342
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 11321 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11321 . . 3  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
3 elxr 11321 . . . . . . . . 9  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4 lttr 9657 . . . . . . . . . . . 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 802 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7 rexr 9635 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  RR* )
8 pnfnlt 11333 . . . . . . . . . . . . . . . 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 4450 . . . . . . . . . . . . . . 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 9635 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  A  e.  RR* )
18 nltmnf 11334 . . . . . . . . . . . . . . . 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 4451 . . . . . . . . . . . . . . 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 802 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
30 ltpnf 11327 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  < +oo )
3130adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  < +oo )
32 breq2 4451 . . . . . . . . . . 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 11334 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -.  B  < -oo )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  < -oo )
39 breq2 4451 . . . . . . . . . . . 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 11333 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -. +oo  <  B )
4847adantl 466 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
49 breq1 4450 . . . . . . . . . 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 11329 . . . . . . . . . . 11  |-  ( C  e.  RR  -> -oo  <  C )
5655adantl 466 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  -> -oo  <  C )
57 breq1 4450 . . . . . . . . . . 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 11331 . . . . . . . . . 10  |- -oo  < +oo
63 breq12 4452 . . . . . . . . . 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 1379    e. wcel 1767   class class class wbr 4447   RRcr 9487   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629
This theorem is referenced by:  xrltso  11343  xrlelttr  11355  xrltletr  11356  xrlttrd  11358  xrub  11499  ioo0  11550  ioojoin  11647  leordtval2  19476  icopnfcld  21007  iocmnfcld  21008  ismbf3d  21793  tanord1  22654  tan2h  29622  asindmre  29677
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