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Theorem xrlttr 11446
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 11423 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11423 . . 3  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
3 elxr 11423 . . . . . . . . 9  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4 lttr 9715 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
543expa 1209 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
65an32s 814 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7 rexr 9691 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  RR* )
8 pnfnlt 11437 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR*  ->  -. +oo  <  C )
97, 8syl 17 . . . . . . . . . . . . . . 15  |-  ( C  e.  RR  ->  -. +oo 
<  C )
109adantr 467 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -. +oo  <  C
)
11 breq1 4408 . . . . . . . . . . . . . . 15  |-  ( B  = +oo  ->  ( B  <  C  <-> +oo  <  C
) )
1211adantl 468 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  <-> +oo 
<  C ) )
1310, 12mtbird 303 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  -.  B  <  C
)
1413pm2.21d 110 . . . . . . . . . . . 12  |-  ( ( C  e.  RR  /\  B  = +oo )  ->  ( B  <  C  ->  A  <  C ) )
1514adantll 721 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( B  < 
C  ->  A  <  C ) )
1615adantld 469 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
17 rexr 9691 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  A  e.  RR* )
18 nltmnf 11438 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR*  ->  -.  A  < -oo )
1917, 18syl 17 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  -.  A  < -oo )
2019adantr 467 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
21 breq2 4409 . . . . . . . . . . . . . . 15  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2221adantl 468 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
2320, 22mtbird 303 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
2423pm2.21d 110 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  A  <  C ) )
2524adantlr 722 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  ->  A  <  C ) )
2625adantrd 470 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
276, 16, 263jaodan 1336 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
283, 27sylan2b 478 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  C  e.  RR )  /\  B  e.  RR* )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
2928an32s 814 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
30 ltpnf 11429 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  < +oo )
3130adantr 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  < +oo )
32 breq2 4409 . . . . . . . . . . 11  |-  ( C  = +oo  ->  ( A  <  C  <->  A  < +oo ) )
3332adantl 468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  ( A  <  C  <->  A  < +oo ) )
3431, 33mpbird 236 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  = +oo )  ->  A  <  C )
3534adantlr 722 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  A  <  C
)
3635a1d 26 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
37 nltmnf 11438 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -.  B  < -oo )
3837adantr 467 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  < -oo )
39 breq2 4409 . . . . . . . . . . . 12  |-  ( C  = -oo  ->  ( B  <  C  <->  B  < -oo ) )
4039adantl 468 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  <->  B  < -oo ) )
4138, 40mtbird 303 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  -.  B  <  C )
4241pm2.21d 110 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  ( B  <  C  ->  A  <  C ) )
4342adantld 469 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  = -oo )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4443adantll 721 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
4529, 36, 443jaodan 1336 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
4645anasss 653 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
47 pnfnlt 11437 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -. +oo  <  B )
4847adantl 468 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
49 breq1 4408 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
5049adantr 467 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
5148, 50mtbird 303 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
5251pm2.21d 110 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <  C ) )
5352adantrd 470 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
5453adantrr 724 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
55 mnflt 11432 . . . . . . . . . . 11  |-  ( C  e.  RR  -> -oo  <  C )
5655adantl 468 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  -> -oo  <  C )
57 breq1 4408 . . . . . . . . . . 11  |-  ( A  = -oo  ->  ( A  <  C  <-> -oo  <  C
) )
5857adantr 467 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( A  <  C  <-> -oo 
<  C ) )
5956, 58mpbird 236 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  A  <  C )
6059a1d 26 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  e.  RR )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6160adantlr 722 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  e.  RR )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
62 mnfltpnf 11435 . . . . . . . . . 10  |- -oo  < +oo
63 breq12 4410 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( A  <  C  <-> -oo 
< +oo ) )
6462, 63mpbiri 237 . . . . . . . . 9  |-  ( ( A  = -oo  /\  C  = +oo )  ->  A  <  C )
6564a1d 26 . . . . . . . 8  |-  ( ( A  = -oo  /\  C  = +oo )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
6665adantlr 722 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = +oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6743adantll 721 . . . . . . 7  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  C  = -oo )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
6861, 66, 673jaodan 1336 . . . . . 6  |-  ( ( ( A  = -oo  /\  B  e.  RR* )  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
6968anasss 653 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR*  /\  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) ) )  ->  ( ( A  <  B  /\  B  <  C )  ->  A  <  C ) )
7046, 54, 693jaoian 1335 . . . 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 1205 . . 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 1307 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
731, 72syl3an1b 1305 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 188    /\ wa 371    \/ w3o 985    /\ w3a 986    = wceq 1446    e. wcel 1889   class class class wbr 4405   RRcr 9543   +oocpnf 9677   -oocmnf 9678   RR*cxr 9679    < clt 9680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-pre-lttrn 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685
This theorem is referenced by:  xrltso  11447  xrlelttr  11460  xrltletr  11461  xrlttrd  11463  xrub  11604  ioo0  11668  ioojoin  11770  leordtval2  20240  icopnfcld  21800  iocmnfcld  21801  ismbf3d  22622  tanord1  23498  tan2h  31949  asindmre  32039  iccpartlt  38748
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