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Theorem xrlttri3 9576
Description: Extended real version of lttri3 7837. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrlttri3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )

Proof of Theorem xrlttri3
StepHypRef Expression
1 elxr 9556 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9556 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 lttri3 7837 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
43ancoms 266 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
5 renepnf 7806 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= +oo )
65adantr 274 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  =/= +oo )
7 neeq2 2320 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( B  =/=  A  <->  B  =/= +oo ) )
87adantl 275 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( B  =/=  A  <->  B  =/= +oo ) )
96, 8mpbird 166 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  =/=  A )
109necomd 2392 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  A  =/=  B )
1110neneqd 2327 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  A  =  B )
12 ltpnf 9560 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  < +oo )
1312adantr 274 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  < +oo )
14 breq2 3928 . . . . . . . . 9  |-  ( A  = +oo  ->  ( B  <  A  <->  B  < +oo ) )
1514adantl 275 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( B  <  A  <->  B  < +oo ) )
1613, 15mpbird 166 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  <  A )
17 notnot 618 . . . . . . . . 9  |-  ( ( A  <  B  \/  B  <  A )  ->  -.  -.  ( A  < 
B  \/  B  < 
A ) )
1817olcs 725 . . . . . . . 8  |-  ( B  <  A  ->  -.  -.  ( A  <  B  \/  B  <  A ) )
19 ioran 741 . . . . . . . 8  |-  ( -.  ( A  <  B  \/  B  <  A )  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) )
2018, 19sylnib 665 . . . . . . 7  |-  ( B  <  A  ->  -.  ( -.  A  <  B  /\  -.  B  < 
A ) )
2116, 20syl 14 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
2211, 212falsed 691 . . . . 5  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
23 renemnf 7807 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= -oo )
2423adantr 274 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  B  =/= -oo )
25 neeq2 2320 . . . . . . . . . 10  |-  ( A  = -oo  ->  ( B  =/=  A  <->  B  =/= -oo ) )
2625adantl 275 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( B  =/=  A  <->  B  =/= -oo ) )
2724, 26mpbird 166 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  B  =/=  A )
2827necomd 2392 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  A  =/=  B )
2928neneqd 2327 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  -.  A  =  B )
30 mnflt 9562 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
3130adantr 274 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  -> -oo  <  B )
32 breq1 3927 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3332adantl 275 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( A  <  B  <-> -oo 
<  B ) )
3431, 33mpbird 166 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  A  <  B )
35 orc 701 . . . . . . 7  |-  ( A  <  B  ->  ( A  <  B  \/  B  <  A ) )
36 oranim 770 . . . . . . 7  |-  ( ( A  <  B  \/  B  <  A )  ->  -.  ( -.  A  < 
B  /\  -.  B  <  A ) )
3734, 35, 363syl 17 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
3829, 372falsed 691 . . . . 5  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
394, 22, 383jaodan 1284 . . . 4  |-  ( ( B  e.  RR  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
4039ancoms 266 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
41 renepnf 7806 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= +oo )
4241adantl 275 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  =/= +oo )
43 neeq2 2320 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  =/=  B  <->  A  =/= +oo ) )
4443adantr 274 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  =/=  B  <->  A  =/= +oo ) )
4542, 44mpbird 166 . . . . . . 7  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  =/=  B )
4645neneqd 2327 . . . . . 6  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  -.  A  =  B )
47 ltpnf 9560 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
4847adantl 275 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  < +oo )
49 breq2 3928 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
5049adantr 274 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  <  B  <->  A  < +oo ) )
5148, 50mpbird 166 . . . . . . 7  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  <  B )
5251, 35, 363syl 17 . . . . . 6  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
5346, 522falsed 691 . . . . 5  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
54 eqtr3 2157 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  B  =  A )
5554eqcomd 2143 . . . . . 6  |-  ( ( B  = +oo  /\  A  = +oo )  ->  A  =  B )
56 pnfxr 7811 . . . . . . . . 9  |- +oo  e.  RR*
57 xrltnr 9559 . . . . . . . . 9  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
5856, 57ax-mp 5 . . . . . . . 8  |-  -. +oo  < +oo
59 breq12 3929 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6059ancoms 266 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6158, 60mtbiri 664 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  -.  A  <  B
)
62 breq12 3929 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( B  <  A  <-> +oo 
< +oo ) )
6358, 62mtbiri 664 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  -.  B  <  A
)
6461, 63jca 304 . . . . . 6  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( -.  A  < 
B  /\  -.  B  <  A ) )
6555, 642thd 174 . . . . 5  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
66 mnfnepnf 7814 . . . . . . . . 9  |- -oo  =/= +oo
67 eqeq12 2150 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  =  B  <-> -oo  = +oo )
)
6867necon3bid 2347 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  =/=  B  <-> -oo 
=/= +oo ) )
6966, 68mpbiri 167 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  =/=  B )
7069ancoms 266 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  A  =/=  B )
7170neneqd 2327 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  A  =  B )
72 mnfltpnf 9564 . . . . . . . . 9  |- -oo  < +oo
73 breq12 3929 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  <-> -oo 
< +oo ) )
7472, 73mpbiri 167 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  <  B )
7574ancoms 266 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  A  <  B )
7675, 35, 363syl 17 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
7771, 762falsed 691 . . . . 5  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
7853, 65, 773jaodan 1284 . . . 4  |-  ( ( B  = +oo  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
7978ancoms 266 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  = +oo )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
80 renemnf 7807 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
8180adantl 275 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  A  =/= -oo )
82 neeq2 2320 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  =/=  B  <->  A  =/= -oo ) )
8382adantr 274 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( A  =/=  B  <->  A  =/= -oo ) )
8481, 83mpbird 166 . . . . . . 7  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  A  =/=  B )
8584neneqd 2327 . . . . . 6  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  -.  A  =  B )
86 mnflt 9562 . . . . . . . . 9  |-  ( A  e.  RR  -> -oo  <  A )
8786adantl 275 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  -> -oo  <  A )
88 breq1 3927 . . . . . . . . 9  |-  ( B  = -oo  ->  ( B  <  A  <-> -oo  <  A
) )
8988adantr 274 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( B  <  A  <-> -oo 
<  A ) )
9087, 89mpbird 166 . . . . . . 7  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  B  <  A )
9190, 20syl 14 . . . . . 6  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
9285, 912falsed 691 . . . . 5  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
9366neii 2308 . . . . . . . . . 10  |-  -. -oo  = +oo
94 eqeq12 2150 . . . . . . . . . 10  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  =  A  <-> -oo  = +oo )
)
9593, 94mtbiri 664 . . . . . . . . 9  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  B  =  A )
9695neneqad 2385 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  =/=  A )
9796necomd 2392 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = +oo )  ->  A  =/=  B )
9897neneqd 2327 . . . . . 6  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  A  =  B )
99 breq12 3929 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  <  A  <-> -oo 
< +oo ) )
10072, 99mpbiri 167 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  <  A )
101100, 20syl 14 . . . . . 6  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
10298, 1012falsed 691 . . . . 5  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
103 eqtr3 2157 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = -oo )  ->  A  =  B )
104103ancoms 266 . . . . . 6  |-  ( ( B  = -oo  /\  A  = -oo )  ->  A  =  B )
105 mnfxr 7815 . . . . . . . . 9  |- -oo  e.  RR*
106 xrltnr 9559 . . . . . . . . 9  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
107105, 106ax-mp 5 . . . . . . . 8  |-  -. -oo  < -oo
108 breq12 3929 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
109108ancoms 266 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
110107, 109mtbiri 664 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = -oo )  ->  -.  A  <  B
)
111 breq12 3929 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( B  <  A  <-> -oo 
< -oo ) )
112107, 111mtbiri 664 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = -oo )  ->  -.  B  <  A
)
113110, 112jca 304 . . . . . 6  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( -.  A  < 
B  /\  -.  B  <  A ) )
114104, 1132thd 174 . . . . 5  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
11592, 102, 1143jaodan 1284 . . . 4  |-  ( ( B  = -oo  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
116115ancoms 266 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  = -oo )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
11740, 79, 1163jaodan 1284 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
1181, 2, 117syl2anb 289 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480    =/= wne 2306   class class class wbr 3924   RRcr 7612   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792    < clt 7793
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725  ax-pre-apti 7728
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798
This theorem is referenced by:  xrletri3  9581  iccid  9701  xrmaxleim  11006  xrmaxif  11013  xrmaxaddlem  11022  infxrnegsupex  11025  bdxmet  12659
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