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Theorem xrlttri3 9754
Description: Extended real version of lttri3 7999. (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 9733 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9733 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 lttri3 7999 . . . . . 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 7967 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= +oo )
65adantr 274 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  =/= +oo )
7 neeq2 2354 . . . . . . . . . 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 2426 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  A  =/=  B )
1110neneqd 2361 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  A  =  B )
12 ltpnf 9737 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  < +oo )
1312adantr 274 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  < +oo )
14 breq2 3993 . . . . . . . . 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 624 . . . . . . . . 9  |-  ( ( A  <  B  \/  B  <  A )  ->  -.  -.  ( A  < 
B  \/  B  < 
A ) )
1817olcs 731 . . . . . . . 8  |-  ( B  <  A  ->  -.  -.  ( A  <  B  \/  B  <  A ) )
19 ioran 747 . . . . . . . 8  |-  ( -.  ( A  <  B  \/  B  <  A )  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) )
2018, 19sylnib 671 . . . . . . 7  |-  ( B  <  A  ->  -.  ( -.  A  <  B  /\  -.  B  < 
A ) )
2116, 20syl 14 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
2211, 212falsed 697 . . . . 5  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
23 renemnf 7968 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= -oo )
2423adantr 274 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  B  =/= -oo )
25 neeq2 2354 . . . . . . . . . 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 2426 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  A  =/=  B )
2928neneqd 2361 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  -.  A  =  B )
30 mnflt 9740 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
3130adantr 274 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  -> -oo  <  B )
32 breq1 3992 . . . . . . . . 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 707 . . . . . . 7  |-  ( A  <  B  ->  ( A  <  B  \/  B  <  A ) )
36 oranim 776 . . . . . . 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 697 . . . . 5  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
394, 22, 383jaodan 1301 . . . 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 7967 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= +oo )
4241adantl 275 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  =/= +oo )
43 neeq2 2354 . . . . . . . . 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 2361 . . . . . 6  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  -.  A  =  B )
47 ltpnf 9737 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
4847adantl 275 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  < +oo )
49 breq2 3993 . . . . . . . . 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 697 . . . . 5  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
54 eqtr3 2190 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  B  =  A )
5554eqcomd 2176 . . . . . 6  |-  ( ( B  = +oo  /\  A  = +oo )  ->  A  =  B )
56 pnfxr 7972 . . . . . . . . 9  |- +oo  e.  RR*
57 xrltnr 9736 . . . . . . . . 9  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
5856, 57ax-mp 5 . . . . . . . 8  |-  -. +oo  < +oo
59 breq12 3994 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6059ancoms 266 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6158, 60mtbiri 670 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  -.  A  <  B
)
62 breq12 3994 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( B  <  A  <-> +oo 
< +oo ) )
6358, 62mtbiri 670 . . . . . . 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 7975 . . . . . . . . 9  |- -oo  =/= +oo
67 eqeq12 2183 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  =  B  <-> -oo  = +oo )
)
6867necon3bid 2381 . . . . . . . . 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 2361 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  A  =  B )
72 mnfltpnf 9742 . . . . . . . . 9  |- -oo  < +oo
73 breq12 3994 . . . . . . . . 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 697 . . . . 5  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
7853, 65, 773jaodan 1301 . . . 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 7968 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
8180adantl 275 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  A  =/= -oo )
82 neeq2 2354 . . . . . . . . 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 2361 . . . . . 6  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  -.  A  =  B )
86 mnflt 9740 . . . . . . . . 9  |-  ( A  e.  RR  -> -oo  <  A )
8786adantl 275 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  -> -oo  <  A )
88 breq1 3992 . . . . . . . . 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 697 . . . . 5  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
9366neii 2342 . . . . . . . . . 10  |-  -. -oo  = +oo
94 eqeq12 2183 . . . . . . . . . 10  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  =  A  <-> -oo  = +oo )
)
9593, 94mtbiri 670 . . . . . . . . 9  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  B  =  A )
9695neneqad 2419 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  =/=  A )
9796necomd 2426 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = +oo )  ->  A  =/=  B )
9897neneqd 2361 . . . . . 6  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  A  =  B )
99 breq12 3994 . . . . . . . 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 697 . . . . 5  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
103 eqtr3 2190 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = -oo )  ->  A  =  B )
104103ancoms 266 . . . . . 6  |-  ( ( B  = -oo  /\  A  = -oo )  ->  A  =  B )
105 mnfxr 7976 . . . . . . . . 9  |- -oo  e.  RR*
106 xrltnr 9736 . . . . . . . . 9  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
107105, 106ax-mp 5 . . . . . . . 8  |-  -. -oo  < -oo
108 breq12 3994 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
109108ancoms 266 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
110107, 109mtbiri 670 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = -oo )  ->  -.  A  <  B
)
111 breq12 3994 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( B  <  A  <-> -oo 
< -oo ) )
112107, 111mtbiri 670 . . . . . . 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 1301 . . . 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 1301 . 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 703    \/ w3o 972    = wceq 1348    e. wcel 2141    =/= wne 2340   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953    < clt 7954
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-apti 7889
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959
This theorem is referenced by:  xrletri3  9761  iccid  9882  xrmaxleim  11207  xrmaxif  11214  xrmaxaddlem  11223  infxrnegsupex  11226  bdxmet  13295
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