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Theorem rpneg 10351
Description: Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.)
Assertion
Ref Expression
rpneg  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  e.  RR+  <->  -.  -u A  e.  RR+ )
)

Proof of Theorem rpneg
StepHypRef Expression
1 0re 8806 . . . . . . . 8  |-  0  e.  RR
2 ltle 8878 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
31, 2mpan 654 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
43imp 420 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <_  A )
54olcd 384 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( -.  -u A  e.  RR  \/  0  <_  A ) )
6 renegcl 9078 . . . . . . . . 9  |-  ( A  e.  RR  ->  -u A  e.  RR )
76pm2.24d 137 . . . . . . . 8  |-  ( A  e.  RR  ->  ( -.  -u A  e.  RR  ->  0  <  A ) )
87adantr 453 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -.  -u A  e.  RR  ->  0  <  A ) )
9 ltlen 8890 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  ( 0  <_  A  /\  A  =/=  0 ) ) )
101, 9mpan 654 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( 0  <_  A  /\  A  =/=  0 ) ) )
1110biimprd 216 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <_  A  /\  A  =/=  0
)  ->  0  <  A ) )
1211exp3acom23 1368 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  =/=  0  ->  (
0  <_  A  ->  0  <  A ) ) )
1312imp 420 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <_  A  ->  0  <  A ) )
148, 13jaod 371 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  ->  0  <  A ) )
15 simpl 445 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  A  e.  RR )
1614, 15jctild 529 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  ->  ( A  e.  RR  /\  0  <  A ) ) )
175, 16impbid2 197 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  ( -.  -u A  e.  RR  \/  0  <_  A ) ) )
18 lenlt 8869 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
191, 18mpan 654 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  A  <  0 ) )
20 lt0neg1 9248 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
2120notbid 287 . . . . . . 7  |-  ( A  e.  RR  ->  ( -.  A  <  0  <->  -.  0  <  -u A
) )
2219, 21bitrd 246 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  0  <  -u A ) )
2322adantr 453 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <_  A  <->  -.  0  <  -u A
) )
2423orbi2d 685 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) ) )
2517, 24bitrd 246 . . 3  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) ) )
26 ianor 476 . . 3  |-  ( -.  ( -u A  e.  RR  /\  0  <  -u A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) )
2725, 26syl6bbr 256 . 2  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  -.  ( -u A  e.  RR  /\  0  <  -u A ) ) )
28 elrp 10324 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
29 elrp 10324 . . 3  |-  ( -u A  e.  RR+  <->  ( -u A  e.  RR  /\  0  <  -u A ) )
3029notbii 289 . 2  |-  ( -.  -u A  e.  RR+  <->  -.  ( -u A  e.  RR  /\  0  <  -u A ) )
3127, 28, 303bitr4g 281 1  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  e.  RR+  <->  -.  -u A  e.  RR+ )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    e. wcel 1621    =/= wne 2421   class class class wbr 3997   RRcr 8704   0cc0 8705    < clt 8835    <_ cle 8836   -ucneg 9006   RR+crp 10322
This theorem is referenced by:  cnpart  11691  angpined  20090
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-rp 10323
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