MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rpneg Unicode version

Theorem rpneg 10262
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 8718 . . . . . . . 8  |-  0  e.  RR
2 ltle 8790 . . . . . . . 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 8990 . . . . . . . . 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 8802 . . . . . . . . . . 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 8781 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
191, 18mpan 654 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  A  <  0 ) )
20 lt0neg1 9160 . . . . . . . 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 10235 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
29 elrp 10235 . . 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 2412   class class class wbr 3920   RRcr 8616   0cc0 8617    < clt 8747    <_ cle 8748   -ucneg 8918   RR+crp 10233
This theorem is referenced by:  cnpart  11602  angpined  19871
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-rp 10234
  Copyright terms: Public domain W3C validator