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Theorem rpneg 11249
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 9596 . . . . . . . 8  |-  0  e.  RR
2 ltle 9673 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
31, 2mpan 670 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
43imp 429 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <_  A )
54olcd 393 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( -.  -u A  e.  RR  \/  0  <_  A ) )
6 renegcl 9882 . . . . . . . . 9  |-  ( A  e.  RR  ->  -u A  e.  RR )
76pm2.24d 143 . . . . . . . 8  |-  ( A  e.  RR  ->  ( -.  -u A  e.  RR  ->  0  <  A ) )
87adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -.  -u A  e.  RR  ->  0  <  A ) )
9 ltlen 9686 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  ( 0  <_  A  /\  A  =/=  0 ) ) )
101, 9mpan 670 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( 0  <_  A  /\  A  =/=  0 ) ) )
1110biimprd 223 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <_  A  /\  A  =/=  0
)  ->  0  <  A ) )
1211expcomd 438 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  =/=  0  ->  (
0  <_  A  ->  0  <  A ) ) )
1312imp 429 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <_  A  ->  0  <  A ) )
148, 13jaod 380 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  ->  0  <  A ) )
15 simpl 457 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  A  e.  RR )
1614, 15jctild 543 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  ->  ( A  e.  RR  /\  0  <  A ) ) )
175, 16impbid2 204 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  ( -.  -u A  e.  RR  \/  0  <_  A ) ) )
18 lenlt 9663 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
191, 18mpan 670 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  A  <  0 ) )
20 lt0neg1 10058 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
2120notbid 294 . . . . . . 7  |-  ( A  e.  RR  ->  ( -.  A  <  0  <->  -.  0  <  -u A
) )
2219, 21bitrd 253 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  0  <  -u A ) )
2322adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <_  A  <->  -.  0  <  -u A
) )
2423orbi2d 701 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( -.  -u A  e.  RR  \/  0  <_  A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) ) )
2517, 24bitrd 253 . . 3  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) ) )
26 ianor 488 . . 3  |-  ( -.  ( -u A  e.  RR  /\  0  <  -u A )  <->  ( -.  -u A  e.  RR  \/  -.  0  <  -u A
) )
2725, 26syl6bbr 263 . 2  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  e.  RR  /\  0  < 
A )  <->  -.  ( -u A  e.  RR  /\  0  <  -u A ) ) )
28 elrp 11222 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
29 elrp 11222 . . 3  |-  ( -u A  e.  RR+  <->  ( -u A  e.  RR  /\  0  <  -u A ) )
3029notbii 296 . 2  |-  ( -.  -u A  e.  RR+  <->  -.  ( -u A  e.  RR  /\  0  <  -u A ) )
3127, 28, 303bitr4g 288 1  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  e.  RR+  <->  -.  -u A  e.  RR+ )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1767    =/= wne 2662   class class class wbr 4447   RRcr 9491   0cc0 9492    < clt 9628    <_ cle 9629   -ucneg 9806   RR+crp 11220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-rp 11221
This theorem is referenced by:  cnpart  13036  angpined  22917  signsply0  28176
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