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Theorem rpneg 11025
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 9391 . . . . . . . 8  |-  0  e.  RR
2 ltle 9468 . . . . . . . 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 9677 . . . . . . . . 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 9481 . . . . . . . . . . 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 9458 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
191, 18mpan 670 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <_  A  <->  -.  A  <  0 ) )
20 lt0neg1 9850 . . . . . . . 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 10998 . 2  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
29 elrp 10998 . . 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 1756    =/= wne 2611   class class class wbr 4297   RRcr 9286   0cc0 9287    < clt 9423    <_ cle 9424   -ucneg 9601   RR+crp 10996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-rp 10997
This theorem is referenced by:  cnpart  12734  angpined  22230  signsply0  26957
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