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Theorem rpneg 11863
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 10040 . . . . . . . 8 |- 0 e. RR
2 ltle 10126 . . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
31, 2mpan 706 . . . . . . 7 |- ( A e. RR -> ( 0 < A -> 0 <_ A ) )
43imp 445 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 <_ A )
54olcd 408 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( -. -u A e. RR \/ 0 <_ A ) )
6 renegcl 10344 . . . . . . . . 9 |- ( A e. RR -> -u A e. RR )
76pm2.24d 147 . . . . . . . 8 |- ( A e. RR -> ( -. -u A e. RR -> 0 < A ) )
87adantr 481 . . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( -. -u A e. RR -> 0 < A ) )
9 ltlen 10138 . . . . . . . . . . 11 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> ( 0 <_ A /\ A =/= 0 ) ) )
101, 9mpan 706 . . . . . . . . . 10 |- ( A e. RR -> ( 0 < A <-> ( 0 <_ A /\ A =/= 0 ) ) )
1110biimprd 238 . . . . . . . . 9 |- ( A e. RR -> ( ( 0 <_ A /\ A =/= 0 ) -> 0 < A ) )
1211expcomd 454 . . . . . . . 8 |- ( A e. RR -> ( A =/= 0 -> ( 0 <_ A -> 0 < A ) ) )
1312imp 445 . . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 <_ A -> 0 < A ) )
148, 13jaod 395 . . . . . 6 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( -. -u A e. RR \/ 0 <_ A ) -> 0 < A ) )
15 simpl 473 . . . . . 6 |- ( ( A e. RR /\ A =/= 0 ) -> A e. RR )
1614, 15jctild 566 . . . . 5 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( -. -u A e. RR \/ 0 <_ A ) -> ( A e. RR /\ 0 < A ) ) )
175, 16impbid2 216 . . . 4 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( A e. RR /\ 0 < A ) <-> ( -. -u A e. RR \/ 0 <_ A ) ) )
18 lenlt 10116 . . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
191, 18mpan 706 . . . . . . 7 |- ( A e. RR -> ( 0 <_ A <-> -. A < 0 ) )
20 lt0neg1 10534 . . . . . . . 8 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
2120notbid 308 . . . . . . 7 |- ( A e. RR -> ( -. A < 0 <-> -. 0 < -u A ) )
2219, 21bitrd 268 . . . . . 6 |- ( A e. RR -> ( 0 <_ A <-> -. 0 < -u A ) )
2322adantr 481 . . . . 5 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 <_ A <-> -. 0 < -u A ) )
2423orbi2d 738 . . . 4 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( -. -u A e. RR \/ 0 <_ A ) <-> ( -. -u A e. RR \/ -. 0 < -u A ) ) )
2517, 24bitrd 268 . . 3 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( A e. RR /\ 0 < A ) <-> ( -. -u A e. RR \/ -. 0 < -u A ) ) )
26 ianor 509 . . 3 |- ( -. ( -u A e. RR /\ 0 < -u A ) <-> ( -. -u A e. RR \/ -. 0 < -u A ) )
2725, 26syl6bbr 278 . 2 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( A e. RR /\ 0 < A ) <-> -. ( -u A e. RR /\ 0 < -u A ) ) )
28 elrp 11834 . 2 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
29 elrp 11834 . . 3 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
3029notbii 310 . 2 |- ( -. -u A e. RR+ <-> -. ( -u A e. RR /\ 0 < -u A ) )
3127, 28, 303bitr4g 303 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 196   \/ wo 383   /\ wa 384   e. wcel 1990   =/= wne 2794   class class class wbr 4653   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   -ucneg 10267   RR+crp 11832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833
This theorem is referenced by:  cnpart  13980  angpined  24557  signsply0  30628
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