Theorem rpnegap 9882

Description: Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)

Assertion

Ref	Expression
rpnegap	|- ( ( A e. RR /\ A # 0 ) -> ( A e. RR+ \/_ -u A e. RR+ ) )

Proof of Theorem rpnegap

Step	Hyp	Ref	Expression
1		0re 8146	. . . . . . 7 |- 0 e. RR
2		reapltxor 8736	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
3	1, 2	mpan2 425	. . . . . 6 |- ( A e. RR -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
4		xorcom 1430	. . . . . 6 |- ( ( A < 0 \/_ 0 < A ) <-> ( 0 < A \/_ A < 0 ) )
5	3, 4	bitrdi 196	. . . . 5 |- ( A e. RR -> ( A # 0 <-> ( 0 < A \/_ A < 0 ) ) )
6	5	pm5.32i 454	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) )
7		anxordi 1442	. . . 4 |- ( ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) <-> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
8	6, 7	bitri 184	. . 3 |- ( ( A e. RR /\ A # 0 ) <-> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
9	8	biimpi 120	. 2 |- ( ( A e. RR /\ A # 0 ) -> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
10		elrp 9851	. . . 4 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
11	10	a1i 9	. . 3 |- ( ( A e. RR /\ A # 0 ) -> ( A e. RR+ <-> ( A e. RR /\ 0 < A ) ) )
12		elrp 9851	. . . . . 6 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
13		renegcl 8407	. . . . . . 7 |- ( A e. RR -> -u A e. RR )
14	13	biantrurd 305	. . . . . 6 |- ( A e. RR -> ( 0 < -u A <-> ( -u A e. RR /\ 0 < -u A ) ) )
15	12, 14	bitr4id 199	. . . . 5 |- ( A e. RR -> ( -u A e. RR+ <-> 0 < -u A ) )
16		lt0neg1 8615	. . . . 5 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
17		ibar 301	. . . . 5 |- ( A e. RR -> ( A < 0 <-> ( A e. RR /\ A < 0 ) ) )
18	15, 16, 17	3bitr2d 216	. . . 4 |- ( A e. RR -> ( -u A e. RR+ <-> ( A e. RR /\ A < 0 ) ) )
19	18	adantr 276	. . 3 |- ( ( A e. RR /\ A # 0 ) -> ( -u A e. RR+ <-> ( A e. RR /\ A < 0 ) ) )
20	11, 19	xorbi12d 1424	. 2 |- ( ( A e. RR /\ A # 0 ) -> ( ( A e. RR+ \/_ -u A e. RR+ ) <-> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) ) )
21	9, 20	mpbird 167	1 |- ( ( A e. RR /\ A # 0 ) -> ( A e. RR+ \/_ -u A e. RR+ ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/_ wxo 1417   e. wcel 2200   class class class wbr 4083   RRcr 7998   0cc0 7999   < clt 8181   -ucneg 8318   # cap 8728   RR+crp 9849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-rp 9850

This theorem is referenced by: (None)