Theorem rpnegap 9481

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 7773	. . . . . . 7 |- 0 e. RR
2		reapltxor 8358	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
3	1, 2	mpan2 421	. . . . . 6 |- ( A e. RR -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
4		xorcom 1366	. . . . . 6 |- ( ( A < 0 \/_ 0 < A ) <-> ( 0 < A \/_ A < 0 ) )
5	3, 4	syl6bb 195	. . . . 5 |- ( A e. RR -> ( A # 0 <-> ( 0 < A \/_ A < 0 ) ) )
6	5	pm5.32i 449	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) )
7		anxordi 1378	. . . 4 |- ( ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) <-> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
8	6, 7	bitri 183	. . 3 |- ( ( A e. RR /\ A # 0 ) <-> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
9	8	biimpi 119	. 2 |- ( ( A e. RR /\ A # 0 ) -> ( ( A e. RR /\ 0 < A ) \/_ ( A e. RR /\ A < 0 ) ) )
10		elrp 9450	. . . 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		renegcl 8030	. . . . . . 7 |- ( A e. RR -> -u A e. RR )
13	12	biantrurd 303	. . . . . 6 |- ( A e. RR -> ( 0 < -u A <-> ( -u A e. RR /\ 0 < -u A ) ) )
14		elrp 9450	. . . . . 6 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
15	13, 14	syl6rbbr 198	. . . . 5 |- ( A e. RR -> ( -u A e. RR+ <-> 0 < -u A ) )
16		lt0neg1 8237	. . . . 5 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
17		ibar 299	. . . . 5 |- ( A e. RR -> ( A < 0 <-> ( A e. RR /\ A < 0 ) ) )
18	15, 16, 17	3bitr2d 215	. . . 4 |- ( A e. RR -> ( -u A e. RR+ <-> ( A e. RR /\ A < 0 ) ) )
19	18	adantr 274	. . 3 |- ( ( A e. RR /\ A # 0 ) -> ( -u A e. RR+ <-> ( A e. RR /\ A < 0 ) ) )
20	11, 19	xorbi12d 1360	. 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 166	1 |- ( ( A e. RR /\ A # 0 ) -> ( A e. RR+ \/_ -u A e. RR+ ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/_ wxo 1353   e. wcel 1480   class class class wbr 3929   RRcr 7626   0cc0 7627   < clt 7807   -ucneg 7941   # cap 8350   RR+crp 9448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-xor 1354  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-ltxr 7812  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-rp 9449

This theorem is referenced by: (None)