Theorem rpnegap 9643

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 7920	. . . . . . 7 |- 0 e. RR
2		reapltxor 8508	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
3	1, 2	mpan2 423	. . . . . 6 |- ( A e. RR -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
4		xorcom 1383	. . . . . 6 |- ( ( A < 0 \/_ 0 < A ) <-> ( 0 < A \/_ A < 0 ) )
5	3, 4	bitrdi 195	. . . . 5 |- ( A e. RR -> ( A # 0 <-> ( 0 < A \/_ A < 0 ) ) )
6	5	pm5.32i 451	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) )
7		anxordi 1395	. . . 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 9612	. . . 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 9612	. . . . . 6 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
13		renegcl 8180	. . . . . . 7 |- ( A e. RR -> -u A e. RR )
14	13	biantrurd 303	. . . . . 6 |- ( A e. RR -> ( 0 < -u A <-> ( -u A e. RR /\ 0 < -u A ) ) )
15	12, 14	bitr4id 198	. . . . 5 |- ( A e. RR -> ( -u A e. RR+ <-> 0 < -u A ) )
16		lt0neg1 8387	. . . . 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 1377	. 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 1370   e. wcel 2141   class class class wbr 3989   RRcr 7773   0cc0 7774   < clt 7954   -ucneg 8091   # cap 8500   RR+crp 9610

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-rp 9611

This theorem is referenced by: (None)