Theorem rpnegap 10087

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 8326	. . . . . . 7 |- 0 e. RR
2		reapltxor 8917	. . . . . . 7 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
3	1, 2	mpan2 429	. . . . . 6 |- ( A e. RR -> ( A # 0 <-> ( A < 0 \/_ 0 < A ) ) )
4		xorcom 1437	. . . . . 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 458	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ ( 0 < A \/_ A < 0 ) ) )
7		anxordi 1449	. . . 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 10056	. . . 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 10056	. . . . . 6 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
13		renegcl 8587	. . . . . . 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 8796	. . . . 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 1431	. 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+ ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/_ wxo 1424   e. wcel 2209   class class class wbr 4130   RRcr 8178   0cc0 8179   < clt 8360   -ucneg 8498   # cap 8909   RR+crp 10054

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-rp 10055

This theorem is used by: (None)