Theorem rpnegap 9982

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 8239	. . . . . . 7 |- 0 e. RR
2		reapltxor 8828	. . . . . . 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 1433	. . . . . 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 1445	. . . 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 9951	. . . 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 9951	. . . . . 6 |- ( -u A e. RR+ <-> ( -u A e. RR /\ 0 < -u A ) )
13		renegcl 8499	. . . . . . 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 8707	. . . . 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 1427	. 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 1420   e. wcel 2202   class class class wbr 4093   RRcr 8091   0cc0 8092   < clt 8273   -ucneg 8410   # cap 8820   RR+crp 9949

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-rp 9950

This theorem is referenced by: (None)