negelrp - Metamath Proof Explorer

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Theorem negelrp 12416

Description: Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)

**Assertion**
Ref	Expression
negelrp	⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ⁺ ↔ 𝐴 < 0))

**Proof of Theorem negelrp**
Step	Hyp	Ref	Expression
1		elrp 12385	. 2 ⊢ (-𝐴 ∈ ℝ⁺ ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴))
2		lt0neg1 11140	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴))
3		renegcl 10943	. . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
4	3	biantrurd 535	. . 3 ⊢ (𝐴 ∈ ℝ → (0 < -𝐴 ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴)))
5	2, 4	bitr2d 282	. 2 ⊢ (𝐴 ∈ ℝ → ((-𝐴 ∈ ℝ ∧ 0 < -𝐴) ↔ 𝐴 < 0))
6	1, 5	syl5bb 285	1 ⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ⁺ ↔ 𝐴 < 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 class class class wbr 5058 ℝcr 10530 0cc0 10531 < clt 10669 -cneg 10865 ℝ⁺crp 12383

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-rp 12384

This theorem is referenced by: negelrpd 12417 mul2lt0rlt0 12485 signsply0 31816