neglt - Mathbox for Glauco Siliprandi

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Theorem neglt 44935

Description: The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
neglt	⊢ (𝐴 ∈ ℝ⁺ → -𝐴 < 𝐴)

**Proof of Theorem neglt**
Step	Hyp	Ref	Expression
1		rpre 13030	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2	1	renegcld 11682	. 2 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
3		0red 11258	. 2 ⊢ (𝐴 ∈ ℝ⁺ → 0 ∈ ℝ)
4		rpgt0 13034	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
5	1	lt0neg2d 11825	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (0 < 𝐴 ↔ -𝐴 < 0))
6	4, 5	mpbid 231	. 2 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 < 0)
7	2, 3, 1, 6, 4	lttrd 11416	1 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5145 0cc0 11149 < clt 11289 -cneg 11486 ℝ⁺crp 13022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-rp 13023

This theorem is referenced by: fourierdlem77 45840 hoicvrrex 46213