Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reclt0 | Structured version Visualization version GIF version |
Description: The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
reclt0.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
reclt0.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
reclt0 | ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reclt0.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
3 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 < 0) | |
4 | 2, 3 | reclt0d 42487 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → (1 / 𝐴) < 0) |
5 | 4 | ex 416 | . 2 ⊢ (𝜑 → (𝐴 < 0 → (1 / 𝐴) < 0)) |
6 | 0red 10725 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ∈ ℝ) | |
7 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | reclt0.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0) | |
9 | 8 | necomd 2990 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≠ 𝐴) |
10 | 9 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ≠ 𝐴) |
11 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0) | |
12 | 6, 7, 10, 11 | lttri5d 42399 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 < 𝐴) |
13 | 0red 10725 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
14 | 1, 8 | rereccld 11548 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
15 | 14 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
16 | 1 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
17 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
18 | 16, 17 | recgt0d 11655 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
19 | 13, 15, 18 | ltled 10869 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ≤ (1 / 𝐴)) |
20 | 13, 15 | lenltd 10867 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
21 | 19, 20 | mpbid 235 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝐴) → ¬ (1 / 𝐴) < 0) |
22 | 12, 21 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ (1 / 𝐴) < 0) |
23 | 22 | ex 416 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 < 0 → ¬ (1 / 𝐴) < 0)) |
24 | 23 | con4d 115 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
25 | 24 | imp 410 | . . 3 ⊢ ((𝜑 ∧ (1 / 𝐴) < 0) → 𝐴 < 0) |
26 | 25 | ex 416 | . 2 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
27 | 5, 26 | impbid 215 | 1 ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5031 (class class class)co 7173 ℝcr 10617 0cc0 10618 1c1 10619 < clt 10756 ≤ cle 10757 / cdiv 11378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 |
This theorem is referenced by: pimrecltneg 43822 smfrec 43885 |
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