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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
3 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 < 0) | |
4 | 2, 3 | reclt0d 41651 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → (1 / 𝐴) < 0) |
5 | 4 | ex 415 | . 2 ⊢ (𝜑 → (𝐴 < 0 → (1 / 𝐴) < 0)) |
6 | 0red 10638 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ∈ ℝ) | |
7 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | reclt0.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0) | |
9 | 8 | necomd 3071 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≠ 𝐴) |
10 | 9 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ≠ 𝐴) |
11 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0) | |
12 | 6, 7, 10, 11 | lttri5d 41559 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 < 𝐴) |
13 | 0red 10638 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
14 | 1, 8 | rereccld 11461 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
15 | 14 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
16 | 1 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
17 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
18 | 16, 17 | recgt0d 11568 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
19 | 13, 15, 18 | ltled 10782 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ≤ (1 / 𝐴)) |
20 | 13, 15 | lenltd 10780 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
21 | 19, 20 | mpbid 234 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝐴) → ¬ (1 / 𝐴) < 0) |
22 | 12, 21 | syldan 593 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ (1 / 𝐴) < 0) |
23 | 22 | ex 415 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 < 0 → ¬ (1 / 𝐴) < 0)) |
24 | 23 | con4d 115 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
25 | 24 | imp 409 | . . 3 ⊢ ((𝜑 ∧ (1 / 𝐴) < 0) → 𝐴 < 0) |
26 | 25 | ex 415 | . 2 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
27 | 5, 26 | impbid 214 | 1 ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 < clt 10669 ≤ cle 10670 / cdiv 11291 |
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-cnex 10587 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 ax-pre-mulgt0 10608 |
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-rmo 3146 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-div 11292 |
This theorem is referenced by: pimrecltneg 42995 smfrec 43058 |
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