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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
3 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 < 0) | |
4 | 2, 3 | reclt0d 39607 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → (1 / 𝐴) < 0) |
5 | 4 | ex 450 | . 2 ⊢ (𝜑 → (𝐴 < 0 → (1 / 𝐴) < 0)) |
6 | 0red 10041 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ∈ ℝ) | |
7 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | reclt0.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0) | |
9 | 8 | necomd 2849 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≠ 𝐴) |
10 | 9 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ≠ 𝐴) |
11 | simpr 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0) | |
12 | 6, 7, 10, 11 | lttri5d 39513 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 < 𝐴) |
13 | 0red 10041 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
14 | 1, 8 | rereccld 10852 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
15 | 14 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
16 | 1 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
17 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
18 | 16, 17 | recgt0d 10958 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
19 | 13, 15, 18 | ltled 10185 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ≤ (1 / 𝐴)) |
20 | 13, 15 | lenltd 10183 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
21 | 19, 20 | mpbid 222 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝐴) → ¬ (1 / 𝐴) < 0) |
22 | 12, 21 | syldan 487 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ (1 / 𝐴) < 0) |
23 | 22 | ex 450 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 < 0 → ¬ (1 / 𝐴) < 0)) |
24 | 23 | con4d 114 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
25 | 24 | imp 445 | . . 3 ⊢ ((𝜑 ∧ (1 / 𝐴) < 0) → 𝐴 < 0) |
26 | 25 | ex 450 | . 2 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
27 | 5, 26 | impbid 202 | 1 ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 < clt 10074 ≤ cle 10075 / cdiv 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 |
This theorem is referenced by: pimrecltneg 40933 smfrec 40996 |
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