![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
3 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 < 0) | |
4 | 2, 3 | reclt0d 44650 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → (1 / 𝐴) < 0) |
5 | 4 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 < 0 → (1 / 𝐴) < 0)) |
6 | 0red 11218 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ∈ ℝ) | |
7 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | reclt0.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0) | |
9 | 8 | necomd 2990 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≠ 𝐴) |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ≠ 𝐴) |
11 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0) | |
12 | 6, 7, 10, 11 | lttri5d 44562 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 < 𝐴) |
13 | 0red 11218 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
14 | 1, 8 | rereccld 12042 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
15 | 14 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
16 | 1 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
17 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
18 | 16, 17 | recgt0d 12149 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
19 | 13, 15, 18 | ltled 11363 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ≤ (1 / 𝐴)) |
20 | 13, 15 | lenltd 11361 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
21 | 19, 20 | mpbid 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝐴) → ¬ (1 / 𝐴) < 0) |
22 | 12, 21 | syldan 590 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ (1 / 𝐴) < 0) |
23 | 22 | ex 412 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 < 0 → ¬ (1 / 𝐴) < 0)) |
24 | 23 | con4d 115 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
25 | 24 | imp 406 | . . 3 ⊢ ((𝜑 ∧ (1 / 𝐴) < 0) → 𝐴 < 0) |
26 | 25 | ex 412 | . 2 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
27 | 5, 26 | impbid 211 | 1 ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 0cc0 11109 1c1 11110 < clt 11249 ≤ cle 11250 / cdiv 11872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 |
This theorem is referenced by: pimrecltneg 45993 smfrec 46058 |
Copyright terms: Public domain | W3C validator |