![]() |
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 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 ∈ ℝ) |
3 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝐴 < 0) | |
4 | 2, 3 | reclt0d 43708 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 0) → (1 / 𝐴) < 0) |
5 | 4 | ex 414 | . 2 ⊢ (𝜑 → (𝐴 < 0 → (1 / 𝐴) < 0)) |
6 | 0red 11163 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ∈ ℝ) | |
7 | 1 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 𝐴 ∈ ℝ) |
8 | reclt0.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 0) | |
9 | 8 | necomd 2996 | . . . . . . . . 9 ⊢ (𝜑 → 0 ≠ 𝐴) |
10 | 9 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 ≠ 𝐴) |
11 | simpr 486 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0) | |
12 | 6, 7, 10, 11 | lttri5d 43620 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → 0 < 𝐴) |
13 | 0red 11163 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ∈ ℝ) | |
14 | 1, 8 | rereccld 11987 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
15 | 14 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → (1 / 𝐴) ∈ ℝ) |
16 | 1 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
17 | simpr 486 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
18 | 16, 17 | recgt0d 12094 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) |
19 | 13, 15, 18 | ltled 11308 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 ≤ (1 / 𝐴)) |
20 | 13, 15 | lenltd 11306 | . . . . . . . 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 592 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ¬ (1 / 𝐴) < 0) |
23 | 22 | ex 414 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 < 0 → ¬ (1 / 𝐴) < 0)) |
24 | 23 | con4d 115 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) < 0 → 𝐴 < 0)) |
25 | 24 | imp 408 | . . 3 ⊢ ((𝜑 ∧ (1 / 𝐴) < 0) → 𝐴 < 0) |
26 | 25 | ex 414 | . 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 397 ∈ wcel 2107 ≠ wne 2940 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 0cc0 11056 1c1 11057 < clt 11194 ≤ cle 11195 / cdiv 11817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 |
This theorem is referenced by: pimrecltneg 45051 smfrec 45116 |
Copyright terms: Public domain | W3C validator |