Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reclt0d | Structured version Visualization version GIF version |
Description: The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
reclt0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
reclt0d.2 | ⊢ (𝜑 → 𝐴 < 0) |
Ref | Expression |
---|---|
reclt0d | ⊢ (𝜑 → (1 / 𝐴) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 11162 | . . 3 ⊢ 0 < 1 | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 < 1) |
3 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → ¬ (1 / 𝐴) < 0) | |
4 | 0red 10644 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 ∈ ℝ) | |
5 | 1red 10642 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
6 | reclt0d.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | reclt0d.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 0) | |
8 | 7 | lt0ne0d 11205 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) |
9 | 5, 6, 8 | redivcld 11468 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
10 | 9 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → (1 / 𝐴) ∈ ℝ) |
11 | 4, 10 | lenltd 10786 | . . . 4 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
12 | 3, 11 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 ≤ (1 / 𝐴)) |
13 | 6 | recnd 10669 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
14 | 13, 8 | recidd 11411 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) |
15 | 14 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → 1 = (𝐴 · (1 / 𝐴))) |
16 | 15 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 = (𝐴 · (1 / 𝐴))) |
17 | 0red 10644 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 6, 17, 7 | ltled 10788 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≤ 0) |
19 | 18 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 𝐴 ≤ 0) |
20 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 0 ≤ (1 / 𝐴)) | |
21 | 19, 20 | jca 514 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴))) |
22 | 21 | orcd 869 | . . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0))) |
23 | mulle0b 11511 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → ((𝐴 · (1 / 𝐴)) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0)))) | |
24 | 6, 9, 23 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 · (1 / 𝐴)) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0)))) |
25 | 24 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ((𝐴 · (1 / 𝐴)) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0)))) |
26 | 22, 25 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (𝐴 · (1 / 𝐴)) ≤ 0) |
27 | 16, 26 | eqbrtrd 5088 | . . . 4 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 ≤ 0) |
28 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 ∈ ℝ) |
29 | 0red 10644 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 0 ∈ ℝ) | |
30 | 28, 29 | lenltd 10786 | . . . 4 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (1 ≤ 0 ↔ ¬ 0 < 1)) |
31 | 27, 30 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ¬ 0 < 1) |
32 | 12, 31 | syldan 593 | . 2 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → ¬ 0 < 1) |
33 | 2, 32 | condan 816 | 1 ⊢ (𝜑 → (1 / 𝐴) < 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 ≤ cle 10676 / cdiv 11297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 |
This theorem is referenced by: reclt0 41683 ltdiv23neg 41686 pimrecltpos 43007 |
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