Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 11240 | . . 3 ⊢ 0 < 1 | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 < 1) |
3 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → ¬ (1 / 𝐴) < 0) | |
4 | 0red 10722 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 ∈ ℝ) | |
5 | 1red 10720 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
6 | reclt0d.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | reclt0d.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 0) | |
8 | 7 | lt0ne0d 11283 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) |
9 | 5, 6, 8 | redivcld 11546 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → (1 / 𝐴) ∈ ℝ) |
11 | 4, 10 | lenltd 10864 | . . . 4 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → (0 ≤ (1 / 𝐴) ↔ ¬ (1 / 𝐴) < 0)) |
12 | 3, 11 | mpbird 260 | . . 3 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → 0 ≤ (1 / 𝐴)) |
13 | 6 | recnd 10747 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
14 | 13, 8 | recidd 11489 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) |
15 | 14 | eqcomd 2744 | . . . . . 6 ⊢ (𝜑 → 1 = (𝐴 · (1 / 𝐴))) |
16 | 15 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 = (𝐴 · (1 / 𝐴))) |
17 | 0red 10722 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ) | |
18 | 6, 17, 7 | ltled 10866 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≤ 0) |
19 | 18 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 𝐴 ≤ 0) |
20 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 0 ≤ (1 / 𝐴)) | |
21 | 19, 20 | jca 515 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴))) |
22 | 21 | orcd 872 | . . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0))) |
23 | mulle0b 11589 | . . . . . . . 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 587 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 · (1 / 𝐴)) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0)))) |
25 | 24 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ((𝐴 · (1 / 𝐴)) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ (1 / 𝐴)) ∨ (0 ≤ 𝐴 ∧ (1 / 𝐴) ≤ 0)))) |
26 | 22, 25 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (𝐴 · (1 / 𝐴)) ≤ 0) |
27 | 16, 26 | eqbrtrd 5052 | . . . 4 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 ≤ 0) |
28 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 1 ∈ ℝ) |
29 | 0red 10722 | . . . . 5 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → 0 ∈ ℝ) | |
30 | 28, 29 | lenltd 10864 | . . . 4 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → (1 ≤ 0 ↔ ¬ 0 < 1)) |
31 | 27, 30 | mpbid 235 | . . 3 ⊢ ((𝜑 ∧ 0 ≤ (1 / 𝐴)) → ¬ 0 < 1) |
32 | 12, 31 | syldan 594 | . 2 ⊢ ((𝜑 ∧ ¬ (1 / 𝐴) < 0) → ¬ 0 < 1) |
33 | 2, 32 | condan 818 | 1 ⊢ (𝜑 → (1 / 𝐴) < 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 1c1 10616 · cmul 10620 < clt 10753 ≤ cle 10754 / cdiv 11375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 |
This theorem is referenced by: reclt0 42469 ltdiv23neg 42472 pimrecltpos 43785 |
Copyright terms: Public domain | W3C validator |