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Mirrors > Home > MPE Home > Th. List > mul2lt0rlt0 | Structured version Visualization version GIF version |
Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.) |
Ref | Expression |
---|---|
mul2lt0.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mul2lt0.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mul2lt0.3 | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Ref | Expression |
---|---|
mul2lt0rlt0 | ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul2lt0.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | mul2lt0.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | remulcld 10660 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℝ) |
5 | 0red 10633 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 ∈ ℝ) | |
6 | negelrp 12410 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) | |
7 | 2, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) |
8 | 7 | biimpar 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℝ+) |
9 | mul2lt0.3 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) < 0) |
11 | 4, 5, 8, 10 | ltdiv1dd 12476 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) < (0 / -𝐵)) |
12 | 1 | recnd 10658 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℂ) |
14 | 2 | recnd 10658 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℂ) |
16 | 13, 15 | mulcld 10650 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℂ) |
17 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 < 0) | |
18 | 17 | lt0ne0d 11194 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ≠ 0) |
19 | 16, 15, 18 | divneg2d 11419 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = ((𝐴 · 𝐵) / -𝐵)) |
20 | 13, 15, 18 | divcan4d 11411 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
21 | 20 | negeqd 10869 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = -𝐴) |
22 | 19, 21 | eqtr3d 2835 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) = -𝐴) |
23 | 15 | negcld 10973 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℂ) |
24 | 15, 18 | negne0d 10984 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ≠ 0) |
25 | 23, 24 | div0d 11404 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 / -𝐵) = 0) |
26 | 11, 22, 25 | 3brtr3d 5061 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐴 < 0) |
27 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℝ) |
28 | 27 | lt0neg2d 11199 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
29 | 26, 28 | mpbird 260 | 1 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 -cneg 10860 / cdiv 11286 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-rp 12378 |
This theorem is referenced by: mul2lt0llt0 12481 mul2lt0bi 12483 sgnmul 31910 signsply0 31931 |
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