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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 10674 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℝ) |
5 | 0red 10647 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 ∈ ℝ) | |
6 | negelrp 12425 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) | |
7 | 2, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) |
8 | 7 | biimpar 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℝ+) |
9 | mul2lt0.3 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) < 0) |
11 | 4, 5, 8, 10 | ltdiv1dd 12491 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) < (0 / -𝐵)) |
12 | 1 | recnd 10672 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℂ) |
14 | 2 | recnd 10672 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 14 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℂ) |
16 | 13, 15 | mulcld 10664 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℂ) |
17 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 < 0) | |
18 | 17 | lt0ne0d 11208 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ≠ 0) |
19 | 16, 15, 18 | divneg2d 11433 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = ((𝐴 · 𝐵) / -𝐵)) |
20 | 13, 15, 18 | divcan4d 11425 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
21 | 20 | negeqd 10883 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = -𝐴) |
22 | 19, 21 | eqtr3d 2861 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) = -𝐴) |
23 | 15 | negcld 10987 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℂ) |
24 | 15, 18 | negne0d 10998 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ≠ 0) |
25 | 23, 24 | div0d 11418 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 / -𝐵) = 0) |
26 | 11, 22, 25 | 3brtr3d 5100 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐴 < 0) |
27 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℝ) |
28 | 27 | lt0neg2d 11213 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
29 | 26, 28 | mpbird 259 | 1 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 class class class wbr 5069 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 · cmul 10545 < clt 10678 -cneg 10874 / cdiv 11300 ℝ+crp 12392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-rp 12393 |
This theorem is referenced by: mul2lt0llt0 12496 mul2lt0bi 12498 sgnmul 31804 signsply0 31825 |
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