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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 10070 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℝ) |
5 | 0red 10041 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 ∈ ℝ) | |
6 | negelrp 11864 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) | |
7 | 2, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (-𝐵 ∈ ℝ+ ↔ 𝐵 < 0)) |
8 | 7 | biimpar 502 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℝ+) |
9 | mul2lt0.3 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) < 0) |
11 | 4, 5, 8, 10 | ltdiv1dd 11929 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) < (0 / -𝐵)) |
12 | 1 | recnd 10068 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℂ) |
14 | 2 | recnd 10068 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℂ) |
16 | 13, 15 | mulcld 10060 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℂ) |
17 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 < 0) | |
18 | 17 | lt0ne0d 10593 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ≠ 0) |
19 | 16, 15, 18 | divneg2d 10815 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = ((𝐴 · 𝐵) / -𝐵)) |
20 | 13, 15, 18 | divcan4d 10807 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
21 | 20 | negeqd 10275 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = -𝐴) |
22 | 19, 21 | eqtr3d 2658 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) = -𝐴) |
23 | 15 | negcld 10379 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℂ) |
24 | 15, 18 | negne0d 10390 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ≠ 0) |
25 | 23, 24 | div0d 10800 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 / -𝐵) = 0) |
26 | 11, 22, 25 | 3brtr3d 4684 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐴 < 0) |
27 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℝ) |
28 | 27 | lt0neg2d 10598 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
29 | 26, 28 | mpbird 247 | 1 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 < clt 10074 -cneg 10267 / cdiv 10684 ℝ+crp 11832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-rp 11833 |
This theorem is referenced by: mul2lt0llt0 11934 mul2lt0bi 11936 sgnmul 30604 signsply0 30628 |
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