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Mirrors > Home > ILE Home > Th. List > mul2lt0rlt0 | 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 7910 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
4 | 3 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℝ) |
5 | 0red 7881 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 ∈ ℝ) | |
6 | 2 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℝ) |
7 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 < 0) | |
8 | 6, 7 | negelrpd 9601 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℝ+) |
9 | mul2lt0.3 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
10 | 9 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) < 0) |
11 | 4, 5, 8, 10 | ltdiv1dd 9667 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) < (0 / -𝐵)) |
12 | 1 | recnd 7908 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℂ) |
14 | 2 | recnd 7908 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 14 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℂ) |
16 | 13, 15 | mulcld 7900 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℂ) |
17 | 6, 7 | lt0ap0d 8528 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 # 0) |
18 | 16, 15, 17 | divneg2apd 8681 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = ((𝐴 · 𝐵) / -𝐵)) |
19 | 13, 15, 17 | divcanap4d 8673 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
20 | 19 | negeqd 8074 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = -𝐴) |
21 | 18, 20 | eqtr3d 2192 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) = -𝐴) |
22 | 15 | negcld 8177 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℂ) |
23 | 15, 17 | negap0d 8510 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 # 0) |
24 | 22, 23 | div0apd 8664 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 / -𝐵) = 0) |
25 | 11, 21, 24 | 3brtr3d 3997 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐴 < 0) |
26 | 1 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℝ) |
27 | 26 | lt0neg2d 8395 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
28 | 25, 27 | mpbird 166 | 1 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2128 class class class wbr 3967 (class class class)co 5826 ℂcc 7732 ℝcr 7733 0cc0 7734 · cmul 7739 < clt 7914 -cneg 8051 / cdiv 8549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-po 4258 df-iso 4259 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-rp 9567 |
This theorem is referenced by: mul2lt0llt0 9674 |
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