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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 7986 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
4 | 3 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℝ) |
5 | 0red 7957 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 ∈ ℝ) | |
6 | 2 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℝ) |
7 | simpr 110 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 < 0) | |
8 | 6, 7 | negelrpd 9686 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℝ+) |
9 | mul2lt0.3 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
10 | 9 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) < 0) |
11 | 4, 5, 8, 10 | ltdiv1dd 9752 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) < (0 / -𝐵)) |
12 | 1 | recnd 7984 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℂ) |
14 | 2 | recnd 7984 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
15 | 14 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 ∈ ℂ) |
16 | 13, 15 | mulcld 7976 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → (𝐴 · 𝐵) ∈ ℂ) |
17 | 6, 7 | lt0ap0d 8604 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐵 # 0) |
18 | 16, 15, 17 | divneg2apd 8759 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = ((𝐴 · 𝐵) / -𝐵)) |
19 | 13, 15, 17 | divcanap4d 8751 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
20 | 19 | negeqd 8150 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -((𝐴 · 𝐵) / 𝐵) = -𝐴) |
21 | 18, 20 | eqtr3d 2212 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → ((𝐴 · 𝐵) / -𝐵) = -𝐴) |
22 | 15 | negcld 8253 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 ∈ ℂ) |
23 | 15, 17 | negap0d 8586 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐵 # 0) |
24 | 22, 23 | div0apd 8742 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 / -𝐵) = 0) |
25 | 11, 21, 24 | 3brtr3d 4034 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → -𝐴 < 0) |
26 | 1 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ 𝐵 < 0) → 𝐴 ∈ ℝ) |
27 | 26 | lt0neg2d 8471 | . 2 ⊢ ((𝜑 ∧ 𝐵 < 0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
28 | 25, 27 | mpbird 167 | 1 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 · cmul 7815 < clt 7990 -cneg 8127 / cdiv 8627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-rp 9652 |
This theorem is referenced by: mul2lt0llt0 9759 |
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