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Mirrors > Home > ILE Home > Th. List > mul2lt0np | GIF version |
Description: The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
Ref | Expression |
---|---|
mul2lt0.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mul2lt0.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mul2lt0.an | ⊢ (𝜑 → 𝐴 < 0) |
mul2lt0.bp | ⊢ (𝜑 → 0 < 𝐵) |
Ref | Expression |
---|---|
mul2lt0np | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul2lt0.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 0red 7767 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | mul2lt0.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | mul2lt0.bp | . . . 4 ⊢ (𝜑 → 0 < 𝐵) | |
5 | 3, 4 | elrpd 9481 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
6 | mul2lt0.an | . . 3 ⊢ (𝜑 → 𝐴 < 0) | |
7 | 1, 2, 5, 6 | ltmul1dd 9539 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) < (0 · 𝐵)) |
8 | 3 | recnd 7794 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | mul02d 8154 | . 2 ⊢ (𝜑 → (0 · 𝐵) = 0) |
10 | 7, 9 | breqtrd 3954 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 · cmul 7625 < clt 7800 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-rp 9442 |
This theorem is referenced by: mul2lt0pn 9551 |
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