Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mul2lt0lgt0 | 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, 2-Oct-2018.) |
Ref | Expression |
---|---|
mul2lt0.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mul2lt0.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mul2lt0.3 | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Ref | Expression |
---|---|
mul2lt0lgt0 | ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul2lt0.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | mul2lt0.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 2 | recnd 10861 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 1 | recnd 10861 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
5 | 3, 4 | mulcomd 10854 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
6 | mul2lt0.3 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
7 | 5, 6 | eqbrtrrd 5077 | . 2 ⊢ (𝜑 → (𝐵 · 𝐴) < 0) |
8 | 1, 2, 7 | mul2lt0rgt0 12689 | 1 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 < 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 0cc0 10729 · cmul 10734 < clt 10867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-sub 11064 df-neg 11065 df-rp 12587 |
This theorem is referenced by: sgnsub 32223 |
Copyright terms: Public domain | W3C validator |