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Mirrors > Home > MPE Home > Th. List > mullt0 | Structured version Visualization version GIF version |
Description: The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.) |
Ref | Expression |
---|---|
mullt0 | ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl 11520 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | 1 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
3 | lt0neg1 11717 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
4 | 3 | biimpa 478 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴) |
5 | 2, 4 | jca 513 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) |
6 | renegcl 11520 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
7 | 6 | adantr 482 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → -𝐵 ∈ ℝ) |
8 | lt0neg1 11717 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 0 ↔ 0 < -𝐵)) | |
9 | 8 | biimpa 478 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → 0 < -𝐵) |
10 | 7, 9 | jca 513 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) |
11 | mulgt0 11288 | . . 3 ⊢ (((-𝐴 ∈ ℝ ∧ 0 < -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) → 0 < (-𝐴 · -𝐵)) | |
12 | 5, 10, 11 | syl2an 597 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (-𝐴 · -𝐵)) |
13 | recn 11197 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | recn 11197 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | mul2neg 11650 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | |
16 | 13, 14, 15 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
17 | 16 | ad2ant2r 746 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
18 | 12, 17 | breqtrd 5174 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 · cmul 11112 < clt 11245 -cneg 11442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 |
This theorem is referenced by: msqgt0 11731 |
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