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Mirrors > Home > ILE Home > Th. List > mullt0 | 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 8016 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | 1 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
3 | lt0neg1 8223 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
4 | 3 | biimpa 294 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴) |
5 | 2, 4 | jca 304 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) |
6 | renegcl 8016 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
7 | 6 | adantr 274 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → -𝐵 ∈ ℝ) |
8 | lt0neg1 8223 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 0 ↔ 0 < -𝐵)) | |
9 | 8 | biimpa 294 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → 0 < -𝐵) |
10 | 7, 9 | jca 304 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) |
11 | mulgt0 7832 | . . 3 ⊢ (((-𝐴 ∈ ℝ ∧ 0 < -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) → 0 < (-𝐴 · -𝐵)) | |
12 | 5, 10, 11 | syl2an 287 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (-𝐴 · -𝐵)) |
13 | recn 7746 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | recn 7746 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | mul2neg 8153 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | |
16 | 13, 14, 15 | syl2an 287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
17 | 16 | ad2ant2r 500 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
18 | 12, 17 | breqtrd 3949 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 · cmul 7618 < clt 7793 -cneg 7927 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 |
This theorem is referenced by: inelr 8339 apsqgt0 8356 |
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