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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 7272 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | 1 | adantr 261 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
3 | lt0neg1 7463 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
4 | 3 | biimpa 280 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴) |
5 | 2, 4 | jca 290 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) |
6 | renegcl 7272 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
7 | 6 | adantr 261 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → -𝐵 ∈ ℝ) |
8 | lt0neg1 7463 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 0 ↔ 0 < -𝐵)) | |
9 | 8 | biimpa 280 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → 0 < -𝐵) |
10 | 7, 9 | jca 290 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) |
11 | mulgt0 7093 | . . 3 ⊢ (((-𝐴 ∈ ℝ ∧ 0 < -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) → 0 < (-𝐴 · -𝐵)) | |
12 | 5, 10, 11 | syl2an 273 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (-𝐴 · -𝐵)) |
13 | recn 7014 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | recn 7014 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | mul2neg 7395 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | |
16 | 13, 14, 15 | syl2an 273 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
17 | 16 | ad2ant2r 478 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
18 | 12, 17 | breqtrd 3788 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℂcc 6887 ℝcr 6888 0cc0 6889 · cmul 6894 < clt 7060 -cneg 7183 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-ltxr 7065 df-sub 7184 df-neg 7185 |
This theorem is referenced by: inelr 7575 apsqgt0 7592 |
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