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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 8216 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | 1 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → -𝐴 ∈ ℝ) |
3 | lt0neg1 8423 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
4 | 3 | biimpa 296 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 < -𝐴) |
5 | 2, 4 | jca 306 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → (-𝐴 ∈ ℝ ∧ 0 < -𝐴)) |
6 | renegcl 8216 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
7 | 6 | adantr 276 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → -𝐵 ∈ ℝ) |
8 | lt0neg1 8423 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 0 ↔ 0 < -𝐵)) | |
9 | 8 | biimpa 296 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → 0 < -𝐵) |
10 | 7, 9 | jca 306 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 0) → (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) |
11 | mulgt0 8030 | . . 3 ⊢ (((-𝐴 ∈ ℝ ∧ 0 < -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 < -𝐵)) → 0 < (-𝐴 · -𝐵)) | |
12 | 5, 10, 11 | syl2an 289 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (-𝐴 · -𝐵)) |
13 | recn 7943 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
14 | recn 7943 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
15 | mul2neg 8353 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | |
16 | 13, 14, 15 | syl2an 289 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
17 | 16 | ad2ant2r 509 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) |
18 | 12, 17 | breqtrd 4029 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 · cmul 7815 < clt 7990 -cneg 8127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-sub 8128 df-neg 8129 |
This theorem is referenced by: inelr 8539 apsqgt0 8556 |
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