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Mirrors > Home > ILE Home > Th. List > mul2lt0np | GIF version |
Description: The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
Ref | Expression |
---|---|
mul2lt0.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mul2lt0.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mul2lt0.an | ⊢ (𝜑 → 𝐴 < 0) |
mul2lt0.bp | ⊢ (𝜑 → 0 < 𝐵) |
Ref | Expression |
---|---|
mul2lt0np | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul2lt0.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 0red 7936 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | mul2lt0.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | mul2lt0.bp | . . . 4 ⊢ (𝜑 → 0 < 𝐵) | |
5 | 3, 4 | elrpd 9667 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
6 | mul2lt0.an | . . 3 ⊢ (𝜑 → 𝐴 < 0) | |
7 | 1, 2, 5, 6 | ltmul1dd 9726 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) < (0 · 𝐵)) |
8 | 3 | recnd 7963 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 8 | mul02d 8326 | . 2 ⊢ (𝜑 → (0 · 𝐵) = 0) |
10 | 7, 9 | breqtrd 4026 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5868 ℝcr 7788 0cc0 7789 · cmul 7794 < clt 7969 |
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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-sub 8107 df-neg 8108 df-rp 9628 |
This theorem is referenced by: mul2lt0pn 9738 |
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