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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con2d | Structured version Visualization version GIF version |
Description: Lemma for mulgt0b2d 41330 and contrapositive of mulgt0 11288. (Contributed by SN, 26-Jun-2024.) |
Ref | Expression |
---|---|
mulgt0con2d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mulgt0con2d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mulgt0con2d.1 | ⊢ (𝜑 → 0 < 𝐴) |
mulgt0con2d.2 | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Ref | Expression |
---|---|
mulgt0con2d | ⊢ (𝜑 → 𝐵 < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgt0con2d.2 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
2 | mulgt0con2d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | mulgt0con2d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3, 2 | remulcld 11241 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
5 | 3 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐴 ∈ ℝ) |
6 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
7 | mulgt0con2d.1 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐴) | |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < 𝐴) |
9 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < 𝐵) | |
10 | 5, 6, 8, 9 | mulgt0d 11366 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
11 | 10 | ex 414 | . . 3 ⊢ (𝜑 → (0 < 𝐵 → 0 < (𝐴 · 𝐵))) |
12 | remul01 41277 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0) | |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 · 0) = 0) |
14 | oveq2 7414 | . . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0)) | |
15 | 14 | eqeq1d 2735 | . . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 · 0) = 0)) |
16 | 13, 15 | syl5ibrcom 246 | . . 3 ⊢ (𝜑 → (𝐵 = 0 → (𝐴 · 𝐵) = 0)) |
17 | 2, 4, 11, 16 | mulgt0con1dlem 41327 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 → 𝐵 < 0)) |
18 | 1, 17 | mpd 15 | 1 ⊢ (𝜑 → 𝐵 < 0) |
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 ℝcr 11106 0cc0 11107 · cmul 11112 < clt 11245 |
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-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-rmo 3377 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-ltxr 11250 df-2 12272 df-3 12273 df-resub 41236 |
This theorem is referenced by: mulgt0b2d 41330 |
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