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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1dlem | Structured version Visualization version GIF version |
Description: Lemma for mulgt0con1d 41633. Contraposes a positive deduction to a negative deduction. (Contributed by SN, 26-Jun-2024.) |
Ref | Expression |
---|---|
mulgt0con1dlem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mulgt0con1dlem.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mulgt0con1dlem.1 | ⊢ (𝜑 → (0 < 𝐴 → 0 < 𝐵)) |
mulgt0con1dlem.2 | ⊢ (𝜑 → (𝐴 = 0 → 𝐵 = 0)) |
Ref | Expression |
---|---|
mulgt0con1dlem | ⊢ (𝜑 → (𝐵 < 0 → 𝐴 < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgt0con1dlem.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 0red 11221 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | 1, 2 | lttrid 11356 | . 2 ⊢ (𝜑 → (𝐵 < 0 ↔ ¬ (𝐵 = 0 ∨ 0 < 𝐵))) |
4 | mulgt0con1dlem.2 | . . . . 5 ⊢ (𝜑 → (𝐴 = 0 → 𝐵 = 0)) | |
5 | mulgt0con1dlem.1 | . . . . 5 ⊢ (𝜑 → (0 < 𝐴 → 0 < 𝐵)) | |
6 | 4, 5 | orim12d 961 | . . . 4 ⊢ (𝜑 → ((𝐴 = 0 ∨ 0 < 𝐴) → (𝐵 = 0 ∨ 0 < 𝐵))) |
7 | 6 | con3d 152 | . . 3 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
8 | mulgt0con1dlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8, 2 | lttrid 11356 | . . 3 ⊢ (𝜑 → (𝐴 < 0 ↔ ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
10 | 7, 9 | sylibrd 258 | . 2 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → 𝐴 < 0)) |
11 | 3, 10 | sylbid 239 | 1 ⊢ (𝜑 → (𝐵 < 0 → 𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 ℝcr 11111 0cc0 11112 < clt 11252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-addrcl 11173 ax-rnegex 11183 ax-cnre 11185 ax-pre-lttri 11186 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 |
This theorem is referenced by: mulgt0con1d 41633 mulgt0con2d 41634 |
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