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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1dlem | Structured version Visualization version GIF version |
Description: Lemma for mulgt0con1d 42465. 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 11262 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | 1, 2 | lttrid 11397 | . 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 966 | . . . 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 11397 | . . 3 ⊢ (𝜑 → (𝐴 < 0 ↔ ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
10 | 7, 9 | sylibrd 259 | . 2 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → 𝐴 < 0)) |
11 | 3, 10 | sylbid 240 | 1 ⊢ (𝜑 → (𝐵 < 0 → 𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ℝcr 11152 0cc0 11153 < clt 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 |
This theorem is referenced by: mulgt0con1d 42465 mulgt0con2d 42466 |
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