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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1dlem | Structured version Visualization version GIF version |
Description: Lemma for mulgt0con1d 40088. 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 10819 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | 1, 2 | lttrid 10953 | . 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 965 | . . . 4 ⊢ (𝜑 → ((𝐴 = 0 ∨ 0 < 𝐴) → (𝐵 = 0 ∨ 0 < 𝐵))) |
7 | 6 | con3d 155 | . . 3 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
8 | mulgt0con1dlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8, 2 | lttrid 10953 | . . 3 ⊢ (𝜑 → (𝐴 < 0 ↔ ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
10 | 7, 9 | sylibrd 262 | . 2 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → 𝐴 < 0)) |
11 | 3, 10 | sylbid 243 | 1 ⊢ (𝜑 → (𝐵 < 0 → 𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ℝcr 10711 0cc0 10712 < clt 10850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-addrcl 10773 ax-rnegex 10783 ax-cnre 10785 ax-pre-lttri 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-ltxr 10855 |
This theorem is referenced by: mulgt0con1d 40088 mulgt0con2d 40089 |
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