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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1dlem | Structured version Visualization version GIF version |
Description: Lemma for mulgt0con1d 39910. 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 10667 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | 1, 2 | lttrid 10801 | . 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 963 | . . . 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 10801 | . . 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 845 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 ℝcr 10559 0cc0 10560 < clt 10698 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-addrcl 10621 ax-rnegex 10631 ax-cnre 10633 ax-pre-lttri 10634 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 |
This theorem is referenced by: mulgt0con1d 39910 mulgt0con2d 39911 |
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