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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1dlem | Structured version Visualization version GIF version |
Description: Lemma for mulgt0con1d 41111. 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 11199 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
3 | 1, 2 | lttrid 11334 | . 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 152 | . . 3 ⊢ (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → ¬ (𝐴 = 0 ∨ 0 < 𝐴))) |
8 | mulgt0con1dlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8, 2 | lttrid 11334 | . . 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 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ℝcr 11091 0cc0 11092 < clt 11230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-addrcl 11153 ax-rnegex 11163 ax-cnre 11165 ax-pre-lttri 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 |
This theorem is referenced by: mulgt0con1d 41111 mulgt0con2d 41112 |
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