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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con1d | Structured version Visualization version GIF version |
Description: Counterpart to mulgt0con2d 40089, though not a lemma of anything. This is the first use of ax-pre-mulgt0 10789. (Contributed by SN, 26-Jun-2024.) |
Ref | Expression |
---|---|
mulgt0con1d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mulgt0con1d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mulgt0con1d.1 | ⊢ (𝜑 → 0 < 𝐵) |
mulgt0con1d.2 | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Ref | Expression |
---|---|
mulgt0con1d | ⊢ (𝜑 → 𝐴 < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgt0con1d.2 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
2 | mulgt0con1d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | mulgt0con1d.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | remulcld 10846 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
5 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) |
6 | 3 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 ∈ ℝ) |
7 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | |
8 | mulgt0con1d.1 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐵) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐵) |
10 | 5, 6, 7, 9 | mulgt0d 10970 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (𝐴 · 𝐵)) |
11 | 10 | ex 416 | . . 3 ⊢ (𝜑 → (0 < 𝐴 → 0 < (𝐴 · 𝐵))) |
12 | remul02 40048 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 · 𝐵) = 0) | |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (0 · 𝐵) = 0) |
14 | oveq1 7209 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵)) | |
15 | 14 | eqeq1d 2736 | . . . 4 ⊢ (𝐴 = 0 → ((𝐴 · 𝐵) = 0 ↔ (0 · 𝐵) = 0)) |
16 | 13, 15 | syl5ibrcom 250 | . . 3 ⊢ (𝜑 → (𝐴 = 0 → (𝐴 · 𝐵) = 0)) |
17 | 2, 4, 11, 16 | mulgt0con1dlem 40087 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 → 𝐴 < 0)) |
18 | 1, 17 | mpd 15 | 1 ⊢ (𝜑 → 𝐴 < 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 ℝcr 10711 0cc0 10712 · cmul 10717 < 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-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 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-reu 3061 df-rmo 3062 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-po 5457 df-so 5458 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-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-ltxr 10855 df-2 11876 df-resub 40009 |
This theorem is referenced by: (None) |
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