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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulgt0con2d | Structured version Visualization version GIF version |
Description: Lemma for mulgt0b2d 39912 and contrapositive of mulgt0 10741. (Contributed by SN, 26-Jun-2024.) |
Ref | Expression |
---|---|
mulgt0con2d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
mulgt0con2d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mulgt0con2d.1 | ⊢ (𝜑 → 0 < 𝐴) |
mulgt0con2d.2 | ⊢ (𝜑 → (𝐴 · 𝐵) < 0) |
Ref | Expression |
---|---|
mulgt0con2d | ⊢ (𝜑 → 𝐵 < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgt0con2d.2 | . 2 ⊢ (𝜑 → (𝐴 · 𝐵) < 0) | |
2 | mulgt0con2d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | mulgt0con2d.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3, 2 | remulcld 10694 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
5 | 3 | adantr 485 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐴 ∈ ℝ) |
6 | 2 | adantr 485 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
7 | mulgt0con2d.1 | . . . . . 6 ⊢ (𝜑 → 0 < 𝐴) | |
8 | 7 | adantr 485 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < 𝐴) |
9 | simpr 489 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < 𝐵) | |
10 | 5, 6, 8, 9 | mulgt0d 10818 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
11 | 10 | ex 417 | . . 3 ⊢ (𝜑 → (0 < 𝐵 → 0 < (𝐴 · 𝐵))) |
12 | remul01 39872 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0) | |
13 | 3, 12 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 · 0) = 0) |
14 | oveq2 7151 | . . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0)) | |
15 | 14 | eqeq1d 2761 | . . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 · 0) = 0)) |
16 | 13, 15 | syl5ibrcom 250 | . . 3 ⊢ (𝜑 → (𝐵 = 0 → (𝐴 · 𝐵) = 0)) |
17 | 2, 4, 11, 16 | mulgt0con1dlem 39909 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 → 𝐵 < 0)) |
18 | 1, 17 | mpd 15 | 1 ⊢ (𝜑 → 𝐵 < 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 (class class class)co 7143 ℝcr 10559 0cc0 10560 · cmul 10565 < 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-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 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-reu 3075 df-rmo 3076 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-po 5436 df-so 5437 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-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 df-2 11722 df-3 11723 df-resub 39831 |
This theorem is referenced by: mulgt0b2d 39912 |
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