Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > prodge0rd | Structured version Visualization version GIF version |
Description: Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Revised by AV, 9-Jul-2022.) |
Ref | Expression |
---|---|
prodge0rd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
prodge0rd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
prodge0rd.3 | ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) |
Ref | Expression |
---|---|
prodge0rd | ⊢ (𝜑 → 0 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10638 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | prodge0rd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | prodge0rd.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
4 | 3 | rpred 12425 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4, 2 | remulcld 10665 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
6 | prodge0rd.3 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | |
7 | 1, 5, 6 | lensymd 10785 | . . 3 ⊢ (𝜑 → ¬ (𝐴 · 𝐵) < 0) |
8 | 4 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐴 ∈ ℝ) |
9 | 2 | renegcld 11061 | . . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
10 | 9 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → -𝐵 ∈ ℝ) |
11 | 3 | rpgt0d 12428 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴) |
12 | 11 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < 𝐴) |
13 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < -𝐵) | |
14 | 8, 10, 12, 13 | mulgt0d 10789 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < (𝐴 · -𝐵)) |
15 | 4 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
16 | 15 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐴 ∈ ℂ) |
17 | 2 | recnd 10663 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | 17 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐵 ∈ ℂ) |
19 | 16, 18 | mulneg2d 11088 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝐵) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
20 | 14, 19 | breqtrd 5084 | . . . . 5 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < -(𝐴 · 𝐵)) |
21 | 20 | ex 415 | . . . 4 ⊢ (𝜑 → (0 < -𝐵 → 0 < -(𝐴 · 𝐵))) |
22 | 2 | lt0neg1d 11203 | . . . 4 ⊢ (𝜑 → (𝐵 < 0 ↔ 0 < -𝐵)) |
23 | 5 | lt0neg1d 11203 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 ↔ 0 < -(𝐴 · 𝐵))) |
24 | 21, 22, 23 | 3imtr4d 296 | . . 3 ⊢ (𝜑 → (𝐵 < 0 → (𝐴 · 𝐵) < 0)) |
25 | 7, 24 | mtod 200 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 0) |
26 | 1, 2, 25 | nltled 10784 | 1 ⊢ (𝜑 → 0 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 · cmul 10536 < clt 10669 ≤ cle 10670 -cneg 10865 ℝ+crp 12383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-rp 12384 |
This theorem is referenced by: prodge0ld 12491 oexpneg 15688 evennn02n 15693 nvge0 28444 oexpnegALTV 43836 |
Copyright terms: Public domain | W3C validator |