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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 11204 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | prodge0rd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | prodge0rd.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
4 | 3 | rpred 13003 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 4, 2 | remulcld 11231 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) |
6 | prodge0rd.3 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | |
7 | 1, 5, 6 | lensymd 11352 | . . 3 ⊢ (𝜑 → ¬ (𝐴 · 𝐵) < 0) |
8 | 4 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐴 ∈ ℝ) |
9 | 2 | renegcld 11628 | . . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
10 | 9 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → -𝐵 ∈ ℝ) |
11 | 3 | rpgt0d 13006 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴) |
12 | 11 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < 𝐴) |
13 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < -𝐵) | |
14 | 8, 10, 12, 13 | mulgt0d 11356 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < (𝐴 · -𝐵)) |
15 | 4 | recnd 11229 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
16 | 15 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐴 ∈ ℂ) |
17 | 2 | recnd 11229 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | 17 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝐵) → 𝐵 ∈ ℂ) |
19 | 16, 18 | mulneg2d 11655 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝐵) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) |
20 | 14, 19 | breqtrd 5170 | . . . . 5 ⊢ ((𝜑 ∧ 0 < -𝐵) → 0 < -(𝐴 · 𝐵)) |
21 | 20 | ex 414 | . . . 4 ⊢ (𝜑 → (0 < -𝐵 → 0 < -(𝐴 · 𝐵))) |
22 | 2 | lt0neg1d 11770 | . . . 4 ⊢ (𝜑 → (𝐵 < 0 ↔ 0 < -𝐵)) |
23 | 5 | lt0neg1d 11770 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 ↔ 0 < -(𝐴 · 𝐵))) |
24 | 21, 22, 23 | 3imtr4d 294 | . . 3 ⊢ (𝜑 → (𝐵 < 0 → (𝐴 · 𝐵) < 0)) |
25 | 7, 24 | mtod 197 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 0) |
26 | 1, 2, 25 | nltled 11351 | 1 ⊢ (𝜑 → 0 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5144 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 · cmul 11102 < clt 11235 ≤ cle 11236 -cneg 11432 ℝ+crp 12961 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-rp 12962 |
This theorem is referenced by: prodge0ld 13069 oexpneg 16275 evennn02n 16280 nvge0 29891 oexpnegALTV 46218 |
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