![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > add20 | Structured version Visualization version GIF version |
Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
add20 | ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 773 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐴) | |
2 | simplrl 774 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ∈ ℝ) | |
3 | simplll 772 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℝ) | |
4 | addge02 11722 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) | |
5 | 2, 3, 4 | syl2anc 583 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) |
6 | 1, 5 | mpbid 231 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ (𝐴 + 𝐵)) |
7 | simpr 484 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = 0) | |
8 | 6, 7 | breqtrd 5164 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ 0) |
9 | simplrr 775 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐵) | |
10 | 0red 11214 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ∈ ℝ) | |
11 | 2, 10 | letri3d 11353 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐵 = 0 ↔ (𝐵 ≤ 0 ∧ 0 ≤ 𝐵))) |
12 | 8, 9, 11 | mpbir2and 710 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 = 0) |
13 | 12 | oveq2d 7417 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = (𝐴 + 0)) |
14 | 3 | recnd 11239 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℂ) |
15 | 14 | addridd 11411 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 0) = 𝐴) |
16 | 13, 7, 15 | 3eqtr3rd 2773 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 = 0) |
17 | 16, 12 | jca 511 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 = 0 ∧ 𝐵 = 0)) |
18 | 17 | ex 412 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))) |
19 | oveq12 7410 | . . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = (0 + 0)) | |
20 | 00id 11386 | . . 3 ⊢ (0 + 0) = 0 | |
21 | 19, 20 | eqtrdi 2780 | . 2 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = 0) |
22 | 18, 21 | impbid1 224 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℝcr 11105 0cc0 11106 + caddc 11109 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: add20i 11754 xnn0xadd0 13223 sumsqeq0 14140 ccat0 14523 4sqlem15 16891 4sqlem16 16892 ang180lem2 26658 mumullem2 27028 2sqlem7 27273 poimirlem23 37001 |
Copyright terms: Public domain | W3C validator |