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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 775 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐴) | |
2 | simplrl 776 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ∈ ℝ) | |
3 | simplll 774 | . . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℝ) | |
4 | addge02 11140 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) | |
5 | 2, 3, 4 | syl2anc 587 | . . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) |
6 | 1, 5 | mpbid 235 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ (𝐴 + 𝐵)) |
7 | simpr 488 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = 0) | |
8 | 6, 7 | breqtrd 5056 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ 0) |
9 | simplrr 777 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐵) | |
10 | 0red 10633 | . . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ∈ ℝ) | |
11 | 2, 10 | letri3d 10771 | . . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐵 = 0 ↔ (𝐵 ≤ 0 ∧ 0 ≤ 𝐵))) |
12 | 8, 9, 11 | mpbir2and 712 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 = 0) |
13 | 12 | oveq2d 7151 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = (𝐴 + 0)) |
14 | 3 | recnd 10658 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℂ) |
15 | 14 | addid1d 10829 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 0) = 𝐴) |
16 | 13, 7, 15 | 3eqtr3rd 2842 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 = 0) |
17 | 16, 12 | jca 515 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 = 0 ∧ 𝐵 = 0)) |
18 | 17 | ex 416 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))) |
19 | oveq12 7144 | . . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = (0 + 0)) | |
20 | 00id 10804 | . . 3 ⊢ (0 + 0) = 0 | |
21 | 19, 20 | eqtrdi 2849 | . 2 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = 0) |
22 | 18, 21 | impbid1 228 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: add20i 11172 xnn0xadd0 12628 sumsqeq0 13538 ccat0 13920 4sqlem15 16285 4sqlem16 16286 ang180lem2 25396 mumullem2 25765 2sqlem7 26008 poimirlem23 35080 |
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