Theorem add20 8802

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.)

Assertion

Ref	Expression
add20	⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))

Proof of Theorem add20

Step	Hyp	Ref	Expression
1		simpllr 540	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐴)
2		simplrl 541	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ∈ ℝ)
3		simplll 539	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℝ)
4		addge02 8801	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵)))
5	2, 3, 4	syl2anc 415	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵)))
6	1, 5	mpbid 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ (𝐴 + 𝐵))
7		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = 0)
8	6, 7	breqtrd 4156	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 ≤ 0)
9		simplrr 542	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ≤ 𝐵)
10		0red 8327	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 0 ∈ ℝ)
11	2, 10	letri3d 8441	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐵 = 0 ↔ (𝐵 ≤ 0 ∧ 0 ≤ 𝐵)))
12	8, 9, 11	mpbir2and 957	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐵 = 0)
13	12	oveq2d 6101	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 𝐵) = (𝐴 + 0))
14	3	recnd 8354	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 ∈ ℂ)
15	14	addridd 8475	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 + 0) = 𝐴)
16	13, 7, 15	3eqtr3rd 2280	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → 𝐴 = 0)
17	16, 12	jca 306	. . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) ∧ (𝐴 + 𝐵) = 0) → (𝐴 = 0 ∧ 𝐵 = 0))
18	17	ex 115	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
19		oveq12 6094	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = (0 + 0))
20		00id 8467	. . 3 ⊢ (0 + 0) = 0
21	19, 20	eqtrdi 2287	. 2 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = 0)
22	18, 21	impbid1 142	1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   class class class wbr 4130  (class class class)co 6085  ℝcr 8178  0cc0 8179   + caddc 8182   ≤ cle 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-iota 5337  df-fv 5385  df-ov 6088  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366

This theorem is used by:  add20i  8820  xnn0xadd0  10271  sumsqeq0  11057  ccat0  11366  4sqlem15  13186  4sqlem16  13187  2sqlem7  16252  vtxd0nedgbfi  16552