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Theorem add20 7224
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 0 ≤ A) (B 0 ≤ B)) → ((A + B) = 0 ↔ (A = 0 B = 0)))

Proof of Theorem add20
StepHypRef Expression
1 simpllr 486 . . . . . . . . 9 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → 0 ≤ A)
2 simplrl 487 . . . . . . . . . 10 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → B ℝ)
3 simplll 485 . . . . . . . . . 10 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → A ℝ)
4 addge02 7223 . . . . . . . . . 10 ((B A ℝ) → (0 ≤ AB ≤ (A + B)))
52, 3, 4syl2anc 391 . . . . . . . . 9 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (0 ≤ AB ≤ (A + B)))
61, 5mpbid 135 . . . . . . . 8 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → B ≤ (A + B))
7 simpr 103 . . . . . . . 8 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (A + B) = 0)
86, 7breqtrd 3779 . . . . . . 7 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → B ≤ 0)
9 simplrr 488 . . . . . . 7 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → 0 ≤ B)
10 0red 6786 . . . . . . . 8 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → 0 ℝ)
112, 10letri3d 6890 . . . . . . 7 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (B = 0 ↔ (B ≤ 0 0 ≤ B)))
128, 9, 11mpbir2and 850 . . . . . 6 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → B = 0)
1312oveq2d 5471 . . . . 5 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (A + B) = (A + 0))
143recnd 6811 . . . . . 6 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → A ℂ)
1514addid1d 6919 . . . . 5 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (A + 0) = A)
1613, 7, 153eqtr3rd 2078 . . . 4 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → A = 0)
1716, 12jca 290 . . 3 ((((A 0 ≤ A) (B 0 ≤ B)) (A + B) = 0) → (A = 0 B = 0))
1817ex 108 . 2 (((A 0 ≤ A) (B 0 ≤ B)) → ((A + B) = 0 → (A = 0 B = 0)))
19 oveq12 5464 . . 3 ((A = 0 B = 0) → (A + B) = (0 + 0))
20 00id 6911 . . 3 (0 + 0) = 0
2119, 20syl6eq 2085 . 2 ((A = 0 B = 0) → (A + B) = 0)
2218, 21impbid1 130 1 (((A 0 ≤ A) (B 0 ≤ B)) → ((A + B) = 0 ↔ (A = 0 B = 0)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390   class class class wbr 3755  (class class class)co 5455  cr 6670  0cc0 6671   + caddc 6674  cle 6818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6734  ax-resscn 6735  ax-1cn 6736  ax-1re 6737  ax-icn 6738  ax-addcl 6739  ax-addrcl 6740  ax-mulcl 6741  ax-addcom 6743  ax-addass 6745  ax-i2m1 6748  ax-0id 6751  ax-rnegex 6752  ax-pre-ltirr 6755  ax-pre-apti 6758  ax-pre-ltadd 6759
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-iota 4810  df-fv 4853  df-ov 5458  df-pnf 6819  df-mnf 6820  df-xr 6821  df-ltxr 6822  df-le 6823
This theorem is referenced by:  add20i  7239  sumsqeq0  8945
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