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Mirrors > Home > ILE Home > Th. List > addge0d | GIF version |
Description: Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
addge0d.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
addge0d.4 | ⊢ (𝜑 → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
addge0d | ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | addge0d.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | addge0d.4 | . 2 ⊢ (𝜑 → 0 ≤ 𝐵) | |
5 | addge0 7927 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1175 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 class class class wbr 3845 (class class class)co 5652 ℝcr 7347 0cc0 7348 + caddc 7351 ≤ cle 7521 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-addcom 7443 ax-addass 7445 ax-i2m1 7448 ax-0id 7451 ax-rnegex 7452 ax-pre-ltwlin 7456 ax-pre-ltadd 7459 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-cnv 4446 df-iota 4980 df-fv 5023 df-ov 5655 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 |
This theorem is referenced by: flqdiv 9724 modaddmodlo 9791 ser3ge0 9948 cjmulge0 10319 abs00ap 10491 absext 10492 absrele 10512 abstri 10533 nn0oddm1d2 11183 |
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