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Mirrors > Home > ILE Home > Th. List > nn0pzuz | GIF version |
Description: The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Ref | Expression |
---|---|
nn0pzuz | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → 𝑍 ∈ ℤ) | |
2 | nn0z 8978 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
3 | zaddcl 8998 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ ℤ) | |
4 | 2, 3 | sylan 279 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ ℤ) |
5 | zre 8962 | . . . 4 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℝ) | |
6 | nn0addge2 8928 | . . . 4 ⊢ ((𝑍 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝑍 ≤ (𝑁 + 𝑍)) | |
7 | 5, 6 | sylan 279 | . . 3 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑍 ≤ (𝑁 + 𝑍)) |
8 | 7 | ancoms 266 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → 𝑍 ≤ (𝑁 + 𝑍)) |
9 | eluz2 9234 | . 2 ⊢ ((𝑁 + 𝑍) ∈ (ℤ≥‘𝑍) ↔ (𝑍 ∈ ℤ ∧ (𝑁 + 𝑍) ∈ ℤ ∧ 𝑍 ≤ (𝑁 + 𝑍))) | |
10 | 1, 4, 8, 9 | syl3anbrc 1148 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1463 class class class wbr 3895 ‘cfv 5081 (class class class)co 5728 ℝcr 7546 + caddc 7550 ≤ cle 7725 ℕ0cn0 8881 ℤcz 8958 ℤ≥cuz 9228 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-inn 8631 df-n0 8882 df-z 8959 df-uz 9229 |
This theorem is referenced by: (None) |
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