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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 110 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → 𝑍 ∈ ℤ) | |
2 | nn0z 9287 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
3 | zaddcl 9307 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ ℤ) | |
4 | 2, 3 | sylan 283 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ ℤ) |
5 | zre 9271 | . . . 4 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℝ) | |
6 | nn0addge2 9237 | . . . 4 ⊢ ((𝑍 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝑍 ≤ (𝑁 + 𝑍)) | |
7 | 5, 6 | sylan 283 | . . 3 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑍 ≤ (𝑁 + 𝑍)) |
8 | 7 | ancoms 268 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → 𝑍 ≤ (𝑁 + 𝑍)) |
9 | eluz2 9548 | . 2 ⊢ ((𝑁 + 𝑍) ∈ (ℤ≥‘𝑍) ↔ (𝑍 ∈ ℤ ∧ (𝑁 + 𝑍) ∈ ℤ ∧ 𝑍 ≤ (𝑁 + 𝑍))) | |
10 | 1, 4, 8, 9 | syl3anbrc 1182 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2158 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 ℝcr 7824 + caddc 7828 ≤ cle 8007 ℕ0cn0 9190 ℤcz 9267 ℤ≥cuz 9542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-inn 8934 df-n0 9191 df-z 9268 df-uz 9543 |
This theorem is referenced by: (None) |
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