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Mirrors > Home > ILE Home > Th. List > uz3m2nn | GIF version |
Description: An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
Ref | Expression |
---|---|
uz3m2nn | ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 9480 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁)) | |
2 | 2lt3 9035 | . . . . . 6 ⊢ 2 < 3 | |
3 | 2re 8935 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 9 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
5 | 3re 8939 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
6 | 5 | a1i 9 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 3 ∈ ℝ) |
7 | zre 9203 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
8 | ltletr 7996 | . . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) | |
9 | 4, 6, 7, 8 | syl3anc 1233 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) |
10 | 2, 9 | mpani 428 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (3 ≤ 𝑁 → 2 < 𝑁)) |
11 | 10 | imp 123 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
12 | 11 | 3adant1 1010 | . . 3 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
13 | 1, 12 | sylbi 120 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁) |
14 | 2nn 9026 | . . 3 ⊢ 2 ∈ ℕ | |
15 | eluzge3nn 9518 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
16 | nnsub 8904 | . . 3 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) | |
17 | 14, 15, 16 | sylancr 412 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) |
18 | 13, 17 | mpbid 146 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 (class class class)co 5850 ℝcr 7760 < clt 7941 ≤ cle 7942 − cmin 8077 ℕcn 8865 2c2 8916 3c3 8917 ℤcz 9199 ℤ≥cuz 9474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-2 8924 df-3 8925 df-z 9200 df-uz 9475 |
This theorem is referenced by: (None) |
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