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Mirrors > Home > MPE Home > Th. List > uz3m2nn | Structured version Visualization version GIF version |
Description: An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12311. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
Ref | Expression |
---|---|
uz3m2nn | ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 12237 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁)) | |
2 | 2lt3 11797 | . . . . . 6 ⊢ 2 < 3 | |
3 | 2re 11699 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
4 | 3re 11705 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
5 | zre 11973 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | ltletr 10721 | . . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) | |
7 | 3, 4, 5, 6 | mp3an12i 1462 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) |
8 | 2, 7 | mpani 695 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (3 ≤ 𝑁 → 2 < 𝑁)) |
9 | 8 | imp 410 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
10 | 9 | 3adant1 1127 | . . 3 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
11 | 1, 10 | sylbi 220 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁) |
12 | 2nn 11698 | . . 3 ⊢ 2 ∈ ℕ | |
13 | eluzge3nn 12278 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
14 | nnsub 11669 | . . 3 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) | |
15 | 12, 13, 14 | sylancr 590 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) |
16 | 11, 15 | mpbid 235 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 2c2 11680 3c3 11681 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-z 11970 df-uz 12232 |
This theorem is referenced by: clwwlknonex2 27894 clwwnrepclwwn 28129 numclwwlk1lem2foa 28139 numclwwlk1lem2fo 28143 numclwlk1lem2 28155 numclwwlk2 28166 numclwwlk3 28170 fltnltalem 39618 |
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