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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 12317. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
Ref | Expression |
---|---|
uz3m2nn | ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 12243 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁)) | |
2 | 2lt3 11803 | . . . . . 6 ⊢ 2 < 3 | |
3 | 2re 11705 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
4 | 3re 11711 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
5 | zre 11979 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | ltletr 10726 | . . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) | |
7 | 3, 4, 5, 6 | mp3an12i 1461 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → ((2 < 3 ∧ 3 ≤ 𝑁) → 2 < 𝑁)) |
8 | 2, 7 | mpani 694 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (3 ≤ 𝑁 → 2 < 𝑁)) |
9 | 8 | imp 409 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
10 | 9 | 3adant1 1126 | . . 3 ⊢ ((3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁) → 2 < 𝑁) |
11 | 1, 10 | sylbi 219 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 < 𝑁) |
12 | 2nn 11704 | . . 3 ⊢ 2 ∈ ℕ | |
13 | eluzge3nn 12284 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
14 | nnsub 11675 | . . 3 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) | |
15 | 12, 13, 14 | sylancr 589 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → (2 < 𝑁 ↔ (𝑁 − 2) ∈ ℕ)) |
16 | 11, 15 | mpbid 234 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 < clt 10669 ≤ cle 10670 − cmin 10864 ℕcn 11632 2c2 11686 3c3 11687 ℤcz 11975 ℤ≥cuz 12237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-z 11976 df-uz 12238 |
This theorem is referenced by: clwwlknonex2 27882 clwwnrepclwwn 28117 numclwwlk1lem2foa 28127 numclwwlk1lem2fo 28131 numclwlk1lem2 28143 numclwwlk2 28154 numclwwlk3 28158 fltnltalem 39267 |
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