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Mirrors > Home > MPE Home > Th. List > nn0ge2m1nn | Structured version Visualization version GIF version |
Description: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
Ref | Expression |
---|---|
nn0ge2m1nn | ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ0) | |
2 | 1red 11243 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
3 | 2re 12314 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
5 | nn0re 12509 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 2, 4, 5 | 3jca 1125 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
7 | 6 | adantr 479 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
8 | simpr 483 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁) | |
9 | 1lt2 12411 | . . . . . 6 ⊢ 1 < 2 | |
10 | 8, 9 | jctil 518 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 < 2 ∧ 2 ≤ 𝑁)) |
11 | ltleletr 11335 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁)) | |
12 | 7, 10, 11 | sylc 65 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁) |
13 | elnnnn0c 12545 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | |
14 | 1, 12, 13 | sylanbrc 581 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | nn1m1nn 12261 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) |
17 | breq2 5147 | . . . . 5 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 ↔ 2 ≤ 1)) | |
18 | 1re 11242 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
19 | 18, 3 | ltnlei 11363 | . . . . . . 7 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1) |
20 | pm2.21 123 | . . . . . . 7 ⊢ (¬ 2 ≤ 1 → (2 ≤ 1 → (𝑁 − 1) ∈ ℕ)) | |
21 | 19, 20 | sylbi 216 | . . . . . 6 ⊢ (1 < 2 → (2 ≤ 1 → (𝑁 − 1) ∈ ℕ)) |
22 | 9, 21 | ax-mp 5 | . . . . 5 ⊢ (2 ≤ 1 → (𝑁 − 1) ∈ ℕ) |
23 | 17, 22 | biimtrdi 252 | . . . 4 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 → (𝑁 − 1) ∈ ℕ)) |
24 | 23 | adantld 489 | . . 3 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)) |
25 | ax-1 6 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)) | |
26 | 24, 25 | jaoi 855 | . 2 ⊢ ((𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ) → ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)) |
27 | 16, 26 | mpcom 38 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 (class class class)co 7415 ℝcr 11135 1c1 11137 < clt 11276 ≤ cle 11277 − cmin 11472 ℕcn 12240 2c2 12295 ℕ0cn0 12500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 |
This theorem is referenced by: nn0ge2m1nn0 12570 wwlksm1edg 29734 clwlkclwwlklem2fv2 29848 clwlkclwwlk 29854 pfxlsw2ccat 32716 fmtnoprmfac1 46967 logbpw2m1 47751 blenpw2m1 47763 nnolog2flm1 47774 |
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