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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 484 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ0) | |
2 | 1red 11161 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
3 | 2re 12232 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
5 | nn0re 12427 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 2, 4, 5 | 3jca 1129 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
7 | 6 | adantr 482 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
8 | simpr 486 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁) | |
9 | 1lt2 12329 | . . . . . 6 ⊢ 1 < 2 | |
10 | 8, 9 | jctil 521 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 < 2 ∧ 2 ≤ 𝑁)) |
11 | ltleletr 11253 | . . . . 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 12463 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | |
14 | 1, 12, 13 | sylanbrc 584 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | nn1m1nn 12179 | . . 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 5110 | . . . . 5 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 ↔ 2 ≤ 1)) | |
18 | 1re 11160 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
19 | 18, 3 | ltnlei 11281 | . . . . . . 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 | syl6bi 253 | . . . 4 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 → (𝑁 − 1) ∈ ℕ)) |
24 | 23 | adantld 492 | . . 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 856 | . 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 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 1c1 11057 < clt 11194 ≤ cle 11195 − cmin 11390 ℕcn 12158 2c2 12213 ℕ0cn0 12418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 |
This theorem is referenced by: nn0ge2m1nn0 12488 wwlksm1edg 28868 clwlkclwwlklem2fv2 28982 clwlkclwwlk 28988 pfxlsw2ccat 31855 fmtnoprmfac1 45843 logbpw2m1 46739 blenpw2m1 46751 nnolog2flm1 46762 |
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