![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > prmlem1 | Structured version Visualization version GIF version |
Description: A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
prmlem1.n | ⊢ 𝑁 ∈ ℕ |
prmlem1.gt | ⊢ 1 < 𝑁 |
prmlem1.2 | ⊢ ¬ 2 ∥ 𝑁 |
prmlem1.3 | ⊢ ¬ 3 ∥ 𝑁 |
prmlem1.lt | ⊢ 𝑁 < ;25 |
Ref | Expression |
---|---|
prmlem1 | ⊢ 𝑁 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmlem1.n | . 2 ⊢ 𝑁 ∈ ℕ | |
2 | prmlem1.gt | . 2 ⊢ 1 < 𝑁 | |
3 | prmlem1.2 | . 2 ⊢ ¬ 2 ∥ 𝑁 | |
4 | prmlem1.3 | . 2 ⊢ ¬ 3 ∥ 𝑁 | |
5 | eluzelre 12242 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 𝑥 ∈ ℝ) | |
6 | 5 | resqcld 13607 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑥↑2) ∈ ℝ) |
7 | eluzle 12244 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 5 ≤ 𝑥) | |
8 | 5re 11712 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
9 | 5nn0 11905 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
10 | 9 | nn0ge0i 11912 | . . . . . . . . 9 ⊢ 0 ≤ 5 |
11 | le2sq2 13496 | . . . . . . . . 9 ⊢ (((5 ∈ ℝ ∧ 0 ≤ 5) ∧ (𝑥 ∈ ℝ ∧ 5 ≤ 𝑥)) → (5↑2) ≤ (𝑥↑2)) | |
12 | 8, 10, 11 | mpanl12 701 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 5 ≤ 𝑥) → (5↑2) ≤ (𝑥↑2)) |
13 | 5, 7, 12 | syl2anc 587 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (5↑2) ≤ (𝑥↑2)) |
14 | 1 | nnrei 11634 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
15 | 8 | resqcli 13545 | . . . . . . . 8 ⊢ (5↑2) ∈ ℝ |
16 | prmlem1.lt | . . . . . . . . . 10 ⊢ 𝑁 < ;25 | |
17 | 5cn 11713 | . . . . . . . . . . . 12 ⊢ 5 ∈ ℂ | |
18 | 17 | sqvali 13539 | . . . . . . . . . . 11 ⊢ (5↑2) = (5 · 5) |
19 | 5t5e25 12189 | . . . . . . . . . . 11 ⊢ (5 · 5) = ;25 | |
20 | 18, 19 | eqtri 2821 | . . . . . . . . . 10 ⊢ (5↑2) = ;25 |
21 | 16, 20 | breqtrri 5057 | . . . . . . . . 9 ⊢ 𝑁 < (5↑2) |
22 | ltletr 10721 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (5↑2) ∧ (5↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2))) | |
23 | 21, 22 | mpani 695 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2))) |
24 | 14, 15, 23 | mp3an12 1448 | . . . . . . 7 ⊢ ((𝑥↑2) ∈ ℝ → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2))) |
25 | 6, 13, 24 | sylc 65 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘5) → 𝑁 < (𝑥↑2)) |
26 | ltnle 10709 | . . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) | |
27 | 14, 6, 26 | sylancr 590 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) |
28 | 25, 27 | mpbid 235 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘5) → ¬ (𝑥↑2) ≤ 𝑁) |
29 | 28 | pm2.21d 121 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
30 | 29 | adantld 494 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
31 | 30 | adantl 485 | . 2 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
32 | 1, 2, 3, 4, 31 | prmlem1a 16432 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 · cmul 10531 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 3c3 11681 5c5 11683 ;cdc 12086 ℤ≥cuz 12231 ↑cexp 13425 ∥ cdvds 15599 ℙcprime 16005 |
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 ax-pre-sup 10604 |
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-rmo 3114 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-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 |
This theorem is referenced by: 5prm 16434 7prm 16436 11prm 16440 13prm 16441 17prm 16442 19prm 16443 23prm 16444 |
Copyright terms: Public domain | W3C validator |