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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 12887 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 𝑥 ∈ ℝ) | |
6 | 5 | resqcld 14162 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑥↑2) ∈ ℝ) |
7 | eluzle 12889 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘5) → 5 ≤ 𝑥) | |
8 | 5re 12351 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
9 | 5nn0 12544 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
10 | 9 | nn0ge0i 12551 | . . . . . . . . 9 ⊢ 0 ≤ 5 |
11 | le2sq2 14172 | . . . . . . . . 9 ⊢ (((5 ∈ ℝ ∧ 0 ≤ 5) ∧ (𝑥 ∈ ℝ ∧ 5 ≤ 𝑥)) → (5↑2) ≤ (𝑥↑2)) | |
12 | 8, 10, 11 | mpanl12 702 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 5 ≤ 𝑥) → (5↑2) ≤ (𝑥↑2)) |
13 | 5, 7, 12 | syl2anc 584 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘5) → (5↑2) ≤ (𝑥↑2)) |
14 | 1 | nnrei 12273 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
15 | 8 | resqcli 14222 | . . . . . . . 8 ⊢ (5↑2) ∈ ℝ |
16 | prmlem1.lt | . . . . . . . . . 10 ⊢ 𝑁 < ;25 | |
17 | 5cn 12352 | . . . . . . . . . . . 12 ⊢ 5 ∈ ℂ | |
18 | 17 | sqvali 14216 | . . . . . . . . . . 11 ⊢ (5↑2) = (5 · 5) |
19 | 5t5e25 12834 | . . . . . . . . . . 11 ⊢ (5 · 5) = ;25 | |
20 | 18, 19 | eqtri 2763 | . . . . . . . . . 10 ⊢ (5↑2) = ;25 |
21 | 16, 20 | breqtrri 5175 | . . . . . . . . 9 ⊢ 𝑁 < (5↑2) |
22 | ltletr 11351 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (5↑2) ∧ (5↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2))) | |
23 | 21, 22 | mpani 696 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ (5↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((5↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2))) |
24 | 14, 15, 23 | mp3an12 1450 | . . . . . . 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 11338 | . . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) | |
27 | 14, 6, 26 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘5) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁)) |
28 | 25, 27 | mpbid 232 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘5) → ¬ (𝑥↑2) ≤ 𝑁) |
29 | 28 | pm2.21d 121 | . . . 4 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
30 | 29 | adantld 490 | . . 3 ⊢ (𝑥 ∈ (ℤ≥‘5) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
31 | 30 | adantl 481 | . 2 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
32 | 1, 2, 3, 4, 31 | prmlem1a 17141 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 ℕcn 12264 2c2 12319 3c3 12320 5c5 12322 ;cdc 12731 ℤ≥cuz 12876 ↑cexp 14099 ∥ cdvds 16287 ℙcprime 16705 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-prm 16706 |
This theorem is referenced by: 5prm 17143 7prm 17145 11prm 17149 13prm 17150 17prm 17151 19prm 17152 23prm 17153 |
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