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| Mirrors > Home > MPE Home > Th. List > prmlem1a | 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 |
| prmlem1a.x | ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
| Ref | Expression |
|---|---|
| prmlem1a | ⊢ 𝑁 ∈ ℙ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prmlem1.n | . . 3 ⊢ 𝑁 ∈ ℕ | |
| 2 | prmlem1.gt | . . 3 ⊢ 1 < 𝑁 | |
| 3 | eluz2b2 12963 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | . 2 ⊢ 𝑁 ∈ (ℤ≥‘2) |
| 5 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 2 → (𝑥 ∥ 𝑁 ↔ 2 ∥ 𝑁)) | |
| 6 | 5 | notbid 318 | . . . . 5 ⊢ (𝑥 = 2 → (¬ 𝑥 ∥ 𝑁 ↔ ¬ 2 ∥ 𝑁)) |
| 7 | 6 | imbi2d 340 | . . . 4 ⊢ (𝑥 = 2 → (((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁) ↔ ((𝑥↑2) ≤ 𝑁 → ¬ 2 ∥ 𝑁))) |
| 8 | prmnn 16711 | . . . . . 6 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → 𝑥 ∈ ℕ) |
| 10 | eldifsn 4786 | . . . . . 6 ⊢ (𝑥 ∈ (ℙ ∖ {2}) ↔ (𝑥 ∈ ℙ ∧ 𝑥 ≠ 2)) | |
| 11 | n2dvds1 16405 | . . . . . . . . 9 ⊢ ¬ 2 ∥ 1 | |
| 12 | prmlem1a.x | . . . . . . . . . . 11 ⊢ ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ≥‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) | |
| 13 | prmlem1.3 | . . . . . . . . . . . 12 ⊢ ¬ 3 ∥ 𝑁 | |
| 14 | 13 | a1i 11 | . . . . . . . . . . 11 ⊢ (3 ∈ ℙ → ¬ 3 ∥ 𝑁) |
| 15 | 3p2e5 12417 | . . . . . . . . . . 11 ⊢ (3 + 2) = 5 | |
| 16 | 12, 14, 15 | prmlem0 17143 | . . . . . . . . . 10 ⊢ ((¬ 2 ∥ 3 ∧ 𝑥 ∈ (ℤ≥‘3)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
| 17 | 1nprm 16716 | . . . . . . . . . . 11 ⊢ ¬ 1 ∈ ℙ | |
| 18 | 17 | pm2.21i 119 | . . . . . . . . . 10 ⊢ (1 ∈ ℙ → ¬ 1 ∥ 𝑁) |
| 19 | 1p2e3 12409 | . . . . . . . . . 10 ⊢ (1 + 2) = 3 | |
| 20 | 16, 18, 19 | prmlem0 17143 | . . . . . . . . 9 ⊢ ((¬ 2 ∥ 1 ∧ 𝑥 ∈ (ℤ≥‘1)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
| 21 | 11, 20 | mpan 690 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘1) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
| 22 | nnuz 12921 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
| 23 | 21, 22 | eleq2s 2859 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
| 24 | 23 | expd 415 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ (ℙ ∖ {2}) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))) |
| 25 | 10, 24 | biimtrrid 243 | . . . . 5 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))) |
| 26 | 9, 25 | mpcom 38 | . . . 4 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
| 27 | prmlem1.2 | . . . . 5 ⊢ ¬ 2 ∥ 𝑁 | |
| 28 | 27 | 2a1i 12 | . . . 4 ⊢ (𝑥 ∈ ℙ → ((𝑥↑2) ≤ 𝑁 → ¬ 2 ∥ 𝑁)) |
| 29 | 7, 26, 28 | pm2.61ne 3027 | . . 3 ⊢ (𝑥 ∈ ℙ → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
| 30 | 29 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁) |
| 31 | isprm5 16744 | . 2 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))) | |
| 32 | 4, 30, 31 | mpbir2an 711 | 1 ⊢ 𝑁 ∈ ℙ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∖ cdif 3948 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 < clt 11295 ≤ cle 11296 ℕcn 12266 2c2 12321 3c3 12322 5c5 12324 ℤ≥cuz 12878 ↑cexp 14102 ∥ cdvds 16290 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 |
| This theorem is referenced by: prmlem1 17145 prmlem2 17157 |
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