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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 11761 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
4 | 1, 2, 3 | mpbir2an 955 | . 2 ⊢ 𝑁 ∈ (ℤ≥‘2) |
5 | breq1 4656 | . . . . . 6 ⊢ (𝑥 = 2 → (𝑥 ∥ 𝑁 ↔ 2 ∥ 𝑁)) | |
6 | 5 | notbid 308 | . . . . 5 ⊢ (𝑥 = 2 → (¬ 𝑥 ∥ 𝑁 ↔ ¬ 2 ∥ 𝑁)) |
7 | 6 | imbi2d 330 | . . . 4 ⊢ (𝑥 = 2 → (((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁) ↔ ((𝑥↑2) ≤ 𝑁 → ¬ 2 ∥ 𝑁))) |
8 | prmnn 15388 | . . . . . 6 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝑥 ∈ ℙ ∧ 𝑥 ≠ 2) → 𝑥 ∈ ℕ) |
10 | eldifsn 4317 | . . . . . 6 ⊢ (𝑥 ∈ (ℙ ∖ {2}) ↔ (𝑥 ∈ ℙ ∧ 𝑥 ≠ 2)) | |
11 | n2dvds1 15104 | . . . . . . . . 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 11160 | . . . . . . . . . . 11 ⊢ (3 + 2) = 5 | |
16 | 12, 14, 15 | prmlem0 15812 | . . . . . . . . . 10 ⊢ ((¬ 2 ∥ 3 ∧ 𝑥 ∈ (ℤ≥‘3)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
17 | 1nprm 15392 | . . . . . . . . . . 11 ⊢ ¬ 1 ∈ ℙ | |
18 | 17 | pm2.21i 116 | . . . . . . . . . 10 ⊢ (1 ∈ ℙ → ¬ 1 ∥ 𝑁) |
19 | 1p2e3 11152 | . . . . . . . . . 10 ⊢ (1 + 2) = 3 | |
20 | 16, 18, 19 | prmlem0 15812 | . . . . . . . . 9 ⊢ ((¬ 2 ∥ 1 ∧ 𝑥 ∈ (ℤ≥‘1)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
21 | 11, 20 | mpan 706 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘1) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
22 | nnuz 11723 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
23 | 21, 22 | eleq2s 2719 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥 ∥ 𝑁)) |
24 | 23 | expd 452 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ (ℙ ∖ {2}) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))) |
25 | 10, 24 | syl5bir 233 | . . . . 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 2879 | . . 3 ⊢ (𝑥 ∈ ℙ → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁)) |
30 | 29 | rgen 2922 | . 2 ⊢ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁) |
31 | isprm5 15419 | . 2 ⊢ (𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ ℙ ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥 ∥ 𝑁))) | |
32 | 4, 30, 31 | mpbir2an 955 | 1 ⊢ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∖ cdif 3571 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 1c1 9937 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 3c3 11071 5c5 11073 ℤ≥cuz 11687 ↑cexp 12860 ∥ cdvds 14983 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 |
This theorem is referenced by: prmlem1 15814 prmlem2 15827 |
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