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Mirrors > Home > MPE Home > Th. List > 5prm | Structured version Visualization version GIF version |
Description: 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
5prm | ⊢ 5 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn 12352 | . 2 ⊢ 5 ∈ ℕ | |
2 | 1lt5 12446 | . 2 ⊢ 1 < 5 | |
3 | 2nn 12339 | . . 3 ⊢ 2 ∈ ℕ | |
4 | 2nn0 12543 | . . 3 ⊢ 2 ∈ ℕ0 | |
5 | 1nn 12277 | . . 3 ⊢ 1 ∈ ℕ | |
6 | 2t2e4 12430 | . . . . 5 ⊢ (2 · 2) = 4 | |
7 | 6 | oveq1i 7436 | . . . 4 ⊢ ((2 · 2) + 1) = (4 + 1) |
8 | df-5 12332 | . . . 4 ⊢ 5 = (4 + 1) | |
9 | 7, 8 | eqtr4i 2757 | . . 3 ⊢ ((2 · 2) + 1) = 5 |
10 | 1lt2 12437 | . . 3 ⊢ 1 < 2 | |
11 | 3, 4, 5, 9, 10 | ndvdsi 16416 | . 2 ⊢ ¬ 2 ∥ 5 |
12 | 3nn 12345 | . . 3 ⊢ 3 ∈ ℕ | |
13 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
14 | 3t1e3 12431 | . . . . 5 ⊢ (3 · 1) = 3 | |
15 | 14 | oveq1i 7436 | . . . 4 ⊢ ((3 · 1) + 2) = (3 + 2) |
16 | 3p2e5 12417 | . . . 4 ⊢ (3 + 2) = 5 | |
17 | 15, 16 | eqtri 2754 | . . 3 ⊢ ((3 · 1) + 2) = 5 |
18 | 2lt3 12438 | . . 3 ⊢ 2 < 3 | |
19 | 12, 13, 3, 17, 18 | ndvdsi 16416 | . 2 ⊢ ¬ 3 ∥ 5 |
20 | 5nn0 12546 | . . 3 ⊢ 5 ∈ ℕ0 | |
21 | 5lt10 12866 | . . 3 ⊢ 5 < ;10 | |
22 | 3, 20, 20, 21 | declti 12769 | . 2 ⊢ 5 < ;25 |
23 | 1, 2, 11, 19, 22 | prmlem1 17112 | 1 ⊢ 5 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 (class class class)co 7426 1c1 11161 + caddc 11163 · cmul 11165 2c2 12321 3c3 12322 4c4 12323 5c5 12324 ℙcprime 16674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-inf 9488 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-rp 13031 df-fz 13541 df-seq 14024 df-exp 14084 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-dvds 16259 df-prm 16675 |
This theorem is referenced by: prmo5 17133 4001prm 17149 lt6abl 19895 bpos1 27315 12gcd5e1 41704 fmtno1prm 47149 fmtnofac1 47160 8gbe 47363 11gbo 47365 nnsum3primesle9 47384 |
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