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Mirrors > Home > MPE Home > Th. List > 23prm | Structured version Visualization version GIF version |
Description: 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
23prm | ⊢ ;23 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11512 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | 3nn 11389 | . . 3 ⊢ 3 ∈ ℕ | |
3 | 1, 2 | decnncl 11721 | . 2 ⊢ ;23 ∈ ℕ |
4 | 2nn 11388 | . . 3 ⊢ 2 ∈ ℕ | |
5 | 3nn0 11513 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 1nn0 11511 | . . 3 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 11883 | . . 3 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 11749 | . 2 ⊢ 1 < ;23 |
9 | 4 | nncni 11233 | . . . 4 ⊢ 2 ∈ ℂ |
10 | 9 | mulid2i 10246 | . . 3 ⊢ (1 · 2) = 2 |
11 | df-3 11283 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 1, 6, 10, 11 | dec2dvds 15975 | . 2 ⊢ ¬ 2 ∥ ;23 |
13 | 7nn0 11517 | . . 3 ⊢ 7 ∈ ℕ0 | |
14 | 7cn 11307 | . . . . 5 ⊢ 7 ∈ ℂ | |
15 | 2 | nncni 11233 | . . . . 5 ⊢ 3 ∈ ℂ |
16 | 7t3e21 11851 | . . . . 5 ⊢ (7 · 3) = ;21 | |
17 | 14, 15, 16 | mulcomli 10250 | . . . 4 ⊢ (3 · 7) = ;21 |
18 | 1p2e3 11355 | . . . 4 ⊢ (1 + 2) = 3 | |
19 | 1, 6, 1, 17, 18 | decaddi 11781 | . . 3 ⊢ ((3 · 7) + 2) = ;23 |
20 | 2lt3 11398 | . . 3 ⊢ 2 < 3 | |
21 | 2, 13, 4, 19, 20 | ndvdsi 15345 | . 2 ⊢ ¬ 3 ∥ ;23 |
22 | 5nn 11391 | . . 3 ⊢ 5 ∈ ℕ | |
23 | 3lt5 11404 | . . 3 ⊢ 3 < 5 | |
24 | 1, 5, 22, 23 | declt 11733 | . 2 ⊢ ;23 < ;25 |
25 | 3, 8, 12, 21, 24 | prmlem1 16022 | 1 ⊢ ;23 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 (class class class)co 6794 1c1 10140 · cmul 10144 2c2 11273 3c3 11274 5c5 11276 7c7 11278 ;cdc 11696 ℙcprime 15593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-pre-sup 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-2o 7715 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-sup 8505 df-inf 8506 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-4 11284 df-5 11285 df-6 11286 df-7 11287 df-8 11288 df-9 11289 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-rp 12037 df-fz 12535 df-seq 13010 df-exp 13069 df-cj 14048 df-re 14049 df-im 14050 df-sqrt 14184 df-abs 14185 df-dvds 15191 df-prm 15594 |
This theorem is referenced by: bpos1 25230 |
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