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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 12543 | . . 3 ⊢ 2 ∈ ℕ0 | |
| 2 | 3nn 12345 | . . 3 ⊢ 3 ∈ ℕ | |
| 3 | 1, 2 | decnncl 12753 | . 2 ⊢ ;23 ∈ ℕ |
| 4 | 2nn 12339 | . . 3 ⊢ 2 ∈ ℕ | |
| 5 | 3nn0 12544 | . . 3 ⊢ 3 ∈ ℕ0 | |
| 6 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 7 | 1lt10 12872 | . . 3 ⊢ 1 < ;10 | |
| 8 | 4, 5, 6, 7 | declti 12771 | . 2 ⊢ 1 < ;23 |
| 9 | 4 | nncni 12276 | . . . 4 ⊢ 2 ∈ ℂ |
| 10 | 9 | mullidi 11266 | . . 3 ⊢ (1 · 2) = 2 |
| 11 | df-3 12330 | . . 3 ⊢ 3 = (2 + 1) | |
| 12 | 1, 6, 10, 11 | dec2dvds 17101 | . 2 ⊢ ¬ 2 ∥ ;23 |
| 13 | 7nn0 12548 | . . 3 ⊢ 7 ∈ ℕ0 | |
| 14 | 7cn 12360 | . . . . 5 ⊢ 7 ∈ ℂ | |
| 15 | 2 | nncni 12276 | . . . . 5 ⊢ 3 ∈ ℂ |
| 16 | 7t3e21 12843 | . . . . 5 ⊢ (7 · 3) = ;21 | |
| 17 | 14, 15, 16 | mulcomli 11270 | . . . 4 ⊢ (3 · 7) = ;21 |
| 18 | 1p2e3 12409 | . . . 4 ⊢ (1 + 2) = 3 | |
| 19 | 1, 6, 1, 17, 18 | decaddi 12793 | . . 3 ⊢ ((3 · 7) + 2) = ;23 |
| 20 | 2lt3 12438 | . . 3 ⊢ 2 < 3 | |
| 21 | 2, 13, 4, 19, 20 | ndvdsi 16449 | . 2 ⊢ ¬ 3 ∥ ;23 |
| 22 | 5nn 12352 | . . 3 ⊢ 5 ∈ ℕ | |
| 23 | 3lt5 12444 | . . 3 ⊢ 3 < 5 | |
| 24 | 1, 5, 22, 23 | declt 12761 | . 2 ⊢ ;23 < ;25 |
| 25 | 3, 8, 12, 21, 24 | prmlem1 17145 | 1 ⊢ ;23 ∈ ℙ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 · cmul 11160 2c2 12321 3c3 12322 5c5 12324 7c7 12326 ;cdc 12733 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-inf 9483 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-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 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: bpos1 27327 |
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