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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 12541 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | 3nn 12343 | . . 3 ⊢ 3 ∈ ℕ | |
3 | 1, 2 | decnncl 12749 | . 2 ⊢ ;23 ∈ ℕ |
4 | 2nn 12337 | . . 3 ⊢ 2 ∈ ℕ | |
5 | 3nn0 12542 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 1nn0 12540 | . . 3 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 12868 | . . 3 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 12767 | . 2 ⊢ 1 < ;23 |
9 | 4 | nncni 12274 | . . . 4 ⊢ 2 ∈ ℂ |
10 | 9 | mullidi 11269 | . . 3 ⊢ (1 · 2) = 2 |
11 | df-3 12328 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 1, 6, 10, 11 | dec2dvds 17065 | . 2 ⊢ ¬ 2 ∥ ;23 |
13 | 7nn0 12546 | . . 3 ⊢ 7 ∈ ℕ0 | |
14 | 7cn 12358 | . . . . 5 ⊢ 7 ∈ ℂ | |
15 | 2 | nncni 12274 | . . . . 5 ⊢ 3 ∈ ℂ |
16 | 7t3e21 12839 | . . . . 5 ⊢ (7 · 3) = ;21 | |
17 | 14, 15, 16 | mulcomli 11273 | . . . 4 ⊢ (3 · 7) = ;21 |
18 | 1p2e3 12407 | . . . 4 ⊢ (1 + 2) = 3 | |
19 | 1, 6, 1, 17, 18 | decaddi 12789 | . . 3 ⊢ ((3 · 7) + 2) = ;23 |
20 | 2lt3 12436 | . . 3 ⊢ 2 < 3 | |
21 | 2, 13, 4, 19, 20 | ndvdsi 16414 | . 2 ⊢ ¬ 3 ∥ ;23 |
22 | 5nn 12350 | . . 3 ⊢ 5 ∈ ℕ | |
23 | 3lt5 12442 | . . 3 ⊢ 3 < 5 | |
24 | 1, 5, 22, 23 | declt 12757 | . 2 ⊢ ;23 < ;25 |
25 | 3, 8, 12, 21, 24 | prmlem1 17110 | 1 ⊢ ;23 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 (class class class)co 7424 1c1 11159 · cmul 11163 2c2 12319 3c3 12320 5c5 12322 7c7 12324 ;cdc 12729 ℙcprime 16672 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
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 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-prm 16673 |
This theorem is referenced by: bpos1 27312 |
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