Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12104 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | 3nn 11906 | . . 3 ⊢ 3 ∈ ℕ | |
3 | 1, 2 | decnncl 12310 | . 2 ⊢ ;23 ∈ ℕ |
4 | 2nn 11900 | . . 3 ⊢ 2 ∈ ℕ | |
5 | 3nn0 12105 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 1nn0 12103 | . . 3 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 12429 | . . 3 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 12328 | . 2 ⊢ 1 < ;23 |
9 | 4 | nncni 11837 | . . . 4 ⊢ 2 ∈ ℂ |
10 | 9 | mulid2i 10835 | . . 3 ⊢ (1 · 2) = 2 |
11 | df-3 11891 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 1, 6, 10, 11 | dec2dvds 16613 | . 2 ⊢ ¬ 2 ∥ ;23 |
13 | 7nn0 12109 | . . 3 ⊢ 7 ∈ ℕ0 | |
14 | 7cn 11921 | . . . . 5 ⊢ 7 ∈ ℂ | |
15 | 2 | nncni 11837 | . . . . 5 ⊢ 3 ∈ ℂ |
16 | 7t3e21 12400 | . . . . 5 ⊢ (7 · 3) = ;21 | |
17 | 14, 15, 16 | mulcomli 10839 | . . . 4 ⊢ (3 · 7) = ;21 |
18 | 1p2e3 11970 | . . . 4 ⊢ (1 + 2) = 3 | |
19 | 1, 6, 1, 17, 18 | decaddi 12350 | . . 3 ⊢ ((3 · 7) + 2) = ;23 |
20 | 2lt3 11999 | . . 3 ⊢ 2 < 3 | |
21 | 2, 13, 4, 19, 20 | ndvdsi 15970 | . 2 ⊢ ¬ 3 ∥ ;23 |
22 | 5nn 11913 | . . 3 ⊢ 5 ∈ ℕ | |
23 | 3lt5 12005 | . . 3 ⊢ 3 < 5 | |
24 | 1, 5, 22, 23 | declt 12318 | . 2 ⊢ ;23 < ;25 |
25 | 3, 8, 12, 21, 24 | prmlem1 16658 | 1 ⊢ ;23 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 (class class class)co 7210 1c1 10727 · cmul 10731 2c2 11882 3c3 11883 5c5 11885 7c7 11887 ;cdc 12290 ℙcprime 16225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-rp 12584 df-fz 13093 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-dvds 15813 df-prm 16226 |
This theorem is referenced by: bpos1 26161 |
Copyright terms: Public domain | W3C validator |