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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 11762 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | 3nn 11564 | . . 3 ⊢ 3 ∈ ℕ | |
3 | 1, 2 | decnncl 11967 | . 2 ⊢ ;23 ∈ ℕ |
4 | 2nn 11558 | . . 3 ⊢ 2 ∈ ℕ | |
5 | 3nn0 11763 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 1nn0 11761 | . . 3 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 12087 | . . 3 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 11985 | . 2 ⊢ 1 < ;23 |
9 | 4 | nncni 11496 | . . . 4 ⊢ 2 ∈ ℂ |
10 | 9 | mulid2i 10492 | . . 3 ⊢ (1 · 2) = 2 |
11 | df-3 11549 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 1, 6, 10, 11 | dec2dvds 16228 | . 2 ⊢ ¬ 2 ∥ ;23 |
13 | 7nn0 11767 | . . 3 ⊢ 7 ∈ ℕ0 | |
14 | 7cn 11579 | . . . . 5 ⊢ 7 ∈ ℂ | |
15 | 2 | nncni 11496 | . . . . 5 ⊢ 3 ∈ ℂ |
16 | 7t3e21 12058 | . . . . 5 ⊢ (7 · 3) = ;21 | |
17 | 14, 15, 16 | mulcomli 10496 | . . . 4 ⊢ (3 · 7) = ;21 |
18 | 1p2e3 11628 | . . . 4 ⊢ (1 + 2) = 3 | |
19 | 1, 6, 1, 17, 18 | decaddi 12007 | . . 3 ⊢ ((3 · 7) + 2) = ;23 |
20 | 2lt3 11657 | . . 3 ⊢ 2 < 3 | |
21 | 2, 13, 4, 19, 20 | ndvdsi 15596 | . 2 ⊢ ¬ 3 ∥ ;23 |
22 | 5nn 11571 | . . 3 ⊢ 5 ∈ ℕ | |
23 | 3lt5 11663 | . . 3 ⊢ 3 < 5 | |
24 | 1, 5, 22, 23 | declt 11975 | . 2 ⊢ ;23 < ;25 |
25 | 3, 8, 12, 21, 24 | prmlem1 16270 | 1 ⊢ ;23 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2081 (class class class)co 7016 1c1 10384 · cmul 10388 2c2 11540 3c3 11541 5c5 11543 7c7 11545 ;cdc 11947 ℙcprime 15844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fz 12743 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-dvds 15441 df-prm 15845 |
This theorem is referenced by: bpos1 25541 |
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