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Mirrors > Home > MPE Home > Th. List > 17prm | Structured version Visualization version GIF version |
Description: 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
17prm | ⊢ ;17 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11901 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 7nn 11717 | . . 3 ⊢ 7 ∈ ℕ | |
3 | 1, 2 | decnncl 12106 | . 2 ⊢ ;17 ∈ ℕ |
4 | 1nn 11636 | . . 3 ⊢ 1 ∈ ℕ | |
5 | 7nn0 11907 | . . 3 ⊢ 7 ∈ ℕ0 | |
6 | 1lt10 12225 | . . 3 ⊢ 1 < ;10 | |
7 | 4, 5, 1, 6 | declti 12124 | . 2 ⊢ 1 < ;17 |
8 | 3nn0 11903 | . . 3 ⊢ 3 ∈ ℕ0 | |
9 | 3t2e6 11791 | . . 3 ⊢ (3 · 2) = 6 | |
10 | df-7 11693 | . . 3 ⊢ 7 = (6 + 1) | |
11 | 1, 8, 9, 10 | dec2dvds 16389 | . 2 ⊢ ¬ 2 ∥ ;17 |
12 | 3nn 11704 | . . 3 ⊢ 3 ∈ ℕ | |
13 | 5nn0 11905 | . . 3 ⊢ 5 ∈ ℕ0 | |
14 | 2nn 11698 | . . 3 ⊢ 2 ∈ ℕ | |
15 | 2nn0 11902 | . . . 4 ⊢ 2 ∈ ℕ0 | |
16 | 5cn 11713 | . . . . 5 ⊢ 5 ∈ ℂ | |
17 | 3cn 11706 | . . . . 5 ⊢ 3 ∈ ℂ | |
18 | 5t3e15 12187 | . . . . 5 ⊢ (5 · 3) = ;15 | |
19 | 16, 17, 18 | mulcomli 10639 | . . . 4 ⊢ (3 · 5) = ;15 |
20 | 5p2e7 11781 | . . . 4 ⊢ (5 + 2) = 7 | |
21 | 1, 13, 15, 19, 20 | decaddi 12146 | . . 3 ⊢ ((3 · 5) + 2) = ;17 |
22 | 2lt3 11797 | . . 3 ⊢ 2 < 3 | |
23 | 12, 13, 14, 21, 22 | ndvdsi 15753 | . 2 ⊢ ¬ 3 ∥ ;17 |
24 | 7lt10 12219 | . . 3 ⊢ 7 < ;10 | |
25 | 1lt2 11796 | . . 3 ⊢ 1 < 2 | |
26 | 1, 15, 5, 13, 24, 25 | decltc 12115 | . 2 ⊢ ;17 < ;25 |
27 | 3, 7, 11, 23, 26 | prmlem1 16433 | 1 ⊢ ;17 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 (class class class)co 7135 1c1 10527 · cmul 10531 2c2 11680 3c3 11681 5c5 11683 6c6 11684 7c7 11685 ;cdc 12086 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 |
This theorem is referenced by: fmtno2prm 44077 |
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