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Mirrors > Home > MPE Home > Th. List > 11prm | Structured version Visualization version GIF version |
Description: 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
11prm | ⊢ ;11 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 12426 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 1nn 12161 | . . 3 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | decnncl 12635 | . 2 ⊢ ;11 ∈ ℕ |
4 | 1lt10 12754 | . . 3 ⊢ 1 < ;10 | |
5 | 2, 1, 1, 4 | declti 12653 | . 2 ⊢ 1 < ;11 |
6 | 0nn0 12425 | . . 3 ⊢ 0 ∈ ℕ0 | |
7 | 2cn 12225 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 7 | mul02i 11341 | . . 3 ⊢ (0 · 2) = 0 |
9 | 1e0p1 12657 | . . 3 ⊢ 1 = (0 + 1) | |
10 | 1, 6, 8, 9 | dec2dvds 16932 | . 2 ⊢ ¬ 2 ∥ ;11 |
11 | 3nn 12229 | . . 3 ⊢ 3 ∈ ℕ | |
12 | 3nn0 12428 | . . 3 ⊢ 3 ∈ ℕ0 | |
13 | 2nn 12223 | . . 3 ⊢ 2 ∈ ℕ | |
14 | 3t3e9 12317 | . . . . 5 ⊢ (3 · 3) = 9 | |
15 | 14 | oveq1i 7364 | . . . 4 ⊢ ((3 · 3) + 2) = (9 + 2) |
16 | 9p2e11 12702 | . . . 4 ⊢ (9 + 2) = ;11 | |
17 | 15, 16 | eqtri 2764 | . . 3 ⊢ ((3 · 3) + 2) = ;11 |
18 | 2lt3 12322 | . . 3 ⊢ 2 < 3 | |
19 | 11, 12, 13, 17, 18 | ndvdsi 16291 | . 2 ⊢ ¬ 3 ∥ ;11 |
20 | 2nn0 12427 | . . 3 ⊢ 2 ∈ ℕ0 | |
21 | 5nn0 12430 | . . 3 ⊢ 5 ∈ ℕ0 | |
22 | 1lt2 12321 | . . 3 ⊢ 1 < 2 | |
23 | 1, 20, 1, 21, 4, 22 | decltc 12644 | . 2 ⊢ ;11 < ;25 |
24 | 3, 5, 10, 19, 23 | prmlem1 16977 | 1 ⊢ ;11 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7354 0cc0 11048 1c1 11049 + caddc 11051 · cmul 11053 2c2 12205 3c3 12206 5c5 12208 9c9 12212 ;cdc 12615 ℙcprime 16544 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-sup 9375 df-inf 9376 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-rp 12913 df-fz 13422 df-seq 13904 df-exp 13965 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-dvds 16134 df-prm 16545 |
This theorem is referenced by: 60gcd7e1 40451 |
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