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Mirrors > Home > MPE Home > Th. List > 19prm | Structured version Visualization version GIF version |
Description: 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
19prm | ⊢ ;19 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 12533 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 9nn 12355 | . . 3 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | decnncl 12742 | . 2 ⊢ ;19 ∈ ℕ |
4 | 1nn 12268 | . . 3 ⊢ 1 ∈ ℕ | |
5 | 9nn0 12541 | . . 3 ⊢ 9 ∈ ℕ0 | |
6 | 1lt10 12861 | . . 3 ⊢ 1 < ;10 | |
7 | 4, 5, 1, 6 | declti 12760 | . 2 ⊢ 1 < ;19 |
8 | 4nn0 12536 | . . 3 ⊢ 4 ∈ ℕ0 | |
9 | 4t2e8 12425 | . . 3 ⊢ (4 · 2) = 8 | |
10 | df-9 12327 | . . 3 ⊢ 9 = (8 + 1) | |
11 | 1, 8, 9, 10 | dec2dvds 17059 | . 2 ⊢ ¬ 2 ∥ ;19 |
12 | 3nn 12336 | . . 3 ⊢ 3 ∈ ℕ | |
13 | 6nn0 12538 | . . 3 ⊢ 6 ∈ ℕ0 | |
14 | 8nn0 12540 | . . . 4 ⊢ 8 ∈ ℕ0 | |
15 | 8p1e9 12407 | . . . 4 ⊢ (8 + 1) = 9 | |
16 | 6cn 12348 | . . . . 5 ⊢ 6 ∈ ℂ | |
17 | 3cn 12338 | . . . . 5 ⊢ 3 ∈ ℂ | |
18 | 6t3e18 12827 | . . . . 5 ⊢ (6 · 3) = ;18 | |
19 | 16, 17, 18 | mulcomli 11263 | . . . 4 ⊢ (3 · 6) = ;18 |
20 | 1, 14, 15, 19 | decsuc 12753 | . . 3 ⊢ ((3 · 6) + 1) = ;19 |
21 | 1lt3 12430 | . . 3 ⊢ 1 < 3 | |
22 | 12, 13, 4, 20, 21 | ndvdsi 16408 | . 2 ⊢ ¬ 3 ∥ ;19 |
23 | 2nn0 12534 | . . 3 ⊢ 2 ∈ ℕ0 | |
24 | 5nn0 12537 | . . 3 ⊢ 5 ∈ ℕ0 | |
25 | 9lt10 12853 | . . 3 ⊢ 9 < ;10 | |
26 | 1lt2 12428 | . . 3 ⊢ 1 < 2 | |
27 | 1, 23, 5, 24, 25, 26 | decltc 12751 | . 2 ⊢ ;19 < ;25 |
28 | 3, 7, 11, 22, 27 | prmlem1 17104 | 1 ⊢ ;19 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 (class class class)co 7415 1c1 11149 · cmul 11153 2c2 12312 3c3 12313 4c4 12314 5c5 12315 6c6 12316 8c8 12318 9c9 12319 ;cdc 12722 ℙcprime 16666 |
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 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
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 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-sup 9477 df-inf 9478 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12604 df-dec 12723 df-uz 12868 df-rp 13022 df-fz 13532 df-seq 14015 df-exp 14075 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-dvds 16251 df-prm 16667 |
This theorem is referenced by: 2503lem3 17135 |
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