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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 11597 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 9nn 11416 | . . 3 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | decnncl 11803 | . 2 ⊢ ;19 ∈ ℕ |
4 | 1nn 11326 | . . 3 ⊢ 1 ∈ ℕ | |
5 | 9nn0 11605 | . . 3 ⊢ 9 ∈ ℕ0 | |
6 | 1lt10 11923 | . . 3 ⊢ 1 < ;10 | |
7 | 4, 5, 1, 6 | declti 11821 | . 2 ⊢ 1 < ;19 |
8 | 4nn0 11600 | . . 3 ⊢ 4 ∈ ℕ0 | |
9 | 4t2e8 11487 | . . 3 ⊢ (4 · 2) = 8 | |
10 | df-9 11382 | . . 3 ⊢ 9 = (8 + 1) | |
11 | 1, 8, 9, 10 | dec2dvds 16099 | . 2 ⊢ ¬ 2 ∥ ;19 |
12 | 3nn 11391 | . . 3 ⊢ 3 ∈ ℕ | |
13 | 6nn0 11602 | . . 3 ⊢ 6 ∈ ℕ0 | |
14 | 8nn0 11604 | . . . 4 ⊢ 8 ∈ ℕ0 | |
15 | 8p1e9 11469 | . . . 4 ⊢ (8 + 1) = 9 | |
16 | 6cn 11406 | . . . . 5 ⊢ 6 ∈ ℂ | |
17 | 3cn 11393 | . . . . 5 ⊢ 3 ∈ ℂ | |
18 | 6t3e18 11889 | . . . . 5 ⊢ (6 · 3) = ;18 | |
19 | 16, 17, 18 | mulcomli 10339 | . . . 4 ⊢ (3 · 6) = ;18 |
20 | 1, 14, 15, 19 | decsuc 11814 | . . 3 ⊢ ((3 · 6) + 1) = ;19 |
21 | 1lt3 11492 | . . 3 ⊢ 1 < 3 | |
22 | 12, 13, 4, 20, 21 | ndvdsi 15470 | . 2 ⊢ ¬ 3 ∥ ;19 |
23 | 2nn0 11598 | . . 3 ⊢ 2 ∈ ℕ0 | |
24 | 5nn0 11601 | . . 3 ⊢ 5 ∈ ℕ0 | |
25 | 9lt10 11915 | . . 3 ⊢ 9 < ;10 | |
26 | 1lt2 11490 | . . 3 ⊢ 1 < 2 | |
27 | 1, 23, 5, 24, 25, 26 | decltc 11812 | . 2 ⊢ ;19 < ;25 |
28 | 3, 7, 11, 22, 27 | prmlem1 16141 | 1 ⊢ ;19 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 (class class class)co 6879 1c1 10226 · cmul 10230 2c2 11367 3c3 11368 4c4 11369 5c5 11370 6c6 11371 8c8 11373 9c9 11374 ;cdc 11782 ℙcprime 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-sup 8591 df-inf 8592 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-rp 12074 df-fz 12580 df-seq 13055 df-exp 13114 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-dvds 15319 df-prm 15719 |
This theorem is referenced by: 2503lem3 16172 |
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