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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 12483 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 9nn 12305 | . . 3 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | decnncl 12692 | . 2 ⊢ ;19 ∈ ℕ |
4 | 1nn 12218 | . . 3 ⊢ 1 ∈ ℕ | |
5 | 9nn0 12491 | . . 3 ⊢ 9 ∈ ℕ0 | |
6 | 1lt10 12811 | . . 3 ⊢ 1 < ;10 | |
7 | 4, 5, 1, 6 | declti 12710 | . 2 ⊢ 1 < ;19 |
8 | 4nn0 12486 | . . 3 ⊢ 4 ∈ ℕ0 | |
9 | 4t2e8 12375 | . . 3 ⊢ (4 · 2) = 8 | |
10 | df-9 12277 | . . 3 ⊢ 9 = (8 + 1) | |
11 | 1, 8, 9, 10 | dec2dvds 16991 | . 2 ⊢ ¬ 2 ∥ ;19 |
12 | 3nn 12286 | . . 3 ⊢ 3 ∈ ℕ | |
13 | 6nn0 12488 | . . 3 ⊢ 6 ∈ ℕ0 | |
14 | 8nn0 12490 | . . . 4 ⊢ 8 ∈ ℕ0 | |
15 | 8p1e9 12357 | . . . 4 ⊢ (8 + 1) = 9 | |
16 | 6cn 12298 | . . . . 5 ⊢ 6 ∈ ℂ | |
17 | 3cn 12288 | . . . . 5 ⊢ 3 ∈ ℂ | |
18 | 6t3e18 12777 | . . . . 5 ⊢ (6 · 3) = ;18 | |
19 | 16, 17, 18 | mulcomli 11218 | . . . 4 ⊢ (3 · 6) = ;18 |
20 | 1, 14, 15, 19 | decsuc 12703 | . . 3 ⊢ ((3 · 6) + 1) = ;19 |
21 | 1lt3 12380 | . . 3 ⊢ 1 < 3 | |
22 | 12, 13, 4, 20, 21 | ndvdsi 16350 | . 2 ⊢ ¬ 3 ∥ ;19 |
23 | 2nn0 12484 | . . 3 ⊢ 2 ∈ ℕ0 | |
24 | 5nn0 12487 | . . 3 ⊢ 5 ∈ ℕ0 | |
25 | 9lt10 12803 | . . 3 ⊢ 9 < ;10 | |
26 | 1lt2 12378 | . . 3 ⊢ 1 < 2 | |
27 | 1, 23, 5, 24, 25, 26 | decltc 12701 | . 2 ⊢ ;19 < ;25 |
28 | 3, 7, 11, 22, 27 | prmlem1 17036 | 1 ⊢ ;19 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 (class class class)co 7403 1c1 11106 · cmul 11110 2c2 12262 3c3 12263 4c4 12264 5c5 12265 6c6 12266 8c8 12268 9c9 12269 ;cdc 12672 ℙcprime 16603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 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 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-rp 12970 df-fz 13480 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16193 df-prm 16604 |
This theorem is referenced by: 2503lem3 17067 |
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