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Mirrors > Home > MPE Home > Th. List > 7prm | Structured version Visualization version GIF version |
Description: 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
7prm | ⊢ 7 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn 12065 | . 2 ⊢ 7 ∈ ℕ | |
2 | 1lt7 12164 | . 2 ⊢ 1 < 7 | |
3 | 2nn 12046 | . . 3 ⊢ 2 ∈ ℕ | |
4 | 3nn0 12251 | . . 3 ⊢ 3 ∈ ℕ0 | |
5 | 1nn 11984 | . . 3 ⊢ 1 ∈ ℕ | |
6 | 3cn 12054 | . . . . . 6 ⊢ 3 ∈ ℂ | |
7 | 2cn 12048 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3t2e6 12139 | . . . . . 6 ⊢ (3 · 2) = 6 | |
9 | 6, 7, 8 | mulcomli 10984 | . . . . 5 ⊢ (2 · 3) = 6 |
10 | 9 | oveq1i 7285 | . . . 4 ⊢ ((2 · 3) + 1) = (6 + 1) |
11 | df-7 12041 | . . . 4 ⊢ 7 = (6 + 1) | |
12 | 10, 11 | eqtr4i 2769 | . . 3 ⊢ ((2 · 3) + 1) = 7 |
13 | 1lt2 12144 | . . 3 ⊢ 1 < 2 | |
14 | 3, 4, 5, 12, 13 | ndvdsi 16121 | . 2 ⊢ ¬ 2 ∥ 7 |
15 | 3nn 12052 | . . 3 ⊢ 3 ∈ ℕ | |
16 | 2nn0 12250 | . . 3 ⊢ 2 ∈ ℕ0 | |
17 | 8 | oveq1i 7285 | . . . 4 ⊢ ((3 · 2) + 1) = (6 + 1) |
18 | 17, 11 | eqtr4i 2769 | . . 3 ⊢ ((3 · 2) + 1) = 7 |
19 | 1lt3 12146 | . . 3 ⊢ 1 < 3 | |
20 | 15, 16, 5, 18, 19 | ndvdsi 16121 | . 2 ⊢ ¬ 3 ∥ 7 |
21 | 5nn0 12253 | . . 3 ⊢ 5 ∈ ℕ0 | |
22 | 7nn0 12255 | . . 3 ⊢ 7 ∈ ℕ0 | |
23 | 7lt10 12570 | . . 3 ⊢ 7 < ;10 | |
24 | 3, 21, 22, 23 | declti 12475 | . 2 ⊢ 7 < ;25 |
25 | 1, 2, 14, 20, 24 | prmlem1 16809 | 1 ⊢ 7 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7275 1c1 10872 + caddc 10874 · cmul 10876 2c2 12028 3c3 12029 5c5 12031 6c6 12032 7c7 12033 ℙcprime 16376 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-prm 16377 |
This theorem is referenced by: bpos1 26431 ex-mod 28813 ex-prmo 28823 60gcd7e1 40013 m3prm 45044 nnsum3primesle9 45246 |
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