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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 11801 | . 2 ⊢ 7 ∈ ℕ | |
2 | 1lt7 11900 | . 2 ⊢ 1 < 7 | |
3 | 2nn 11782 | . . 3 ⊢ 2 ∈ ℕ | |
4 | 3nn0 11987 | . . 3 ⊢ 3 ∈ ℕ0 | |
5 | 1nn 11720 | . . 3 ⊢ 1 ∈ ℕ | |
6 | 3cn 11790 | . . . . . 6 ⊢ 3 ∈ ℂ | |
7 | 2cn 11784 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3t2e6 11875 | . . . . . 6 ⊢ (3 · 2) = 6 | |
9 | 6, 7, 8 | mulcomli 10721 | . . . . 5 ⊢ (2 · 3) = 6 |
10 | 9 | oveq1i 7174 | . . . 4 ⊢ ((2 · 3) + 1) = (6 + 1) |
11 | df-7 11777 | . . . 4 ⊢ 7 = (6 + 1) | |
12 | 10, 11 | eqtr4i 2764 | . . 3 ⊢ ((2 · 3) + 1) = 7 |
13 | 1lt2 11880 | . . 3 ⊢ 1 < 2 | |
14 | 3, 4, 5, 12, 13 | ndvdsi 15850 | . 2 ⊢ ¬ 2 ∥ 7 |
15 | 3nn 11788 | . . 3 ⊢ 3 ∈ ℕ | |
16 | 2nn0 11986 | . . 3 ⊢ 2 ∈ ℕ0 | |
17 | 8 | oveq1i 7174 | . . . 4 ⊢ ((3 · 2) + 1) = (6 + 1) |
18 | 17, 11 | eqtr4i 2764 | . . 3 ⊢ ((3 · 2) + 1) = 7 |
19 | 1lt3 11882 | . . 3 ⊢ 1 < 3 | |
20 | 15, 16, 5, 18, 19 | ndvdsi 15850 | . 2 ⊢ ¬ 3 ∥ 7 |
21 | 5nn0 11989 | . . 3 ⊢ 5 ∈ ℕ0 | |
22 | 7nn0 11991 | . . 3 ⊢ 7 ∈ ℕ0 | |
23 | 7lt10 12305 | . . 3 ⊢ 7 < ;10 | |
24 | 3, 21, 22, 23 | declti 12210 | . 2 ⊢ 7 < ;25 |
25 | 1, 2, 14, 20, 24 | prmlem1 16537 | 1 ⊢ 7 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 (class class class)co 7164 1c1 10609 + caddc 10611 · cmul 10613 2c2 11764 3c3 11765 5c5 11767 6c6 11768 7c7 11769 ℙcprime 16105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-rp 12466 df-fz 12975 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-dvds 15693 df-prm 16106 |
This theorem is referenced by: bpos1 26011 ex-mod 28378 ex-prmo 28388 60gcd7e1 39622 m3prm 44562 nnsum3primesle9 44764 |
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