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Mirrors > Home > MPE Home > Th. List > dirith | Structured version Visualization version GIF version |
Description: Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to 𝑁. Theorem 9.4.1 of [Shapiro], p. 375. See http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html for an informal exposition. This is Metamath 100 proof #48. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
dirith | ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → {𝑝 ∈ ℙ ∣ 𝑁 ∥ (𝑝 − 𝐴)} ≈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1130 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ) | |
2 | 1 | nnnn0d 11557 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ0) |
3 | 2 | adantr 466 | . . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ0) |
4 | eqid 2771 | . . . . . . 7 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
5 | eqid 2771 | . . . . . . 7 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
6 | eqid 2771 | . . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁)) | |
7 | 4, 5, 6 | znzrhfo 20110 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤ/nℤ‘𝑁)):ℤ–onto→(Base‘(ℤ/nℤ‘𝑁))) |
8 | fofn 6259 | . . . . . 6 ⊢ ((ℤRHom‘(ℤ/nℤ‘𝑁)):ℤ–onto→(Base‘(ℤ/nℤ‘𝑁)) → (ℤRHom‘(ℤ/nℤ‘𝑁)) Fn ℤ) | |
9 | 3, 7, 8 | 3syl 18 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → (ℤRHom‘(ℤ/nℤ‘𝑁)) Fn ℤ) |
10 | prmz 15595 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ) | |
11 | 10 | adantl 467 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ) |
12 | fniniseg 6483 | . . . . . 6 ⊢ ((ℤRHom‘(ℤ/nℤ‘𝑁)) Fn ℤ → (𝑝 ∈ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) ↔ (𝑝 ∈ ℤ ∧ ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑝) = ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))) | |
13 | 12 | baibd 529 | . . . . 5 ⊢ (((ℤRHom‘(ℤ/nℤ‘𝑁)) Fn ℤ ∧ 𝑝 ∈ ℤ) → (𝑝 ∈ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) ↔ ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑝) = ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) |
14 | 9, 11, 13 | syl2anc 573 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) ↔ ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑝) = ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))) |
15 | simp2 1131 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℤ) | |
16 | 15 | adantr 466 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ) |
17 | 4, 6 | zndvds 20112 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑝) = ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ↔ 𝑁 ∥ (𝑝 − 𝐴))) |
18 | 3, 11, 16, 17 | syl3anc 1476 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → (((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝑝) = ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ↔ 𝑁 ∥ (𝑝 − 𝐴))) |
19 | 14, 18 | bitrd 268 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) ↔ 𝑁 ∥ (𝑝 − 𝐴))) |
20 | 19 | rabbi2dva 3970 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ℙ ∩ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)})) = {𝑝 ∈ ℙ ∣ 𝑁 ∥ (𝑝 − 𝐴)}) |
21 | eqid 2771 | . . 3 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁)) | |
22 | simp3 1132 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 gcd 𝑁) = 1) | |
23 | 4, 21, 6 | znunit 20126 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ∈ (Unit‘(ℤ/nℤ‘𝑁)) ↔ (𝐴 gcd 𝑁) = 1)) |
24 | 2, 15, 23 | syl2anc 573 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ∈ (Unit‘(ℤ/nℤ‘𝑁)) ↔ (𝐴 gcd 𝑁) = 1)) |
25 | 22, 24 | mpbird 247 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ∈ (Unit‘(ℤ/nℤ‘𝑁))) |
26 | eqid 2771 | . . 3 ⊢ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) = (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)}) | |
27 | 4, 6, 1, 21, 25, 26 | dirith2 25437 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ℙ ∩ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)})) ≈ ℕ) |
28 | 20, 27 | eqbrtrrd 4811 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → {𝑝 ∈ ℙ ∣ 𝑁 ∥ (𝑝 − 𝐴)} ≈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 {crab 3065 ∩ cin 3722 {csn 4317 class class class wbr 4787 ◡ccnv 5249 “ cima 5253 Fn wfn 6025 –onto→wfo 6028 ‘cfv 6030 (class class class)co 6795 ≈ cen 8109 1c1 10142 − cmin 10471 ℕcn 11225 ℕ0cn0 11498 ℤcz 11583 ∥ cdvds 15188 gcd cgcd 15423 ℙcprime 15591 Basecbs 16063 Unitcui 18846 ℤRHomczrh 20062 ℤ/nℤczn 20065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-disj 4756 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-rpss 7087 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-tpos 7507 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-omul 7721 df-er 7899 df-ec 7901 df-qs 7905 df-map 8014 df-pm 8015 df-ixp 8066 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-fi 8476 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-acn 8971 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-xnn0 11570 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-word 13494 df-concat 13496 df-s1 13497 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-o1 14428 df-lo1 14429 df-sum 14624 df-ef 15003 df-e 15004 df-sin 15005 df-cos 15006 df-tan 15007 df-pi 15008 df-dvds 15189 df-gcd 15424 df-prm 15592 df-numer 15649 df-denom 15650 df-phi 15677 df-pc 15748 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-qus 16376 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-mulg 17748 df-subg 17798 df-nsg 17799 df-eqg 17800 df-ghm 17865 df-gim 17908 df-ga 17929 df-cntz 17956 df-oppg 17982 df-od 18154 df-gex 18155 df-pgp 18156 df-lsm 18257 df-pj1 18258 df-cmn 18401 df-abl 18402 df-cyg 18486 df-dprd 18601 df-dpj 18602 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-rnghom 18924 df-drng 18958 df-subrg 18987 df-lmod 19074 df-lss 19142 df-lsp 19184 df-sra 19386 df-rgmod 19387 df-lidl 19388 df-rsp 19389 df-2idl 19446 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-zring 20033 df-zrh 20066 df-zn 20069 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-cmp 21410 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-0p 23656 df-limc 23849 df-dv 23850 df-ply 24163 df-idp 24164 df-coe 24165 df-dgr 24166 df-quot 24265 df-log 24523 df-cxp 24524 df-em 24939 df-cht 25043 df-vma 25044 df-chp 25045 df-ppi 25046 df-mu 25047 df-dchr 25178 |
This theorem is referenced by: (None) |
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