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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 1132 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ) | |
2 | 1 | nnnn0d 11958 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ0) |
3 | 2 | adantr 483 | . . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ0) |
4 | eqid 2823 | . . . . . . 7 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁) | |
5 | eqid 2823 | . . . . . . 7 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁)) | |
6 | eqid 2823 | . . . . . . 7 ⊢ (ℤRHom‘(ℤ/nℤ‘𝑁)) = (ℤRHom‘(ℤ/nℤ‘𝑁)) | |
7 | 4, 5, 6 | znzrhfo 20696 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘(ℤ/nℤ‘𝑁)):ℤ–onto→(Base‘(ℤ/nℤ‘𝑁))) |
8 | fofn 6594 | . . . . . 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 16021 | . . . . . 6 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ) | |
11 | 10 | adantl 484 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ) |
12 | fniniseg 6832 | . . . . . 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 542 | . . . . 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 586 | . . . 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 1133 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℤ) | |
16 | 15 | adantr 483 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ) |
17 | 4, 6 | zndvds 20698 | . . . . 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 1367 | . . . 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 281 | . . 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 4196 | . 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 2823 | . . 3 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁)) | |
22 | simp3 1134 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 gcd 𝑁) = 1) | |
23 | 4, 21, 6 | znunit 20712 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ∈ (Unit‘(ℤ/nℤ‘𝑁)) ↔ (𝐴 gcd 𝑁) = 1)) |
24 | 2, 15, 23 | syl2anc 586 | . . . 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 259 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴) ∈ (Unit‘(ℤ/nℤ‘𝑁))) |
26 | eqid 2823 | . . 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 26106 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ℙ ∩ (◡(ℤRHom‘(ℤ/nℤ‘𝑁)) “ {((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)})) ≈ ℕ) |
28 | 20, 27 | eqbrtrrd 5092 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {crab 3144 ∩ cin 3937 {csn 4569 class class class wbr 5068 ◡ccnv 5556 “ cima 5560 Fn wfn 6352 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 ≈ cen 8508 1c1 10540 − cmin 10872 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ∥ cdvds 15609 gcd cgcd 15845 ℙcprime 16017 Basecbs 16485 Unitcui 19391 ℤRHomczrh 20649 ℤ/nℤczn 20652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-rpss 7451 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-o1 14849 df-lo1 14850 df-sum 15045 df-ef 15423 df-e 15424 df-sin 15425 df-cos 15426 df-tan 15427 df-pi 15428 df-dvds 15610 df-gcd 15846 df-prm 16018 df-numer 16077 df-denom 16078 df-phi 16105 df-pc 16176 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-qus 16784 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-gim 18401 df-ga 18422 df-cntz 18449 df-oppg 18476 df-od 18658 df-gex 18659 df-pgp 18660 df-lsm 18763 df-pj1 18764 df-cmn 18910 df-abl 18911 df-cyg 18999 df-dprd 19119 df-dpj 19120 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-0p 24273 df-limc 24466 df-dv 24467 df-ply 24780 df-idp 24781 df-coe 24782 df-dgr 24783 df-quot 24882 df-ulm 24967 df-log 25142 df-cxp 25143 df-atan 25447 df-em 25572 df-cht 25676 df-vma 25677 df-chp 25678 df-ppi 25679 df-mu 25680 df-dchr 25811 |
This theorem is referenced by: (None) |
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