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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 π. Theorem 9.4.1 of [Shapiro], p. 375. See https://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 | β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β {π β β β£ π β₯ (π β π΄)} β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . . . . . 8 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β π β β) | |
2 | 1 | nnnn0d 12568 | . . . . . . 7 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β π β β0) |
3 | 2 | adantr 479 | . . . . . 6 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β π β β0) |
4 | eqid 2727 | . . . . . . 7 β’ (β€/nβ€βπ) = (β€/nβ€βπ) | |
5 | eqid 2727 | . . . . . . 7 β’ (Baseβ(β€/nβ€βπ)) = (Baseβ(β€/nβ€βπ)) | |
6 | eqid 2727 | . . . . . . 7 β’ (β€RHomβ(β€/nβ€βπ)) = (β€RHomβ(β€/nβ€βπ)) | |
7 | 4, 5, 6 | znzrhfo 21486 | . . . . . 6 β’ (π β β0 β (β€RHomβ(β€/nβ€βπ)):β€βontoβ(Baseβ(β€/nβ€βπ))) |
8 | fofn 6816 | . . . . . 6 β’ ((β€RHomβ(β€/nβ€βπ)):β€βontoβ(Baseβ(β€/nβ€βπ)) β (β€RHomβ(β€/nβ€βπ)) Fn β€) | |
9 | 3, 7, 8 | 3syl 18 | . . . . 5 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β (β€RHomβ(β€/nβ€βπ)) Fn β€) |
10 | prmz 16651 | . . . . . 6 β’ (π β β β π β β€) | |
11 | 10 | adantl 480 | . . . . 5 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β π β β€) |
12 | fniniseg 7072 | . . . . . 6 β’ ((β€RHomβ(β€/nβ€βπ)) Fn β€ β (π β (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) β (π β β€ β§ ((β€RHomβ(β€/nβ€βπ))βπ) = ((β€RHomβ(β€/nβ€βπ))βπ΄)))) | |
13 | 12 | baibd 538 | . . . . 5 β’ (((β€RHomβ(β€/nβ€βπ)) Fn β€ β§ π β β€) β (π β (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) β ((β€RHomβ(β€/nβ€βπ))βπ) = ((β€RHomβ(β€/nβ€βπ))βπ΄))) |
14 | 9, 11, 13 | syl2anc 582 | . . . 4 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β (π β (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) β ((β€RHomβ(β€/nβ€βπ))βπ) = ((β€RHomβ(β€/nβ€βπ))βπ΄))) |
15 | simp2 1134 | . . . . . 6 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β π΄ β β€) | |
16 | 15 | adantr 479 | . . . . 5 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β π΄ β β€) |
17 | 4, 6 | zndvds 21488 | . . . . 5 β’ ((π β β0 β§ π β β€ β§ π΄ β β€) β (((β€RHomβ(β€/nβ€βπ))βπ) = ((β€RHomβ(β€/nβ€βπ))βπ΄) β π β₯ (π β π΄))) |
18 | 3, 11, 16, 17 | syl3anc 1368 | . . . 4 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β (((β€RHomβ(β€/nβ€βπ))βπ) = ((β€RHomβ(β€/nβ€βπ))βπ΄) β π β₯ (π β π΄))) |
19 | 14, 18 | bitrd 278 | . . 3 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β (π β (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) β π β₯ (π β π΄))) |
20 | 19 | rabbi2dva 4218 | . 2 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β (β β© (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)})) = {π β β β£ π β₯ (π β π΄)}) |
21 | eqid 2727 | . . 3 β’ (Unitβ(β€/nβ€βπ)) = (Unitβ(β€/nβ€βπ)) | |
22 | simp3 1135 | . . . 4 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β (π΄ gcd π) = 1) | |
23 | 4, 21, 6 | znunit 21502 | . . . . 5 β’ ((π β β0 β§ π΄ β β€) β (((β€RHomβ(β€/nβ€βπ))βπ΄) β (Unitβ(β€/nβ€βπ)) β (π΄ gcd π) = 1)) |
24 | 2, 15, 23 | syl2anc 582 | . . . 4 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β (((β€RHomβ(β€/nβ€βπ))βπ΄) β (Unitβ(β€/nβ€βπ)) β (π΄ gcd π) = 1)) |
25 | 22, 24 | mpbird 256 | . . 3 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β ((β€RHomβ(β€/nβ€βπ))βπ΄) β (Unitβ(β€/nβ€βπ))) |
26 | eqid 2727 | . . 3 β’ (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) = (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)}) | |
27 | 4, 6, 1, 21, 25, 26 | dirith2 27479 | . 2 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β (β β© (β‘(β€RHomβ(β€/nβ€βπ)) β {((β€RHomβ(β€/nβ€βπ))βπ΄)})) β β) |
28 | 20, 27 | eqbrtrrd 5174 | 1 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β {π β β β£ π β₯ (π β π΄)} β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3428 β© cin 3946 {csn 4630 class class class wbr 5150 β‘ccnv 5679 β cima 5683 Fn wfn 6546 βontoβwfo 6549 βcfv 6551 (class class class)co 7424 β cen 8965 1c1 11145 β cmin 11480 βcn 12248 β0cn0 12508 β€cz 12594 β₯ cdvds 16236 gcd cgcd 16474 βcprime 16647 Basecbs 17185 Unitcui 20299 β€RHomczrh 21430 β€/nβ€czn 21433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5116 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-rpss 7732 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-oadd 8495 df-omul 8496 df-er 8729 df-ec 8731 df-qs 8735 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-dju 9930 df-card 9968 df-acn 9971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-xnn0 12581 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-fac 14271 df-bc 14300 df-hash 14328 df-word 14503 df-concat 14559 df-s1 14584 df-shft 15052 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-limsup 15453 df-clim 15470 df-rlim 15471 df-o1 15472 df-lo1 15473 df-sum 15671 df-ef 16049 df-e 16050 df-sin 16051 df-cos 16052 df-tan 16053 df-pi 16054 df-dvds 16237 df-gcd 16475 df-prm 16648 df-numer 16712 df-denom 16713 df-phi 16740 df-pc 16811 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-qus 17496 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-nsg 19084 df-eqg 19085 df-ghm 19173 df-gim 19218 df-ga 19246 df-cntz 19273 df-oppg 19302 df-od 19488 df-gex 19489 df-pgp 19490 df-lsm 19596 df-pj1 19597 df-cmn 19742 df-abl 19743 df-cyg 19838 df-dprd 19957 df-dpj 19958 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-cring 20181 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-rhm 20416 df-subrng 20488 df-subrg 20513 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-sra 21063 df-rgmod 21064 df-lidl 21109 df-rsp 21110 df-2idl 21149 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-zring 21378 df-zrh 21434 df-zn 21437 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-cmp 23309 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-0p 25617 df-limc 25813 df-dv 25814 df-ply 26140 df-idp 26141 df-coe 26142 df-dgr 26143 df-quot 26244 df-ulm 26331 df-log 26508 df-cxp 26509 df-atan 26817 df-em 26943 df-cht 27047 df-vma 27048 df-chp 27049 df-ppi 27050 df-mu 27051 df-dchr 27184 |
This theorem is referenced by: (None) |
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