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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 12533 | . . . . . . 7 β’ ((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β π β β0) |
3 | 2 | adantr 480 | . . . . . 6 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β π β β0) |
4 | eqid 2726 | . . . . . . 7 β’ (β€/nβ€βπ) = (β€/nβ€βπ) | |
5 | eqid 2726 | . . . . . . 7 β’ (Baseβ(β€/nβ€βπ)) = (Baseβ(β€/nβ€βπ)) | |
6 | eqid 2726 | . . . . . . 7 β’ (β€RHomβ(β€/nβ€βπ)) = (β€RHomβ(β€/nβ€βπ)) | |
7 | 4, 5, 6 | znzrhfo 21438 | . . . . . 6 β’ (π β β0 β (β€RHomβ(β€/nβ€βπ)):β€βontoβ(Baseβ(β€/nβ€βπ))) |
8 | fofn 6800 | . . . . . 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 16617 | . . . . . 6 β’ (π β β β π β β€) | |
11 | 10 | adantl 481 | . . . . 5 β’ (((π β β β§ π΄ β β€ β§ (π΄ gcd π) = 1) β§ π β β) β π β β€) |
12 | fniniseg 7054 | . . . . . 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 539 | . . . . 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 583 | . . . 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 480 | . . . . 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 21440 | . . . . 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 279 | . . 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 4212 | . 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 2726 | . . 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 21454 | . . . . 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 583 | . . . 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 257 | . . 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 2726 | . . 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 27412 | . 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 5165 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3426 β© cin 3942 {csn 4623 class class class wbr 5141 β‘ccnv 5668 β cima 5672 Fn wfn 6531 βontoβwfo 6534 βcfv 6536 (class class class)co 7404 β cen 8935 1c1 11110 β cmin 11445 βcn 12213 β0cn0 12473 β€cz 12559 β₯ cdvds 16202 gcd cgcd 16440 βcprime 16613 Basecbs 17151 Unitcui 20255 β€RHomczrh 21382 β€/nβ€czn 21385 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-rpss 7709 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-ec 8704 df-qs 8708 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-o1 15438 df-lo1 15439 df-sum 15637 df-ef 16015 df-e 16016 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-dvds 16203 df-gcd 16441 df-prm 16614 df-numer 16678 df-denom 16679 df-phi 16706 df-pc 16777 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-qus 17462 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-nsg 19049 df-eqg 19050 df-ghm 19137 df-gim 19182 df-ga 19204 df-cntz 19231 df-oppg 19260 df-od 19446 df-gex 19447 df-pgp 19448 df-lsm 19554 df-pj1 19555 df-cmn 19700 df-abl 19701 df-cyg 19796 df-dprd 19915 df-dpj 19916 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-rhm 20372 df-subrng 20444 df-subrg 20469 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-sra 21019 df-rgmod 21020 df-lidl 21065 df-rsp 21066 df-2idl 21105 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-zring 21330 df-zrh 21386 df-zn 21389 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-0p 25550 df-limc 25746 df-dv 25747 df-ply 26073 df-idp 26074 df-coe 26075 df-dgr 26076 df-quot 26177 df-ulm 26264 df-log 26441 df-cxp 26442 df-atan 26750 df-em 26876 df-cht 26980 df-vma 26981 df-chp 26982 df-ppi 26983 df-mu 26984 df-dchr 27117 |
This theorem is referenced by: (None) |
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