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Mirrors > Home > MPE Home > Th. List > znchr | Structured version Visualization version GIF version |
Description: Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
znchr.y | β’ π = (β€/nβ€βπ) |
Ref | Expression |
---|---|
znchr | β’ (π β β0 β (chrβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znchr.y | . . . . . . 7 β’ π = (β€/nβ€βπ) | |
2 | 1 | zncrng 20954 | . . . . . 6 β’ (π β β0 β π β CRing) |
3 | crngring 19977 | . . . . . 6 β’ (π β CRing β π β Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 β’ (π β β0 β π β Ring) |
5 | nn0z 12525 | . . . . 5 β’ (π₯ β β0 β π₯ β β€) | |
6 | eqid 2737 | . . . . . 6 β’ (chrβπ) = (chrβπ) | |
7 | eqid 2737 | . . . . . 6 β’ (β€RHomβπ) = (β€RHomβπ) | |
8 | eqid 2737 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
9 | 6, 7, 8 | chrdvds 20934 | . . . . 5 β’ ((π β Ring β§ π₯ β β€) β ((chrβπ) β₯ π₯ β ((β€RHomβπ)βπ₯) = (0gβπ))) |
10 | 4, 5, 9 | syl2an 597 | . . . 4 β’ ((π β β0 β§ π₯ β β0) β ((chrβπ) β₯ π₯ β ((β€RHomβπ)βπ₯) = (0gβπ))) |
11 | 1, 7, 8 | zndvds0 20960 | . . . . 5 β’ ((π β β0 β§ π₯ β β€) β (((β€RHomβπ)βπ₯) = (0gβπ) β π β₯ π₯)) |
12 | 5, 11 | sylan2 594 | . . . 4 β’ ((π β β0 β§ π₯ β β0) β (((β€RHomβπ)βπ₯) = (0gβπ) β π β₯ π₯)) |
13 | 10, 12 | bitrd 279 | . . 3 β’ ((π β β0 β§ π₯ β β0) β ((chrβπ) β₯ π₯ β π β₯ π₯)) |
14 | 13 | ralrimiva 3144 | . 2 β’ (π β β0 β βπ₯ β β0 ((chrβπ) β₯ π₯ β π β₯ π₯)) |
15 | 6 | chrcl 20932 | . . . 4 β’ (π β Ring β (chrβπ) β β0) |
16 | 4, 15 | syl 17 | . . 3 β’ (π β β0 β (chrβπ) β β0) |
17 | dvdsext 16204 | . . 3 β’ (((chrβπ) β β0 β§ π β β0) β ((chrβπ) = π β βπ₯ β β0 ((chrβπ) β₯ π₯ β π β₯ π₯))) | |
18 | 16, 17 | mpancom 687 | . 2 β’ (π β β0 β ((chrβπ) = π β βπ₯ β β0 ((chrβπ) β₯ π₯ β π β₯ π₯))) |
19 | 14, 18 | mpbird 257 | 1 β’ (π β β0 β (chrβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 class class class wbr 5106 βcfv 6497 β0cn0 12414 β€cz 12500 β₯ cdvds 16137 0gc0g 17322 Ringcrg 19965 CRingccrg 19966 β€RHomczrh 20903 chrcchr 20905 β€/nβ€czn 20906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-ec 8651 df-qs 8655 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-rp 12917 df-fz 13426 df-fl 13698 df-mod 13776 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-dvds 16138 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-0g 17324 df-imas 17391 df-qus 17392 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-minusg 18753 df-sbg 18754 df-mulg 18874 df-subg 18926 df-nsg 18927 df-eqg 18928 df-ghm 19007 df-od 19311 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-rnghom 20147 df-subrg 20223 df-lmod 20327 df-lss 20396 df-lsp 20436 df-sra 20636 df-rgmod 20637 df-lidl 20638 df-rsp 20639 df-2idl 20705 df-cnfld 20800 df-zring 20873 df-zrh 20907 df-chr 20909 df-zn 20910 |
This theorem is referenced by: ply1fermltl 32287 |
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