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Mirrors > Home > MPE Home > Th. List > zncrng | Structured version Visualization version GIF version |
Description: β€/nβ€ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
zncrng.y | β’ π = (β€/nβ€βπ) |
Ref | Expression |
---|---|
zncrng | β’ (π β β0 β π β CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 12582 | . . 3 β’ (π β β0 β π β β€) | |
2 | eqid 2724 | . . . 4 β’ (RSpanββ€ring) = (RSpanββ€ring) | |
3 | eqid 2724 | . . . 4 β’ (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) = (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) | |
4 | 2, 3 | zncrng2 21414 | . . 3 β’ (π β β€ β (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) β CRing) |
5 | 1, 4 | syl 17 | . 2 β’ (π β β0 β (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) β CRing) |
6 | eqidd 2725 | . . 3 β’ (π β β0 β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) = (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))))) | |
7 | zncrng.y | . . . 4 β’ π = (β€/nβ€βπ) | |
8 | 2, 3, 7 | znbas2 21420 | . . 3 β’ (π β β0 β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) = (Baseβπ)) |
9 | 2, 3, 7 | znadd 21422 | . . . 4 β’ (π β β0 β (+gβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) = (+gβπ)) |
10 | 9 | oveqdr 7430 | . . 3 β’ ((π β β0 β§ (π₯ β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) β§ π¦ β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))))) β (π₯(+gβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))))π¦) = (π₯(+gβπ)π¦)) |
11 | 2, 3, 7 | znmul 21424 | . . . 4 β’ (π β β0 β (.rβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) = (.rβπ)) |
12 | 11 | oveqdr 7430 | . . 3 β’ ((π β β0 β§ (π₯ β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) β§ π¦ β (Baseβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))))) β (π₯(.rβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))))π¦) = (π₯(.rβπ)π¦)) |
13 | 6, 8, 10, 12 | crngpropd 20184 | . 2 β’ (π β β0 β ((β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) β CRing β π β CRing)) |
14 | 5, 13 | mpbid 231 | 1 β’ (π β β0 β π β CRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {csn 4621 βcfv 6534 (class class class)co 7402 β0cn0 12471 β€cz 12557 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 /s cqus 17456 ~QG cqg 19045 CRingccrg 20135 RSpancrsp 21062 β€ringczring 21322 β€/nβ€czn 21378 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-ec 8702 df-qs 8706 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-imas 17459 df-qus 17460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-nsg 19047 df-eqg 19048 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-2idl 21103 df-cnfld 21235 df-zring 21323 df-zn 21382 |
This theorem is referenced by: zncyg 21432 zndvds0 21434 znf1o 21435 zzngim 21436 znfld 21444 znchr 21446 znunit 21447 znrrg 21449 cygznlem3 21453 dchrelbas3 27112 dchrelbasd 27113 dchrzrh1 27118 dchrzrhmul 27120 dchrmulcl 27123 dchrn0 27124 dchrfi 27129 dchrghm 27130 dchrabs 27134 dchrinv 27135 dchrptlem1 27138 dchrptlem2 27139 dchrptlem3 27140 dchrpt 27141 dchrsum2 27142 dchrhash 27145 sum2dchr 27148 lgsdchr 27229 dchrisum0flblem1 27382 dchrisum0re 27387 znfermltl 32975 ply1fermltl 33157 hashscontpowcl 41488 hashscontpow 41490 frlmpwfi 42392 isnumbasgrplem3 42399 cznabel 47184 cznrng 47185 |
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