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Mirrors > Home > MPE Home > Th. List > zncrng | Structured version Visualization version GIF version |
Description: ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
zncrng.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
Ref | Expression |
---|---|
zncrng | ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 11729 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | eqid 2826 | . . . 4 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring) | |
3 | eqid 2826 | . . . 4 ⊢ (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) | |
4 | 2, 3 | zncrng2 20243 | . . 3 ⊢ (𝑁 ∈ ℤ → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) ∈ CRing) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) ∈ CRing) |
6 | eqidd 2827 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))))) | |
7 | zncrng.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
8 | 2, 3, 7 | znbas2 20248 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (Base‘𝑌)) |
9 | 2, 3, 7 | znadd 20249 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (+g‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (+g‘𝑌)) |
10 | 9 | oveqdr 6934 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ∧ 𝑦 ∈ (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))))) → (𝑥(+g‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))))𝑦) = (𝑥(+g‘𝑌)𝑦)) |
11 | 2, 3, 7 | znmul 20250 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (.r‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) = (.r‘𝑌)) |
12 | 11 | oveqdr 6934 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))) ∧ 𝑦 ∈ (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁})))))) → (𝑥(.r‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))))𝑦) = (𝑥(.r‘𝑌)𝑦)) |
13 | 6, 8, 10, 12 | crngpropd 18938 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑁}))) ∈ CRing ↔ 𝑌 ∈ CRing)) |
14 | 5, 13 | mpbid 224 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4398 ‘cfv 6124 (class class class)co 6906 ℕ0cn0 11619 ℤcz 11705 Basecbs 16223 +gcplusg 16306 .rcmulr 16307 /s cqus 16519 ~QG cqg 17942 CRingccrg 18903 RSpancrsp 19533 ℤringzring 20179 ℤ/nℤczn 20212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-ec 8012 df-qs 8016 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-imas 16522 df-qus 16523 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-nsg 17944 df-eqg 17945 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-oppr 18978 df-subrg 19135 df-lmod 19222 df-lss 19290 df-lsp 19332 df-sra 19534 df-rgmod 19535 df-lidl 19536 df-rsp 19537 df-2idl 19594 df-cnfld 20108 df-zring 20180 df-zn 20216 |
This theorem is referenced by: zncyg 20257 zndvds0 20259 znf1o 20260 zzngim 20261 znfld 20269 znchr 20271 znunit 20272 znrrg 20274 cygznlem3 20278 dchrelbas3 25377 dchrelbasd 25378 dchrzrh1 25383 dchrzrhmul 25385 dchrmulcl 25388 dchrn0 25389 dchrfi 25394 dchrghm 25395 dchrabs 25399 dchrinv 25400 dchrptlem1 25403 dchrptlem2 25404 dchrptlem3 25405 dchrpt 25406 dchrsum2 25407 dchrhash 25410 sum2dchr 25413 lgsdchr 25494 dchrisum0flblem1 25611 dchrisum0re 25616 frlmpwfi 38512 isnumbasgrplem3 38519 cznabel 42802 cznrng 42803 |
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