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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cntrcrng | Structured version Visualization version GIF version |
Description: The center of a ring is a commutative ring. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
Ref | Expression |
---|---|
cntrcrng.z | ⊢ 𝑍 = (𝑅 ↾s (Cntr‘(mulGrp‘𝑅))) |
Ref | Expression |
---|---|
cntrcrng | ⊢ (𝑅 ∈ Ring → 𝑍 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19238 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
4 | eqid 2798 | . . . . 5 ⊢ (Cntz‘(mulGrp‘𝑅)) = (Cntz‘(mulGrp‘𝑅)) | |
5 | 3, 4 | cntrval 18441 | . . . 4 ⊢ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) = (Cntr‘(mulGrp‘𝑅)) |
6 | ssid 3937 | . . . . 5 ⊢ (Base‘𝑅) ⊆ (Base‘𝑅) | |
7 | 2, 1, 4 | cntzsubr 19561 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (Base‘𝑅) ⊆ (Base‘𝑅)) → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubRing‘𝑅)) |
8 | 6, 7 | mpan2 690 | . . . 4 ⊢ (𝑅 ∈ Ring → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubRing‘𝑅)) |
9 | 5, 8 | eqeltrrid 2895 | . . 3 ⊢ (𝑅 ∈ Ring → (Cntr‘(mulGrp‘𝑅)) ∈ (SubRing‘𝑅)) |
10 | cntrcrng.z | . . . 4 ⊢ 𝑍 = (𝑅 ↾s (Cntr‘(mulGrp‘𝑅))) | |
11 | 10 | subrgring 19531 | . . 3 ⊢ ((Cntr‘(mulGrp‘𝑅)) ∈ (SubRing‘𝑅) → 𝑍 ∈ Ring) |
12 | 9, 11 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → 𝑍 ∈ Ring) |
13 | fvex 6658 | . . . 4 ⊢ (Cntr‘(mulGrp‘𝑅)) ∈ V | |
14 | 10, 1 | mgpress 19243 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (Cntr‘(mulGrp‘𝑅)) ∈ V) → ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) = (mulGrp‘𝑍)) |
15 | 13, 14 | mpan2 690 | . . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) = (mulGrp‘𝑍)) |
16 | 1 | ringmgp 19296 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
17 | eqid 2798 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) = ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) | |
18 | 17 | cntrcmnd 18955 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) ∈ CMnd) |
19 | 16, 18 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) ∈ CMnd) |
20 | 15, 19 | eqeltrrd 2891 | . 2 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑍) ∈ CMnd) |
21 | eqid 2798 | . . 3 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
22 | 21 | iscrng 19297 | . 2 ⊢ (𝑍 ∈ CRing ↔ (𝑍 ∈ Ring ∧ (mulGrp‘𝑍) ∈ CMnd)) |
23 | 12, 20, 22 | sylanbrc 586 | 1 ⊢ (𝑅 ∈ Ring → 𝑍 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 Mndcmnd 17903 Cntzccntz 18437 Cntrccntr 18438 CMndccmn 18898 mulGrpcmgp 19232 Ringcrg 19290 CRingccrg 19291 SubRingcsubrg 19524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-subg 18268 df-cntz 18439 df-cntr 18440 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 |
This theorem is referenced by: (None) |
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