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 2820 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19238 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
4 | eqid 2820 | . . . . 5 ⊢ (Cntz‘(mulGrp‘𝑅)) = (Cntz‘(mulGrp‘𝑅)) | |
5 | 3, 4 | cntrval 18442 | . . . 4 ⊢ ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) = (Cntr‘(mulGrp‘𝑅)) |
6 | ssid 3982 | . . . . 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 689 | . . . 4 ⊢ (𝑅 ∈ Ring → ((Cntz‘(mulGrp‘𝑅))‘(Base‘𝑅)) ∈ (SubRing‘𝑅)) |
9 | 5, 8 | eqeltrrid 2917 | . . 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 6676 | . . . 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 689 | . . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Cntr‘(mulGrp‘𝑅))) = (mulGrp‘𝑍)) |
16 | 1 | ringmgp 19296 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
17 | eqid 2820 | . . . . 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 2913 | . 2 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑍) ∈ CMnd) |
21 | eqid 2820 | . . 3 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
22 | 21 | iscrng 19297 | . 2 ⊢ (𝑍 ∈ CRing ↔ (𝑍 ∈ Ring ∧ (mulGrp‘𝑍) ∈ CMnd)) |
23 | 12, 20, 22 | sylanbrc 585 | 1 ⊢ (𝑅 ∈ Ring → 𝑍 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 ↾s cress 16477 Mndcmnd 17904 Cntzccntz 18438 Cntrccntr 18439 CMndccmn 18899 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 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-submnd 17950 df-grp 18099 df-minusg 18100 df-subg 18269 df-cntz 18440 df-cntr 18441 df-cmn 18901 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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