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Mirrors > Home > MPE Home > Th. List > dmatsrng | Structured version Visualization version GIF version |
Description: The set of diagonal matrices is a subring of the matrix ring/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
dmatid.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
dmatid.b | ⊢ 𝐵 = (Base‘𝐴) |
dmatid.0 | ⊢ 0 = (0g‘𝑅) |
dmatid.d | ⊢ 𝐷 = (𝑁 DMat 𝑅) |
Ref | Expression |
---|---|
dmatsrng | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmatid.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | dmatid.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | dmatid.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | dmatid.d | . . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅) | |
5 | 1, 2, 3, 4 | dmatsgrp 21107 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴)) |
6 | 1, 2, 3, 4 | dmatid 21103 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝐷) |
7 | 6 | ancoms 461 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (1r‘𝐴) ∈ 𝐷) |
8 | 1, 2, 3, 4 | dmatmulcl 21108 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥(.r‘𝐴)𝑦) ∈ 𝐷) |
9 | 8 | ralrimivva 3191 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐴)𝑦) ∈ 𝐷) |
10 | 9 | ancoms 461 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐴)𝑦) ∈ 𝐷) |
11 | 1 | matring 21051 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
12 | 11 | ancoms 461 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Ring) |
13 | eqid 2821 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
14 | eqid 2821 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
15 | 2, 13, 14 | issubrg2 19554 | . . 3 ⊢ (𝐴 ∈ Ring → (𝐷 ∈ (SubRing‘𝐴) ↔ (𝐷 ∈ (SubGrp‘𝐴) ∧ (1r‘𝐴) ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐴)𝑦) ∈ 𝐷))) |
16 | 12, 15 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝐷 ∈ (SubRing‘𝐴) ↔ (𝐷 ∈ (SubGrp‘𝐴) ∧ (1r‘𝐴) ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥(.r‘𝐴)𝑦) ∈ 𝐷))) |
17 | 5, 7, 10, 16 | mpbir3and 1338 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 Basecbs 16482 .rcmulr 16565 0gc0g 16712 SubGrpcsubg 18272 1rcur 19250 Ringcrg 19296 SubRingcsubrg 19530 Mat cmat 21015 DMat cdmat 21096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-subrg 19532 df-lmod 19635 df-lss 19703 df-sra 19943 df-rgmod 19944 df-dsmm 20875 df-frlm 20890 df-mamu 20994 df-mat 21016 df-dmat 21098 |
This theorem is referenced by: dmatcrng 21110 scmatsgrp1 21130 scmatsrng1 21131 |
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