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Mirrors > Home > MPE Home > Th. List > scmatsrng1 | Structured version Visualization version GIF version |
Description: The set of scalar matrices is a subring of the ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
Ref | Expression |
---|---|
scmatid.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatid.b | ⊢ 𝐵 = (Base‘𝐴) |
scmatid.e | ⊢ 𝐸 = (Base‘𝑅) |
scmatid.0 | ⊢ 0 = (0g‘𝑅) |
scmatid.s | ⊢ 𝑆 = (𝑁 ScMat 𝑅) |
scmatsgrp1.d | ⊢ 𝐷 = (𝑁 DMat 𝑅) |
scmatsgrp1.c | ⊢ 𝐶 = (𝐴 ↾s 𝐷) |
Ref | Expression |
---|---|
scmatsrng1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatid.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | scmatid.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | scmatid.e | . . 3 ⊢ 𝐸 = (Base‘𝑅) | |
4 | scmatid.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | scmatid.s | . . 3 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
6 | scmatsgrp1.d | . . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅) | |
7 | scmatsgrp1.c | . . 3 ⊢ 𝐶 = (𝐴 ↾s 𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | scmatsgrp1 20850 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
9 | 1, 2, 4, 6 | dmatsrng 20829 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) |
10 | 9 | ancoms 451 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴)) |
11 | eqid 2780 | . . . . . 6 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
12 | 7, 11 | subrg1 19280 | . . . . 5 ⊢ (𝐷 ∈ (SubRing‘𝐴) → (1r‘𝐴) = (1r‘𝐶)) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (1r‘𝐶)) |
14 | 13 | eqcomd 2786 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (1r‘𝐴)) |
15 | 1, 2, 3, 4, 5 | scmatid 20842 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝑆) |
16 | 14, 15 | eqeltrd 2868 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
17 | eqid 2780 | . . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
18 | 7, 17 | ressmulr 16487 | . . . . . . 7 ⊢ (𝐷 ∈ (SubRing‘𝐴) → (.r‘𝐴) = (.r‘𝐶)) |
19 | 10, 18 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (.r‘𝐴) = (.r‘𝐶)) |
20 | 19 | eqcomd 2786 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (.r‘𝐶) = (.r‘𝐴)) |
21 | 20 | oveqdr 7010 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐶)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
22 | 1, 2, 3, 4, 5 | scmatmulcl 20846 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐴)𝑦) ∈ 𝑆) |
23 | 21, 22 | eqeltrd 2868 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) |
24 | 23 | ralrimivva 3143 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) |
25 | 7 | subrgring 19273 | . . 3 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring) |
26 | eqid 2780 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
27 | eqid 2780 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
28 | eqid 2780 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
29 | 26, 27, 28 | issubrg2 19290 | . . 3 ⊢ (𝐶 ∈ Ring → (𝑆 ∈ (SubRing‘𝐶) ↔ (𝑆 ∈ (SubGrp‘𝐶) ∧ (1r‘𝐶) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆))) |
30 | 10, 25, 29 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubRing‘𝐶) ↔ (𝑆 ∈ (SubGrp‘𝐶) ∧ (1r‘𝐶) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆))) |
31 | 8, 16, 24, 30 | mpbir3and 1323 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3090 ‘cfv 6193 (class class class)co 6982 Fincfn 8312 Basecbs 16345 ↾s cress 16346 .rcmulr 16428 0gc0g 16575 SubGrpcsubg 18069 1rcur 18986 Ringcrg 19032 SubRingcsubrg 19266 Mat cmat 20735 DMat cdmat 20816 ScMat cscmat 20817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-ot 4453 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-of 7233 df-om 7403 df-1st 7507 df-2nd 7508 df-supp 7640 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-oadd 7915 df-er 8095 df-map 8214 df-ixp 8266 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-fsupp 8635 df-sup 8707 df-oi 8775 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-z 11800 df-dec 11918 df-uz 12065 df-fz 12715 df-fzo 12856 df-seq 13191 df-hash 13512 df-struct 16347 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-ress 16353 df-plusg 16440 df-mulr 16441 df-sca 16443 df-vsca 16444 df-ip 16445 df-tset 16446 df-ple 16447 df-ds 16449 df-hom 16451 df-cco 16452 df-0g 16577 df-gsum 16578 df-prds 16583 df-pws 16585 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-subrg 19268 df-lmod 19370 df-lss 19438 df-sra 19678 df-rgmod 19679 df-dsmm 20593 df-frlm 20608 df-mamu 20712 df-mat 20736 df-dmat 20818 df-scmat 20819 |
This theorem is referenced by: (None) |
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