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Mirrors > Home > MPE Home > Th. List > scmatsrng | Structured version Visualization version GIF version |
Description: The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-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) |
Ref | Expression |
---|---|
scmatsrng | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴)) |
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 | 1, 2, 3, 4, 5 | scmatsgrp 21664 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐴)) |
7 | 1, 2, 3, 4, 5 | scmatid 21659 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝑆) |
8 | 1, 2, 3, 4, 5 | scmatmulcl 21663 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐴)𝑦) ∈ 𝑆) |
9 | 8 | ralrimivva 3117 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) ∈ 𝑆) |
10 | 1 | matring 21588 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
11 | eqid 2740 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
12 | eqid 2740 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
13 | 2, 11, 12 | issubrg2 20040 | . . 3 ⊢ (𝐴 ∈ Ring → (𝑆 ∈ (SubRing‘𝐴) ↔ (𝑆 ∈ (SubGrp‘𝐴) ∧ (1r‘𝐴) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) ∈ 𝑆))) |
14 | 10, 13 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubRing‘𝐴) ↔ (𝑆 ∈ (SubGrp‘𝐴) ∧ (1r‘𝐴) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) ∈ 𝑆))) |
15 | 6, 7, 9, 14 | mpbir3and 1341 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ‘cfv 6431 (class class class)co 7269 Fincfn 8714 Basecbs 16908 .rcmulr 16959 0gc0g 17146 SubGrpcsubg 18745 1rcur 19733 Ringcrg 19779 SubRingcsubrg 20016 Mat cmat 21550 ScMat cscmat 21634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-sup 9177 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-fz 13237 df-fzo 13380 df-seq 13718 df-hash 14041 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-hom 16982 df-cco 16983 df-0g 17148 df-gsum 17149 df-prds 17154 df-pws 17156 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mhm 18426 df-submnd 18427 df-grp 18576 df-minusg 18577 df-sbg 18578 df-mulg 18697 df-subg 18748 df-ghm 18828 df-cntz 18919 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-subrg 20018 df-lmod 20121 df-lss 20190 df-sra 20430 df-rgmod 20431 df-dsmm 20935 df-frlm 20950 df-mamu 21529 df-mat 21551 df-dmat 21635 df-scmat 21636 |
This theorem is referenced by: scmatcrng 21666 scmatstrbas 21671 scmatmhm 21679 scmatrhm 21680 scmatrngiso 21681 |
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