Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21391 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
9 | 1, 2, 4, 6 | dmatsrng 21370 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) |
10 | 9 | ancoms 462 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐷 ∈ (SubRing‘𝐴)) |
11 | eqid 2734 | . . . . . 6 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
12 | 7, 11 | subrg1 19782 | . . . . 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 2740 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (1r‘𝐴)) |
15 | 1, 2, 3, 4, 5 | scmatid 21383 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝑆) |
16 | 14, 15 | eqeltrd 2834 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
17 | eqid 2734 | . . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
18 | 7, 17 | ressmulr 16827 | . . . . . . 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 2740 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (.r‘𝐶) = (.r‘𝐴)) |
21 | 20 | oveqdr 7230 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐶)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
22 | 1, 2, 3, 4, 5 | scmatmulcl 21387 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐴)𝑦) ∈ 𝑆) |
23 | 21, 22 | eqeltrd 2834 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) |
24 | 23 | ralrimivva 3105 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) |
25 | 7 | subrgring 19775 | . . 3 ⊢ (𝐷 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring) |
26 | eqid 2734 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
27 | eqid 2734 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
28 | eqid 2734 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
29 | 26, 27, 28 | issubrg2 19792 | . . 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 1344 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 Basecbs 16684 ↾s cress 16685 .rcmulr 16768 0gc0g 16916 SubGrpcsubg 18509 1rcur 19488 Ringcrg 19534 SubRingcsubrg 19768 Mat cmat 21276 DMat cdmat 21357 ScMat cscmat 21358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-ot 4540 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 df-mamu 21255 df-mat 21277 df-dmat 21359 df-scmat 21360 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |