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Mirrors > Home > MPE Home > Th. List > cpmatsrgpmat | Structured version Visualization version GIF version |
Description: The set of all constant polynomial matrices over a ring 𝑅 is a subring of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 18-Nov-2019.) |
Ref | Expression |
---|---|
cpmatsrngpmat.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
cpmatsrngpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmatsrngpmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
Ref | Expression |
---|---|
cpmatsrgpmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmatsrngpmat.s | . . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | cpmatsrngpmat.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | cpmatsrngpmat.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
4 | 1, 2, 3 | cpmatsubgpmat 22021 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
5 | 1, 2, 3 | 1elcpmat 22016 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
6 | 1, 2, 3 | cpmatmcl 22020 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) |
7 | 2, 3 | pmatring 21993 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
8 | eqid 2738 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2738 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
10 | eqid 2738 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
11 | 8, 9, 10 | issubrg2 20195 | . . 3 ⊢ (𝐶 ∈ Ring → (𝑆 ∈ (SubRing‘𝐶) ↔ (𝑆 ∈ (SubGrp‘𝐶) ∧ (1r‘𝐶) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆))) |
12 | 7, 11 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubRing‘𝐶) ↔ (𝑆 ∈ (SubGrp‘𝐶) ∧ (1r‘𝐶) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆))) |
13 | 4, 5, 6, 12 | mpbir3and 1343 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ‘cfv 6494 (class class class)co 7352 Fincfn 8842 Basecbs 17043 .rcmulr 17094 SubGrpcsubg 18881 1rcur 19872 Ringcrg 19918 SubRingcsubrg 20171 Poly1cpl1 21500 Mat cmat 21706 ConstPolyMat ccpmat 22004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-ofr 7611 df-om 7796 df-1st 7914 df-2nd 7915 df-supp 8086 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-pm 8727 df-ixp 8795 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fsupp 9265 df-sup 9337 df-oi 9405 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-fz 13380 df-fzo 13523 df-seq 13862 df-hash 14185 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-hom 17117 df-cco 17118 df-0g 17283 df-gsum 17284 df-prds 17289 df-pws 17291 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cntz 19056 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-srg 19877 df-ring 19920 df-subrg 20173 df-lmod 20277 df-lss 20346 df-sra 20586 df-rgmod 20587 df-dsmm 21091 df-frlm 21106 df-ascl 21214 df-psr 21264 df-mvr 21265 df-mpl 21266 df-opsr 21268 df-psr1 21503 df-vr1 21504 df-ply1 21505 df-coe1 21506 df-mamu 21685 df-mat 21707 df-cpmat 22007 |
This theorem is referenced by: m2cpmmhm 22046 m2cpmrhm 22047 chfacfisfcpmat 22156 |
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