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Mirrors > Home > MPE Home > Th. List > cpmatsubgpmat | Structured version Visualization version GIF version |
Description: The set of all constant polynomial matrices over a ring 𝑅 is an additive subgroup of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 15-Nov-2019.) |
Ref | Expression |
---|---|
cpmatsrngpmat.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
cpmatsrngpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmatsrngpmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
Ref | Expression |
---|---|
cpmatsubgpmat | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmatsrngpmat.s | . . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | cpmatsrngpmat.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | cpmatsrngpmat.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
4 | eqid 2726 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmat 22697 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | ssrab2 4074 | . . 3 ⊢ {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} ⊆ (Base‘𝐶) | |
7 | 5, 6 | eqsstrdi 4034 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
8 | 1, 2, 3 | 1elcpmat 22703 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
9 | 8 | ne0d 4336 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅) |
10 | 1, 2, 3 | cpmatacl 22704 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆) |
11 | 1, 2, 3 | cpmatinvcl 22705 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆) |
12 | r19.26 3101 | . . 3 ⊢ (∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆) ↔ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆)) | |
13 | 10, 11, 12 | sylanbrc 581 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆)) |
14 | 2, 3 | pmatring 22680 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
15 | ringgrp 20215 | . . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp) | |
16 | eqid 2726 | . . . 4 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
17 | eqid 2726 | . . . 4 ⊢ (invg‘𝐶) = (invg‘𝐶) | |
18 | 4, 16, 17 | issubg2 19129 | . . 3 ⊢ (𝐶 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆)))) |
19 | 14, 15, 18 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆)))) |
20 | 7, 9, 13, 19 | mpbir3and 1339 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 {crab 3420 ⊆ wss 3947 ∅c0 4323 ‘cfv 6544 (class class class)co 7414 Fincfn 8964 ℕcn 12256 Basecbs 17206 +gcplusg 17259 0gc0g 17447 Grpcgrp 18921 invgcminusg 18922 SubGrpcsubg 19108 1rcur 20158 Ringcrg 20210 Poly1cpl1 22160 coe1cco1 22161 Mat cmat 22393 ConstPolyMat ccpmat 22691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-ofr 7681 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-sup 9476 df-oi 9544 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-fz 13531 df-fzo 13674 df-seq 14014 df-hash 14341 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-hom 17283 df-cco 17284 df-0g 17449 df-gsum 17450 df-prds 17455 df-pws 17457 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-srg 20164 df-ring 20212 df-subrng 20522 df-subrg 20547 df-lmod 20832 df-lss 20903 df-sra 21145 df-rgmod 21146 df-dsmm 21724 df-frlm 21739 df-ascl 21847 df-psr 21900 df-mvr 21901 df-mpl 21902 df-opsr 21904 df-psr1 22163 df-vr1 22164 df-ply1 22165 df-coe1 22166 df-mamu 22377 df-mat 22394 df-cpmat 22694 |
This theorem is referenced by: cpmatsrgpmat 22709 0elcpmat 22710 m2cpmghm 22732 chfacfisfcpmat 22843 |
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