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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 2734 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmat 21578 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | ssrab2 3983 | . . 3 ⊢ {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} ⊆ (Base‘𝐶) | |
7 | 5, 6 | eqsstrdi 3945 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
8 | 1, 2, 3 | 1elcpmat 21584 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
9 | 8 | ne0d 4240 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅) |
10 | 1, 2, 3 | cpmatacl 21585 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆) |
11 | 1, 2, 3 | cpmatinvcl 21586 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆) |
12 | r19.26 3085 | . . 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 586 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆)) |
14 | 2, 3 | pmatring 21561 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
15 | ringgrp 19539 | . . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp) | |
16 | eqid 2734 | . . . 4 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
17 | eqid 2734 | . . . 4 ⊢ (invg‘𝐶) = (invg‘𝐶) | |
18 | 4, 16, 17 | issubg2 18530 | . . 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 1344 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 {crab 3058 ⊆ wss 3857 ∅c0 4227 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 ℕcn 11813 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Grpcgrp 18337 invgcminusg 18338 SubGrpcsubg 18509 1rcur 19488 Ringcrg 19534 Poly1cpl1 21070 coe1cco1 21071 Mat cmat 21276 ConstPolyMat ccpmat 21572 |
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-ofr 7459 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-pm 8500 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-srg 19493 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-ascl 20789 df-psr 20840 df-mvr 20841 df-mpl 20842 df-opsr 20844 df-psr1 21073 df-vr1 21074 df-ply1 21075 df-coe1 21076 df-mamu 21255 df-mat 21277 df-cpmat 21575 |
This theorem is referenced by: cpmatsrgpmat 21590 0elcpmat 21591 m2cpmghm 21613 m2cpmrngiso 21627 chfacfisfcpmat 21724 |
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