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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 2733 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmat 22081 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) |
6 | ssrab2 4041 | . . 3 ⊢ {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)} ⊆ (Base‘𝐶) | |
7 | 5, 6 | eqsstrdi 4002 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
8 | 1, 2, 3 | 1elcpmat 22087 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) |
9 | 8 | ne0d 4299 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅) |
10 | 1, 2, 3 | cpmatacl 22088 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆) |
11 | 1, 2, 3 | cpmatinvcl 22089 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆) |
12 | r19.26 3111 | . . 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 584 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆 ∧ ((invg‘𝐶)‘𝑥) ∈ 𝑆)) |
14 | 2, 3 | pmatring 22064 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
15 | ringgrp 19977 | . . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp) | |
16 | eqid 2733 | . . . 4 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
17 | eqid 2733 | . . . 4 ⊢ (invg‘𝐶) = (invg‘𝐶) | |
18 | 4, 16, 17 | issubg2 18951 | . . 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 1343 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 {crab 3406 ⊆ wss 3914 ∅c0 4286 ‘cfv 6500 (class class class)co 7361 Fincfn 8889 ℕcn 12161 Basecbs 17091 +gcplusg 17141 0gc0g 17329 Grpcgrp 18756 invgcminusg 18757 SubGrpcsubg 18930 1rcur 19921 Ringcrg 19972 Poly1cpl1 21571 coe1cco1 21572 Mat cmat 21777 ConstPolyMat ccpmat 22075 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-srg 19926 df-ring 19974 df-subrg 20262 df-lmod 20367 df-lss 20437 df-sra 20678 df-rgmod 20679 df-dsmm 21161 df-frlm 21176 df-ascl 21284 df-psr 21334 df-mvr 21335 df-mpl 21336 df-opsr 21338 df-psr1 21574 df-vr1 21575 df-ply1 21576 df-coe1 21577 df-mamu 21756 df-mat 21778 df-cpmat 22078 |
This theorem is referenced by: cpmatsrgpmat 22093 0elcpmat 22094 m2cpmghm 22116 chfacfisfcpmat 22227 |
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