![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > matlmod | Structured version Visualization version GIF version |
Description: The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matlmod.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
Ref | Expression |
---|---|
matlmod | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqxpexg 7005 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ V) | |
2 | eqid 2651 | . . . . 5 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
3 | 2 | frlmlmod 20141 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 × 𝑁) ∈ V) → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ LMod) |
4 | 3 | ancoms 468 | . . 3 ⊢ (((𝑁 × 𝑁) ∈ V ∧ 𝑅 ∈ Ring) → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ LMod) |
5 | 1, 4 | sylan 487 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ LMod) |
6 | eqidd 2652 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))) | |
7 | matlmod.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | 7, 2 | matbas 20267 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴)) |
9 | 7, 2 | matplusg 20268 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴)) |
10 | 9 | oveqdr 6714 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))) → (𝑥(+g‘(𝑅 freeLMod (𝑁 × 𝑁)))𝑦) = (𝑥(+g‘𝐴)𝑦)) |
11 | eqidd 2652 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) | |
12 | 7, 2 | matsca 20269 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Scalar‘𝐴)) |
13 | eqid 2651 | . . 3 ⊢ (Base‘(Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) = (Base‘(Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) | |
14 | 7, 2 | matvsca 20270 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ( ·𝑠 ‘(𝑅 freeLMod (𝑁 × 𝑁))) = ( ·𝑠 ‘𝐴)) |
15 | 14 | oveqdr 6714 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘(Scalar‘(𝑅 freeLMod (𝑁 × 𝑁)))) ∧ 𝑦 ∈ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))) → (𝑥( ·𝑠 ‘(𝑅 freeLMod (𝑁 × 𝑁)))𝑦) = (𝑥( ·𝑠 ‘𝐴)𝑦)) |
16 | 6, 8, 10, 11, 12, 13, 15 | lmodpropd 18974 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑅 freeLMod (𝑁 × 𝑁)) ∈ LMod ↔ 𝐴 ∈ LMod)) |
17 | 5, 16 | mpbid 222 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 × cxp 5141 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 Basecbs 15904 +gcplusg 15988 Scalarcsca 15991 ·𝑠 cvsca 15992 Ringcrg 18593 LModclmod 18911 freeLMod cfrlm 20138 Mat cmat 20261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-lmod 18913 df-lss 18981 df-sra 19220 df-rgmod 19221 df-dsmm 20124 df-frlm 20139 df-mat 20262 |
This theorem is referenced by: matgrp 20284 matvscl 20285 matassa 20298 mat0dimscm 20323 scmatid 20368 scmataddcl 20370 scmatsubcl 20371 smatvscl 20378 scmatghm 20387 scmatmhm 20388 pmatlmod 20547 pm2mp 20678 chpmat1dlem 20688 chpmat1d 20689 cpmidpmatlem3 20725 cpmadugsumlemB 20727 cpmadugsumlemC 20728 chcoeffeqlem 20738 |
Copyright terms: Public domain | W3C validator |