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Mirrors > Home > MPE Home > Th. List > scmatlss | Structured version Visualization version GIF version |
Description: The set of scalar matrices is a linear subspace of the matrix algebra. (Contributed by AV, 25-Dec-2019.) |
Ref | Expression |
---|---|
scmatlss.a | β’ π΄ = (π Mat π ) |
scmatlss.s | β’ π = (π ScMat π ) |
Ref | Expression |
---|---|
scmatlss | β’ ((π β Fin β§ π β Ring) β π β (LSubSpβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatlss.a | . . 3 β’ π΄ = (π Mat π ) | |
2 | 1 | matsca2 21792 | . 2 β’ ((π β Fin β§ π β Ring) β π = (Scalarβπ΄)) |
3 | eqidd 2734 | . 2 β’ ((π β Fin β§ π β Ring) β (Baseβπ ) = (Baseβπ )) | |
4 | eqidd 2734 | . 2 β’ ((π β Fin β§ π β Ring) β (Baseβπ΄) = (Baseβπ΄)) | |
5 | eqidd 2734 | . 2 β’ ((π β Fin β§ π β Ring) β (+gβπ΄) = (+gβπ΄)) | |
6 | eqidd 2734 | . 2 β’ ((π β Fin β§ π β Ring) β ( Β·π βπ΄) = ( Β·π βπ΄)) | |
7 | eqidd 2734 | . 2 β’ ((π β Fin β§ π β Ring) β (LSubSpβπ΄) = (LSubSpβπ΄)) | |
8 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
9 | eqid 2733 | . . . 4 β’ (Baseβπ΄) = (Baseβπ΄) | |
10 | eqid 2733 | . . . 4 β’ (1rβπ΄) = (1rβπ΄) | |
11 | eqid 2733 | . . . 4 β’ ( Β·π βπ΄) = ( Β·π βπ΄) | |
12 | scmatlss.s | . . . 4 β’ π = (π ScMat π ) | |
13 | 8, 1, 9, 10, 11, 12 | scmatval 21876 | . . 3 β’ ((π β Fin β§ π β Ring) β π = {π β (Baseβπ΄) β£ βπ β (Baseβπ )π = (π( Β·π βπ΄)(1rβπ΄))}) |
14 | ssrab2 4041 | . . 3 β’ {π β (Baseβπ΄) β£ βπ β (Baseβπ )π = (π( Β·π βπ΄)(1rβπ΄))} β (Baseβπ΄) | |
15 | 13, 14 | eqsstrdi 4002 | . 2 β’ ((π β Fin β§ π β Ring) β π β (Baseβπ΄)) |
16 | eqid 2733 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
17 | 1, 9, 8, 16, 12 | scmatid 21886 | . . 3 β’ ((π β Fin β§ π β Ring) β (1rβπ΄) β π) |
18 | 17 | ne0d 4299 | . 2 β’ ((π β Fin β§ π β Ring) β π β β ) |
19 | 8, 1, 12, 11 | smatvscl 21896 | . . . . 5 β’ (((π β Fin β§ π β Ring) β§ (π β (Baseβπ ) β§ π₯ β π)) β (π( Β·π βπ΄)π₯) β π) |
20 | 19 | 3adantr3 1172 | . . . 4 β’ (((π β Fin β§ π β Ring) β§ (π β (Baseβπ ) β§ π₯ β π β§ π¦ β π)) β (π( Β·π βπ΄)π₯) β π) |
21 | simpr3 1197 | . . . 4 β’ (((π β Fin β§ π β Ring) β§ (π β (Baseβπ ) β§ π₯ β π β§ π¦ β π)) β π¦ β π) | |
22 | 20, 21 | jca 513 | . . 3 β’ (((π β Fin β§ π β Ring) β§ (π β (Baseβπ ) β§ π₯ β π β§ π¦ β π)) β ((π( Β·π βπ΄)π₯) β π β§ π¦ β π)) |
23 | 1, 9, 8, 16, 12 | scmataddcl 21888 | . . 3 β’ (((π β Fin β§ π β Ring) β§ ((π( Β·π βπ΄)π₯) β π β§ π¦ β π)) β ((π( Β·π βπ΄)π₯)(+gβπ΄)π¦) β π) |
24 | 22, 23 | syldan 592 | . 2 β’ (((π β Fin β§ π β Ring) β§ (π β (Baseβπ ) β§ π₯ β π β§ π¦ β π)) β ((π( Β·π βπ΄)π₯)(+gβπ΄)π¦) β π) |
25 | 2, 3, 4, 5, 6, 7, 15, 18, 24 | islssd 20440 | 1 β’ ((π β Fin β§ π β Ring) β π β (LSubSpβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwrex 3070 {crab 3406 βcfv 6500 (class class class)co 7361 Fincfn 8889 Basecbs 17091 +gcplusg 17141 Β·π cvsca 17145 0gc0g 17329 1rcur 19921 Ringcrg 19972 LSubSpclss 20436 Mat cmat 21777 ScMat cscmat 21861 |
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-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-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-ring 19974 df-subrg 20262 df-lmod 20367 df-lss 20437 df-sra 20678 df-rgmod 20679 df-dsmm 21161 df-frlm 21176 df-mamu 21756 df-mat 21778 df-scmat 21863 |
This theorem is referenced by: scmatghm 21905 |
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