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Mirrors > Home > MPE Home > Th. List > mat0dimscm | Structured version Visualization version GIF version |
Description: The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
mat0dim.a | β’ π΄ = (β Mat π ) |
Ref | Expression |
---|---|
mat0dimscm | β’ ((π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ΄)β ) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . 2 β’ ((π β Ring β§ π β (Baseβπ )) β π β Ring) | |
2 | 0fin 9192 | . . . 4 β’ β β Fin | |
3 | mat0dim.a | . . . . 5 β’ π΄ = (β Mat π ) | |
4 | 3 | matlmod 22347 | . . . 4 β’ ((β β Fin β§ π β Ring) β π΄ β LMod) |
5 | 2, 1, 4 | sylancr 585 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β π΄ β LMod) |
6 | 3 | matsca2 22338 | . . . . . . 7 β’ ((β β Fin β§ π β Ring) β π = (Scalarβπ΄)) |
7 | 2, 6 | mpan 688 | . . . . . 6 β’ (π β Ring β π = (Scalarβπ΄)) |
8 | 7 | fveq2d 6895 | . . . . 5 β’ (π β Ring β (Baseβπ ) = (Baseβ(Scalarβπ΄))) |
9 | 8 | eleq2d 2811 | . . . 4 β’ (π β Ring β (π β (Baseβπ ) β π β (Baseβ(Scalarβπ΄)))) |
10 | 9 | biimpa 475 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β π β (Baseβ(Scalarβπ΄))) |
11 | 0ex 5302 | . . . . . 6 β’ β β V | |
12 | 11 | snid 4660 | . . . . 5 β’ β β {β } |
13 | 3 | fveq2i 6894 | . . . . . 6 β’ (Baseβπ΄) = (Baseβ(β Mat π )) |
14 | mat0dimbas0 22384 | . . . . . 6 β’ (π β Ring β (Baseβ(β Mat π )) = {β }) | |
15 | 13, 14 | eqtrid 2777 | . . . . 5 β’ (π β Ring β (Baseβπ΄) = {β }) |
16 | 12, 15 | eleqtrrid 2832 | . . . 4 β’ (π β Ring β β β (Baseβπ΄)) |
17 | 16 | adantr 479 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β β β (Baseβπ΄)) |
18 | eqid 2725 | . . . 4 β’ (Baseβπ΄) = (Baseβπ΄) | |
19 | eqid 2725 | . . . 4 β’ (Scalarβπ΄) = (Scalarβπ΄) | |
20 | eqid 2725 | . . . 4 β’ ( Β·π βπ΄) = ( Β·π βπ΄) | |
21 | eqid 2725 | . . . 4 β’ (Baseβ(Scalarβπ΄)) = (Baseβ(Scalarβπ΄)) | |
22 | 18, 19, 20, 21 | lmodvscl 20763 | . . 3 β’ ((π΄ β LMod β§ π β (Baseβ(Scalarβπ΄)) β§ β β (Baseβπ΄)) β (π( Β·π βπ΄)β ) β (Baseβπ΄)) |
23 | 5, 10, 17, 22 | syl3anc 1368 | . 2 β’ ((π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ΄)β ) β (Baseβπ΄)) |
24 | 15 | eleq2d 2811 | . . 3 β’ (π β Ring β ((π( Β·π βπ΄)β ) β (Baseβπ΄) β (π( Β·π βπ΄)β ) β {β })) |
25 | elsni 4641 | . . 3 β’ ((π( Β·π βπ΄)β ) β {β } β (π( Β·π βπ΄)β ) = β ) | |
26 | 24, 25 | biimtrdi 252 | . 2 β’ (π β Ring β ((π( Β·π βπ΄)β ) β (Baseβπ΄) β (π( Β·π βπ΄)β ) = β )) |
27 | 1, 23, 26 | sylc 65 | 1 β’ ((π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ΄)β ) = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β c0 4318 {csn 4624 βcfv 6542 (class class class)co 7415 Fincfn 8960 Basecbs 17177 Scalarcsca 17233 Β·π cvsca 17234 Ringcrg 20175 LModclmod 20745 Mat cmat 22323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-prds 17426 df-pws 17428 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-subrg 20510 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-mat 22324 |
This theorem is referenced by: mat0scmat 22456 chpmat0d 22752 |
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