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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 482 | . 2 β’ ((π β Ring β§ π β (Baseβπ )) β π β Ring) | |
2 | 0fin 9173 | . . . 4 β’ β β Fin | |
3 | mat0dim.a | . . . . 5 β’ π΄ = (β Mat π ) | |
4 | 3 | matlmod 22286 | . . . 4 β’ ((β β Fin β§ π β Ring) β π΄ β LMod) |
5 | 2, 1, 4 | sylancr 586 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β π΄ β LMod) |
6 | 3 | matsca2 22277 | . . . . . . 7 β’ ((β β Fin β§ π β Ring) β π = (Scalarβπ΄)) |
7 | 2, 6 | mpan 687 | . . . . . 6 β’ (π β Ring β π = (Scalarβπ΄)) |
8 | 7 | fveq2d 6889 | . . . . 5 β’ (π β Ring β (Baseβπ ) = (Baseβ(Scalarβπ΄))) |
9 | 8 | eleq2d 2813 | . . . 4 β’ (π β Ring β (π β (Baseβπ ) β π β (Baseβ(Scalarβπ΄)))) |
10 | 9 | biimpa 476 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β π β (Baseβ(Scalarβπ΄))) |
11 | 0ex 5300 | . . . . . 6 β’ β β V | |
12 | 11 | snid 4659 | . . . . 5 β’ β β {β } |
13 | 3 | fveq2i 6888 | . . . . . 6 β’ (Baseβπ΄) = (Baseβ(β Mat π )) |
14 | mat0dimbas0 22323 | . . . . . 6 β’ (π β Ring β (Baseβ(β Mat π )) = {β }) | |
15 | 13, 14 | eqtrid 2778 | . . . . 5 β’ (π β Ring β (Baseβπ΄) = {β }) |
16 | 12, 15 | eleqtrrid 2834 | . . . 4 β’ (π β Ring β β β (Baseβπ΄)) |
17 | 16 | adantr 480 | . . 3 β’ ((π β Ring β§ π β (Baseβπ )) β β β (Baseβπ΄)) |
18 | eqid 2726 | . . . 4 β’ (Baseβπ΄) = (Baseβπ΄) | |
19 | eqid 2726 | . . . 4 β’ (Scalarβπ΄) = (Scalarβπ΄) | |
20 | eqid 2726 | . . . 4 β’ ( Β·π βπ΄) = ( Β·π βπ΄) | |
21 | eqid 2726 | . . . 4 β’ (Baseβ(Scalarβπ΄)) = (Baseβ(Scalarβπ΄)) | |
22 | 18, 19, 20, 21 | lmodvscl 20724 | . . 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 2813 | . . 3 β’ (π β Ring β ((π( Β·π βπ΄)β ) β (Baseβπ΄) β (π( Β·π βπ΄)β ) β {β })) |
25 | elsni 4640 | . . 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 395 = wceq 1533 β wcel 2098 β c0 4317 {csn 4623 βcfv 6537 (class class class)co 7405 Fincfn 8941 Basecbs 17153 Scalarcsca 17209 Β·π cvsca 17210 Ringcrg 20138 LModclmod 20706 Mat cmat 22262 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-subrg 20471 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-mat 22263 |
This theorem is referenced by: mat0scmat 22395 chpmat0d 22691 |
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