![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > assalmod | Structured version Visualization version GIF version |
Description: An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.) |
Ref | Expression |
---|---|
assalmod | β’ (π β AssAlg β π β LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2732 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
3 | eqid 2732 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
4 | eqid 2732 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
5 | eqid 2732 | . . . 4 β’ (.rβπ) = (.rβπ) | |
6 | 1, 2, 3, 4, 5 | isassa 21417 | . . 3 β’ (π β AssAlg β ((π β LMod β§ π β Ring) β§ βπ§ β (Baseβ(Scalarβπ))βπ₯ β (Baseβπ)βπ¦ β (Baseβπ)(((π§( Β·π βπ)π₯)(.rβπ)π¦) = (π§( Β·π βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π§( Β·π βπ)π¦)) = (π§( Β·π βπ)(π₯(.rβπ)π¦))))) |
7 | 6 | simplbi 498 | . 2 β’ (π β AssAlg β (π β LMod β§ π β Ring)) |
8 | 7 | simpld 495 | 1 β’ (π β AssAlg β π β LMod) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 βcfv 6543 (class class class)co 7411 Basecbs 17146 .rcmulr 17200 Scalarcsca 17202 Β·π cvsca 17203 Ringcrg 20058 LModclmod 20475 AssAlgcasa 21411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-assa 21414 |
This theorem is referenced by: assasca 21423 assa2ass 21424 issubassa3 21426 issubassa 21427 assapropd 21432 aspval 21433 asplss 21434 ascldimul 21448 asclrhm 21450 rnascl 21451 issubassa2 21452 aspval2 21458 assamulgscmlem1 21459 assamulgscmlem2 21460 mplmon2mul 21636 mplind 21637 matinv 22186 asclmulg 32680 assaascl0 47139 assaascl1 47140 |
Copyright terms: Public domain | W3C validator |