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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 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
3 | eqid 2733 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
4 | eqid 2733 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
5 | eqid 2733 | . . . 4 β’ (.rβπ) = (.rβπ) | |
6 | 1, 2, 3, 4, 5 | isassa 21411 | . . 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 499 | . 2 β’ (π β AssAlg β (π β LMod β§ π β Ring)) |
8 | 7 | simpld 496 | 1 β’ (π β AssAlg β π β LMod) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 βcfv 6544 (class class class)co 7409 Basecbs 17144 .rcmulr 17198 Scalarcsca 17200 Β·π cvsca 17201 Ringcrg 20056 LModclmod 20471 AssAlgcasa 21405 |
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-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-assa 21408 |
This theorem is referenced by: assasca 21417 assa2ass 21418 issubassa3 21420 issubassa 21421 assapropd 21426 aspval 21427 asplss 21428 ascldimul 21442 asclrhm 21444 rnascl 21445 issubassa2 21446 aspval2 21452 assamulgscmlem1 21453 assamulgscmlem2 21454 mplmon2mul 21630 mplind 21631 matinv 22179 asclmulg 32635 assaascl0 47060 assaascl1 47061 |
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