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Mirrors > Home > MPE Home > Th. List > assasca | Structured version Visualization version GIF version |
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
assasca.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
assasca | β’ (π β AssAlg β πΉ β CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | assasca.f | . . . 4 β’ πΉ = (Scalarβπ) | |
3 | eqid 2733 | . . . 4 β’ (BaseβπΉ) = (BaseβπΉ) | |
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 21278 | . . 3 β’ (π β AssAlg β ((π β LMod β§ π β Ring β§ πΉ β CRing) β§ βπ§ β (BaseβπΉ)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 β§ πΉ β CRing)) |
8 | 7 | simp3d 1145 | 1 β’ (π β AssAlg β πΉ β CRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 Ringcrg 19969 CRingccrg 19970 LModclmod 20336 AssAlgcasa 21272 |
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 5264 |
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 2941 df-ral 3062 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-assa 21275 |
This theorem is referenced by: assa2ass 21285 issubassa3 21287 asclrhm 21309 assamulgscmlem2 21319 asclmulg 32311 |
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