![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > assaring | Structured version Visualization version GIF version |
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.) |
Ref | Expression |
---|---|
assaring | β’ (π β AssAlg β π β Ring) |
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 21278 | . . 3 β’ (π β AssAlg β ((π β LMod β§ π β Ring β§ (Scalarβπ) β CRing) 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 β§ (Scalarβπ) β CRing)) |
8 | 7 | simp2d 1144 | 1 β’ (π β AssAlg β π β Ring) |
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: issubassa 21288 assapropd 21291 aspval 21292 asclmul1 21305 asclmul2 21306 ascldimul 21307 asclrhm 21309 rnascl 21310 aspval2 21317 assamulgscmlem1 21318 assamulgscmlem2 21319 zlmassa 21321 mplind 21494 evlseu 21509 pf1subrg 21730 matinv 22042 asclmulg 32311 irngnzply1lem 32421 assaascl0 46546 assaascl1 46547 |
Copyright terms: Public domain | W3C validator |