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Mirrors > Home > MPE Home > Th. List > nlmnrg | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
nlmnrg | β’ (π β NrmMod β πΉ β NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . . 4 β’ (normβπ) = (normβπ) | |
3 | eqid 2733 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | nlmnrg.1 | . . . 4 β’ πΉ = (Scalarβπ) | |
5 | eqid 2733 | . . . 4 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | eqid 2733 | . . . 4 β’ (normβπΉ) = (normβπΉ) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 24062 | . . 3 β’ (π β NrmMod β ((π β NrmGrp β§ π β LMod β§ πΉ β NrmRing) β§ βπ₯ β (BaseβπΉ)βπ¦ β (Baseβπ)((normβπ)β(π₯( Β·π βπ)π¦)) = (((normβπΉ)βπ₯) Β· ((normβπ)βπ¦)))) |
8 | 7 | simplbi 499 | . 2 β’ (π β NrmMod β (π β NrmGrp β§ π β LMod β§ πΉ β NrmRing)) |
9 | 8 | simp3d 1145 | 1 β’ (π β NrmMod β πΉ β NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6500 (class class class)co 7361 Β· cmul 11064 Basecbs 17091 Scalarcsca 17144 Β·π cvsca 17145 LModclmod 20365 normcnm 23955 NrmGrpcngp 23956 NrmRingcnrg 23958 NrmModcnlm 23959 |
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 5267 |
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 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 df-nlm 23965 |
This theorem is referenced by: nlmngp2 24067 nlmtlm 24081 nvctvc 24087 lssnlm 24088 |
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