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Mirrors > Home > MPE Home > Th. List > nlmlmod | Structured version Visualization version GIF version |
Description: A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmlmod | β’ (π β NrmMod β π β LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2730 | . . . 4 β’ (normβπ) = (normβπ) | |
3 | eqid 2730 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | eqid 2730 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2730 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | eqid 2730 | . . . 4 β’ (normβ(Scalarβπ)) = (normβ(Scalarβπ)) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 24412 | . . 3 β’ (π β NrmMod β ((π β NrmGrp β§ π β LMod β§ (Scalarβπ) β NrmRing) β§ βπ₯ β (Baseβ(Scalarβπ))βπ¦ β (Baseβπ)((normβπ)β(π₯( Β·π βπ)π¦)) = (((normβ(Scalarβπ))βπ₯) Β· ((normβπ)βπ¦)))) |
8 | 7 | simplbi 496 | . 2 β’ (π β NrmMod β (π β NrmGrp β§ π β LMod β§ (Scalarβπ) β NrmRing)) |
9 | 8 | simp2d 1141 | 1 β’ (π β NrmMod β π β LMod) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 βwral 3059 βcfv 6542 (class class class)co 7411 Β· cmul 11117 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 LModclmod 20614 normcnm 24305 NrmGrpcngp 24306 NrmRingcnrg 24308 NrmModcnlm 24309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-nlm 24315 |
This theorem is referenced by: nlmdsdi 24418 nlmdsdir 24419 nlmmul0or 24420 nlmvscnlem2 24422 nlmvscn 24424 nlmtlm 24431 nvclmod 24435 isnvc2 24436 lssnlm 24438 ngpocelbl 24441 idnmhm 24491 0nmhm 24492 nmhmplusg 24494 nmhmcn 24867 cphlmod 24922 bnlmod 25091 nmmulg 33246 |
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