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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 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . . 4 β’ (normβπ) = (normβπ) | |
3 | eqid 2733 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | eqid 2733 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2733 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | eqid 2733 | . . . 4 β’ (normβ(Scalarβπ)) = (normβ(Scalarβπ)) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 24192 | . . 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 499 | . 2 β’ (π β NrmMod β (π β NrmGrp β§ π β LMod β§ (Scalarβπ) β NrmRing)) |
9 | 8 | simp2d 1144 | 1 β’ (π β NrmMod β π β LMod) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 βcfv 6544 (class class class)co 7409 Β· cmul 11115 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 LModclmod 20471 normcnm 24085 NrmGrpcngp 24086 NrmRingcnrg 24088 NrmModcnlm 24089 |
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-nlm 24095 |
This theorem is referenced by: nlmdsdi 24198 nlmdsdir 24199 nlmmul0or 24200 nlmvscnlem2 24202 nlmvscn 24204 nlmtlm 24211 nvclmod 24215 isnvc2 24216 lssnlm 24218 ngpocelbl 24221 idnmhm 24271 0nmhm 24272 nmhmplusg 24274 nmhmcn 24636 cphlmod 24691 bnlmod 24860 nmmulg 32948 |
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