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Mirrors > Home > MPE Home > Th. List > nlmngp | Structured version Visualization version GIF version |
Description: A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmngp | β’ (π β NrmMod β π β NrmGrp) |
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 24062 | . . 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 | simp1d 1143 | 1 β’ (π β NrmMod β π β NrmGrp) |
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: nlmdsdi 24068 nlmdsdir 24069 nlmmul0or 24070 nlmvscnlem2 24072 nlmvscnlem1 24073 nlmvscn 24074 nlmtlm 24081 lssnlm 24088 ngpocelbl 24091 isnmhm2 24139 idnmhm 24141 0nmhm 24142 nmoleub2lem 24500 nmoleub2lem3 24501 nmoleub2lem2 24502 nmoleub3 24505 nmhmcn 24506 ncvsm1 24541 ncvsdif 24542 ncvspi 24543 ncvs1 24544 ncvspds 24548 cphngp 24560 ipcnlem2 24631 ipcnlem1 24632 csscld 24636 bnngp 24729 cssbn 24762 |
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