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Mirrors > Home > MPE Home > Th. List > isnvc2 | Structured version Visualization version GIF version |
Description: A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnvc2.1 | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
isnvc2 | β’ (π β NrmVec β (π β NrmMod β§ πΉ β DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvc 24082 | . 2 β’ (π β NrmVec β (π β NrmMod β§ π β LVec)) | |
2 | nlmlmod 24065 | . . . 4 β’ (π β NrmMod β π β LMod) | |
3 | isnvc2.1 | . . . . . 6 β’ πΉ = (Scalarβπ) | |
4 | 3 | islvec 20609 | . . . . 5 β’ (π β LVec β (π β LMod β§ πΉ β DivRing)) |
5 | 4 | baib 537 | . . . 4 β’ (π β LMod β (π β LVec β πΉ β DivRing)) |
6 | 2, 5 | syl 17 | . . 3 β’ (π β NrmMod β (π β LVec β πΉ β DivRing)) |
7 | 6 | pm5.32i 576 | . 2 β’ ((π β NrmMod β§ π β LVec) β (π β NrmMod β§ πΉ β DivRing)) |
8 | 1, 7 | bitri 275 | 1 β’ (π β NrmVec β (π β NrmMod β§ πΉ β DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6500 Scalarcsca 17144 DivRingcdr 20219 LModclmod 20365 LVecclvec 20607 NrmModcnlm 23959 NrmVeccnvc 23960 |
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-lvec 20608 df-nlm 23965 df-nvc 23966 |
This theorem is referenced by: lssnvc 24089 srabn 24747 |
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