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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 | ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2818 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2818 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2818 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | eqid 2818 | . . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊)) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 23211 | . . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘(Scalar‘𝑊))‘𝑥) · ((norm‘𝑊)‘𝑦)))) |
8 | 7 | simplbi 498 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing)) |
9 | 8 | simp2d 1135 | 1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 · cmul 10530 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 LModclmod 19563 normcnm 23113 NrmGrpcngp 23114 NrmRingcnrg 23116 NrmModcnlm 23117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-nlm 23123 |
This theorem is referenced by: nlmdsdi 23217 nlmdsdir 23218 nlmmul0or 23219 nlmvscnlem2 23221 nlmvscn 23223 nlmtlm 23230 nvclmod 23234 isnvc2 23235 lssnlm 23237 ngpocelbl 23240 idnmhm 23290 0nmhm 23291 nmhmplusg 23293 nmhmcn 23651 cphlmod 23705 bnlmod 23873 nmmulg 31108 |
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