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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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | eqid 2821 | . . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊)) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 23278 | . . 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 500 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing)) |
9 | 8 | simp2d 1139 | 1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6349 (class class class)co 7150 · cmul 10536 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 LModclmod 19628 normcnm 23180 NrmGrpcngp 23181 NrmRingcnrg 23183 NrmModcnlm 23184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-nlm 23190 |
This theorem is referenced by: nlmdsdi 23284 nlmdsdir 23285 nlmmul0or 23286 nlmvscnlem2 23288 nlmvscn 23290 nlmtlm 23297 nvclmod 23301 isnvc2 23302 lssnlm 23304 ngpocelbl 23307 idnmhm 23357 0nmhm 23358 nmhmplusg 23360 nmhmcn 23718 cphlmod 23772 bnlmod 23940 nmmulg 31204 |
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