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Mirrors > Home > MPE Home > Th. List > nlmnrg | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
nlmnrg | ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2736 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
3 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | nlmnrg.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | eqid 2736 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2736 | . . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 23527 | . . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘𝐹)‘𝑥) · ((norm‘𝑊)‘𝑦)))) |
8 | 7 | simplbi 501 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing)) |
9 | 8 | simp3d 1146 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ‘cfv 6358 (class class class)co 7191 · cmul 10699 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 LModclmod 19853 normcnm 23428 NrmGrpcngp 23429 NrmRingcnrg 23431 NrmModcnlm 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-nlm 23438 |
This theorem is referenced by: nlmngp2 23532 nlmtlm 23546 nvctvc 23552 lssnlm 23553 |
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