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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 2818 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2818 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | nlmnrg.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | eqid 2818 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2818 | . . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isnlm 23211 | . . 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 498 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing)) |
9 | 8 | simp3d 1136 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
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: nlmngp2 23216 nlmtlm 23230 nvctvc 23236 lssnlm 23237 |
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