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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 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
| 3 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | nlmnrg.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 5 | eqid 2737 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 6 | eqid 2737 | . . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
| 7 | 1, 2, 3, 4, 5, 6 | isnlm 24696 | . . 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 497 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing)) |
| 9 | 8 | simp3d 1145 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 · cmul 11160 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 NrmModcnlm 24593 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-nlm 24599 |
| This theorem is referenced by: nlmngp2 24701 nlmtlm 24715 nvctvc 24721 lssnlm 24722 |
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