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Mirrors > Home > MPE Home > Th. List > nlmngp2 | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
nlmngp2 | ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmnrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | nlmnrg 24640 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
3 | nrgngp 24623 | . 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 Scalarcsca 17239 NrmGrpcngp 24530 NrmRingcnrg 24532 NrmModcnlm 24533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-nrg 24538 df-nlm 24539 |
This theorem is referenced by: nlmdsdir 24643 nlmmul0or 24644 nlmvscnlem2 24646 nlmvscnlem1 24647 nlmvscn 24648 |
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