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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 22891 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
3 | nrgngp 22874 | . 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 1601 ∈ wcel 2106 ‘cfv 6135 Scalarcsca 16341 NrmGrpcngp 22790 NrmRingcnrg 22792 NrmModcnlm 22793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-nrg 22798 df-nlm 22799 |
This theorem is referenced by: nlmdsdir 22894 nlmmul0or 22895 nlmvscnlem2 22897 nlmvscnlem1 22898 nlmvscn 22899 |
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