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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 24715 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
3 | nrgngp 24698 | . 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 1536 ∈ wcel 2105 ‘cfv 6562 Scalarcsca 17300 NrmGrpcngp 24605 NrmRingcnrg 24607 NrmModcnlm 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-nrg 24613 df-nlm 24614 |
This theorem is referenced by: nlmdsdir 24718 nlmmul0or 24719 nlmvscnlem2 24721 nlmvscnlem1 24722 nlmvscn 24723 |
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