MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlmngp2 Structured version   Visualization version   GIF version

Theorem nlmngp2 22892
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
nlmngp2 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Proof of Theorem nlmngp2
StepHypRef Expression
1 nlmnrg.1 . . 3 𝐹 = (Scalar‘𝑊)
21nlmnrg 22891 . 2 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3 nrgngp 22874 . 2 (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
42, 3syl 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
  Copyright terms: Public domain W3C validator