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Theorem nlmnrg 24715
Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
nlmnrg (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

Proof of Theorem nlmnrg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2734 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2734 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 nlmnrg.1 . . . 4 𝐹 = (Scalar‘𝑊)
5 eqid 2734 . . . 4 (Base‘𝐹) = (Base‘𝐹)
6 eqid 2734 . . . 4 (norm‘𝐹) = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 24711 . . 3 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠𝑊)𝑦)) = (((norm‘𝐹)‘𝑥) · ((norm‘𝑊)‘𝑦))))
87simplbi 497 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
98simp3d 1143 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  wral 3058  cfv 6562  (class class class)co 7430   · cmul 11157  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  LModclmod 20874  normcnm 24604  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-nlm 24614
This theorem is referenced by:  nlmngp2  24716  nlmtlm  24730  nvctvc  24736  lssnlm  24737
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