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Theorem nlmnrg 24700
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 2737 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2737 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2737 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 nlmnrg.1 . . . 4 𝐹 = (Scalar‘𝑊)
5 eqid 2737 . . . 4 (Base‘𝐹) = (Base‘𝐹)
6 eqid 2737 . . . 4 (norm‘𝐹) = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 24696 . . 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 1145 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431   · cmul 11160  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  LModclmod 20858  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592  NrmModcnlm 24593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-nlm 24599
This theorem is referenced by:  nlmngp2  24701  nlmtlm  24715  nvctvc  24721  lssnlm  24722
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