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Theorem nlmnrg 23285
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 2798 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2798 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2798 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 nlmnrg.1 . . . 4 𝐹 = (Scalar‘𝑊)
5 eqid 2798 . . . 4 (Base‘𝐹) = (Base‘𝐹)
6 eqid 2798 . . . 4 (norm‘𝐹) = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 23281 . . 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 501 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
98simp3d 1141 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2111  wral 3106  cfv 6324  (class class class)co 7135   · cmul 10531  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561  LModclmod 19627  normcnm 23183  NrmGrpcngp 23184  NrmRingcnrg 23186  NrmModcnlm 23187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-nlm 23193
This theorem is referenced by:  nlmngp2  23286  nlmtlm  23300  nvctvc  23306  lssnlm  23307
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