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Theorem nlmnrg 22477
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 2621 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2621 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 nlmnrg.1 . . . 4 𝐹 = (Scalar‘𝑊)
5 eqid 2621 . . . 4 (Base‘𝐹) = (Base‘𝐹)
6 eqid 2621 . . . 4 (norm‘𝐹) = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 22473 . . 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 476 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
98simp3d 1074 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1037   = wceq 1482  wcel 1989  wral 2911  cfv 5886  (class class class)co 6647   · cmul 9938  Basecbs 15851  Scalarcsca 15938   ·𝑠 cvsca 15939  LModclmod 18857  normcnm 22375  NrmGrpcngp 22376  NrmRingcnrg 22378  NrmModcnlm 22379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-nlm 22385
This theorem is referenced by:  nlmngp2  22478  nlmtlm  22492  nvctvc  22498  lssnlm  22499
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