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Theorem nlmnrg 23215
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 2818 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2818 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2818 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 nlmnrg.1 . . . 4 𝐹 = (Scalar‘𝑊)
5 eqid 2818 . . . 4 (Base‘𝐹) = (Base‘𝐹)
6 eqid 2818 . . . 4 (norm‘𝐹) = (norm‘𝐹)
71, 2, 3, 4, 5, 6isnlm 23211 . . 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 498 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
98simp3d 1136 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145   · cmul 10530  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  LModclmod 19563  normcnm 23113  NrmGrpcngp 23114  NrmRingcnrg 23116  NrmModcnlm 23117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-nlm 23123
This theorem is referenced by:  nlmngp2  23216  nlmtlm  23230  nvctvc  23236  lssnlm  23237
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