Theorem nlmnrg 23749

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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2738	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2738	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		nlmnrg.1	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		eqid 2738	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
6		eqid 2738	. . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 23745	. . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘𝐹)‘𝑥) · ((norm‘𝑊)‘𝑦))))
8	7	simplbi 497	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
9	8	simp3d 1142	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255   · cmul 10807  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  LModclmod 20038  normcnm 23638  NrmGrpcngp 23639  NrmRingcnrg 23641  NrmModcnlm 23642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-nlm 23648

This theorem is referenced by:  nlmngp2  23750  nlmtlm  23764  nvctvc  23770  lssnlm  23771