Theorem nlmnrg 24721

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 2740	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2740	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2740	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		nlmnrg.1	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		eqid 2740	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
6		eqid 2740	. . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 24717	. . 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 1144	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6573  (class class class)co 7448   · cmul 11189  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  LModclmod 20880  normcnm 24610  NrmGrpcngp 24611  NrmRingcnrg 24613  NrmModcnlm 24614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-nlm 24620

This theorem is referenced by:  nlmngp2  24722  nlmtlm  24736  nvctvc  24742  lssnlm  24743