Theorem nlmnrg 24700

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 2737	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2737	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		nlmnrg.1	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		eqid 2737	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
6		eqid 2737	. . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 24696	. . 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 1145	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ‘cfv 6561  (class class class)co 7431   · cmul 11160  Basecbs 17247  Scalarcsca 17300   ·_𝑠 cvsca 17301  LModclmod 20858  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592  NrmModcnlm 24593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-nlm 24599

This theorem is referenced by:  nlmngp2  24701  nlmtlm  24715  nvctvc  24721  lssnlm  24722