Theorem nlmnrg 24959

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 2760	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2760	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2760	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		nlmnrg.1	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		eqid 2760	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
6		eqid 2760	. . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 24955	. . 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 502	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing))
9	8	simp3d 1162	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ‘cfv 6527  (class class class)co 7408   · cmul 11176  Basecbs 17348  Scalarcsca 17392   ·_𝑠 cvsca 17393  LModclmod 21096  normcnm 24856  NrmGrpcngp 24857  NrmRingcnrg 24859  NrmModcnlm 24860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-ov 7411  df-nlm 24866

This theorem is used by:  nlmngp2  24960  nlmtlm  24974  nvctvc  24980  lssnlm  24981