Theorem nlmnrg 24625

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 2735	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2735	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2735	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		nlmnrg.1	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		eqid 2735	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
6		eqid 2735	. . . 4 ⊢ (norm‘𝐹) = (norm‘𝐹)
7	1, 2, 3, 4, 5, 6	isnlm 24621	. . 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 1542   ∈ wcel 2114  ∀wral 3050  ‘cfv 6491  (class class class)co 7358   · cmul 11033  Basecbs 17138  Scalarcsca 17182   ·_𝑠 cvsca 17183  LModclmod 20813  normcnm 24522  NrmGrpcngp 24523  NrmRingcnrg 24525  NrmModcnlm 24526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-nul 5250

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499  df-ov 7361  df-nlm 24532

This theorem is referenced by:  nlmngp2  24626  nlmtlm  24640  nvctvc  24646  lssnlm  24647