Theorem lsslmod

Description: A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014)

	Ref	Expression
Hypotheses	lsslss.x	⊢ 𝑋 = ( 𝑊 ↾s 𝑈 )
	lsslss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
Assertion	lsslmod	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → 𝑋 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		lsslss.x	⊢ 𝑋 = ( 𝑊 ↾s 𝑈 )
2		lsslss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4	1 3 2	islss3	⊢ ( 𝑊 ∈ LMod → ( 𝑈 ∈ 𝑆 ↔ ( 𝑈 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑋 ∈ LMod ) ) )
5	4	simplbda	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → 𝑋 ∈ LMod )