Theorem dihrnlss

Description: The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014)

	Ref	Expression
Hypotheses	dihsslss.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihsslss.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihsslss.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihsslss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion	dihrnlss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → 𝑋 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		dihsslss.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dihsslss.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		dihsslss.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4		dihsslss.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
5	1 2 3 4	dihsslss	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ran 𝐼 ⊆ 𝑆 )
6	5	sselda	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → 𝑋 ∈ 𝑆 )