Theorem dihsmsnrn

Description: The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015)

	Ref	Expression
Hypotheses	dihsmsnrn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihsmsnrn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihsmsnrn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	dihsmsnrn.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
	dihsmsnrn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	dihsmsnrn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihsmsnrn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dihsmsnrn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	dihsmsnrn.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	dihsmsnrn	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊕ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ran 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dihsmsnrn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dihsmsnrn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		dihsmsnrn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		dihsmsnrn.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
5		dihsmsnrn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6		dihsmsnrn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7		dihsmsnrn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		dihsmsnrn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
9		dihsmsnrn.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
10	1 2 3 5 6	dihlsprn	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 )
11	7 8 10	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 )
12	1 2 3 4 5 6 7 11 9	dihsmsprn	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊕ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ran 𝐼 )