Theorem dihcl

Description: Closure of isomorphism H. (Contributed by NM, 8-Mar-2014)

	Ref	Expression
Hypotheses	dihfn.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihfn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihfn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihcl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dihfn.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihfn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dihfn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		eqid	⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
6	1 2 3 4 5	dihf11	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
7	6	adantr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐼 : 𝐵 –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
8		f1fn	⊢ ( 𝐼 : 𝐵 –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) → 𝐼 Fn 𝐵 )
9	7 8	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐼 Fn 𝐵 )
10		fnfvelrn	⊢ ( ( 𝐼 Fn 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
11	9 10	sylancom	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )