Theorem dih1rn

Description: The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014)

	Ref	Expression
Hypotheses	dih1rn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dih1rn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dih1rn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dih1rn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
Assertion	dih1rn	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 ∈ ran 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dih1rn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dih1rn.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3		dih1rn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		dih1rn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		eqid	⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
6	5 1 2 3 4	dih1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) = 𝑉 )
7		hlop	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
8	7	adantr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP )
9		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10	9 5	op1cl	⊢ ( 𝐾 ∈ OP → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
11	8 10	syl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
12	9 1 2	dihcl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) ∈ ran 𝐼 )
13	11 12	mpdan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) ∈ ran 𝐼 )
14	6 13	eqeltrrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 ∈ ran 𝐼 )