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Metamath Proof Explorer

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 )