This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dihcnvid2

Description: The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014)

Ref Expression
Hypotheses dihcnvid2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihcnvid2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihcnvid2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 dihcnvid2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dihcnvid2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
4 eqid ⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
6 3 1 2 4 5 dihf11 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : ( Base ‘ 𝐾 ) –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
7 f1f1orn ⊢ ( 𝐼 : ( Base ‘ 𝐾 ) –1-1→ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) → 𝐼 : ( Base ‘ 𝐾 ) –1-1-onto→ ran 𝐼 )
8 6 7 syl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : ( Base ‘ 𝐾 ) –1-1-onto→ ran 𝐼 )
9 f1ocnvfv2 ⊢ ( ( 𝐼 : ( Base ‘ 𝐾 ) –1-1-onto→ ran 𝐼 ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
10 8 9 sylan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )