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Metamath Proof Explorer

Theorem doca3N

Description: Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses doca2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
doca2.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
doca2.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
Assertion doca3N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 doca2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 doca2.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
3 doca2.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
4 1 2 diacnvclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑋 ) ∈ dom 𝐼 )
5 1 2 3 doca2N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ◡ 𝐼 ‘ 𝑋 ) ∈ dom 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ) ) = ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) )
6 4 5 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ) ) = ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) )
7 1 2 diaf11N ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 )
8 f1ocnvfv2 ⊢ ( ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
9 7 8 sylan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) = 𝑋 )
10 9 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ⊥ ‘ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) )
11 10 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑋 ) ) ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) )
12 6 11 9 3eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )