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 )