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Metamath Proof Explorer

Theorem dochvalr2

Description: Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014)

Ref Expression
Hypotheses dochvalr2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dochvalr2.o ⊢ ⊥ = ( oc ‘ 𝐾 )
dochvalr2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dochvalr2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dochvalr2.n ⊢ 𝑁 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dochvalr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 dochvalr2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dochvalr2.o ⊢ ⊥ = ( oc ‘ 𝐾 )
3 dochvalr2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dochvalr2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5 dochvalr2.n ⊢ 𝑁 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
6 1 3 4 dihcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
7 2 3 4 5 dochvalr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ⊥ ‘ ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) ) )
8 6 7 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ⊥ ‘ ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) ) )
9 1 3 4 dihcnvid1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 )
10 9 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) )
11 10 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( ⊥ ‘ ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) ) = ( 𝐼 ‘ ( ⊥ ‘ 𝑋 ) ) )
12 8 11 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ⊥ ‘ 𝑋 ) ) )