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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 -1 ⁡ I ⁡ X
8 6 7 syldan ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B → N ⁡ I ⁡ X = I ⁡ ⊥ ˙ ⁡ I -1 ⁡ I ⁡ X
9 1 3 4 dihcnvid1 ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B → I -1 ⁡ I ⁡ X = X
10 9 fveq2d ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B → ⊥ ˙ ⁡ I -1 ⁡ I ⁡ X = ⊥ ˙ ⁡ X
11 10 fveq2d ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B → I ⁡ ⊥ ˙ ⁡ I -1 ⁡ I ⁡ X = I ⁡ ⊥ ˙ ⁡ X
12 8 11 eqtrd ⊢ K ∈ HL ∧ W ∈ H ∧ X ∈ B → N ⁡ I ⁡ X = I ⁡ ⊥ ˙ ⁡ X