Metamath Proof Explorer


Theorem doch11

Description: Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014)

Ref Expression
Hypotheses doch11.h 𝐻 = ( LHyp ‘ 𝐾 )
doch11.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
doch11.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
doch11.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
doch11.x ( 𝜑𝑋 ∈ ran 𝐼 )
doch11.y ( 𝜑𝑌 ∈ ran 𝐼 )
Assertion doch11 ( 𝜑 → ( ( 𝑋 ) = ( 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 doch11.h 𝐻 = ( LHyp ‘ 𝐾 )
2 doch11.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3 doch11.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4 doch11.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
5 doch11.x ( 𝜑𝑋 ∈ ran 𝐼 )
6 doch11.y ( 𝜑𝑌 ∈ ran 𝐼 )
7 1 2 3 4 6 5 dochord ( 𝜑 → ( 𝑌𝑋 ↔ ( 𝑋 ) ⊆ ( 𝑌 ) ) )
8 1 2 3 4 5 6 dochord ( 𝜑 → ( 𝑋𝑌 ↔ ( 𝑌 ) ⊆ ( 𝑋 ) ) )
9 7 8 anbi12d ( 𝜑 → ( ( 𝑌𝑋𝑋𝑌 ) ↔ ( ( 𝑋 ) ⊆ ( 𝑌 ) ∧ ( 𝑌 ) ⊆ ( 𝑋 ) ) ) )
10 eqcom ( 𝑋 = 𝑌𝑌 = 𝑋 )
11 eqss ( 𝑌 = 𝑋 ↔ ( 𝑌𝑋𝑋𝑌 ) )
12 10 11 bitri ( 𝑋 = 𝑌 ↔ ( 𝑌𝑋𝑋𝑌 ) )
13 eqss ( ( 𝑋 ) = ( 𝑌 ) ↔ ( ( 𝑋 ) ⊆ ( 𝑌 ) ∧ ( 𝑌 ) ⊆ ( 𝑋 ) ) )
14 9 12 13 3bitr4g ( 𝜑 → ( 𝑋 = 𝑌 ↔ ( 𝑋 ) = ( 𝑌 ) ) )
15 14 bicomd ( 𝜑 → ( ( 𝑋 ) = ( 𝑌 ) ↔ 𝑋 = 𝑌 ) )