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 ∧ 𝑊 ∈ 𝐻 ) )
	doch11.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
	doch11.y	⊢ ( 𝜑 → 𝑌 ∈ 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 ∧ 𝑊 ∈ 𝐻 ) )
5		doch11.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
6		doch11.y	⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
7	1 2 3 4 6 5	dochord	⊢ ( 𝜑 → ( 𝑌 ⊆ 𝑋 ↔ ( ⊥ ‘ 𝑋 ) ⊆ ( ⊥ ‘ 𝑌 ) ) )
8	1 2 3 4 5 6	dochord	⊢ ( 𝜑 → ( 𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )
9	7 8	anbi12d	⊢ ( 𝜑 → ( ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) ↔ ( ( ⊥ ‘ 𝑋 ) ⊆ ( ⊥ ‘ 𝑌 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) ) )
10		eqcom	⊢ ( 𝑋 = 𝑌 ↔ 𝑌 = 𝑋 )
11		eqss	⊢ ( 𝑌 = 𝑋 ↔ ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) )
12	10 11	bitri	⊢ ( 𝑋 = 𝑌 ↔ ( 𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ 𝑌 ) )
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 ) )