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Metamath Proof Explorer

Theorem dochsordN

Description: Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014) (New usage is discouraged.)

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 dochsordN ( 𝜑 → ( 𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ) )

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 5 6 dochord ⊢ ( 𝜑 → ( 𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )
8 1 2 3 4 6 5 doch11 ⊢ ( 𝜑 → ( ( ⊥ ‘ 𝑌 ) = ( ⊥ ‘ 𝑋 ) ↔ 𝑌 = 𝑋 ) )
9 eqcom ⊢ ( 𝑌 = 𝑋 ↔ 𝑋 = 𝑌 )
10 8 9 bitr2di ⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ( ⊥ ‘ 𝑌 ) = ( ⊥ ‘ 𝑋 ) ) )
11 10 necon3bid ⊢ ( 𝜑 → ( 𝑋 ≠ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) )
12 7 11 anbi12d ⊢ ( 𝜑 → ( ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ↔ ( ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) ) )
13 df-pss ⊢ ( 𝑋 ⊊ 𝑌 ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) )
14 df-pss ⊢ ( ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ↔ ( ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) )
15 12 13 14 3bitr4g ⊢ ( 𝜑 → ( 𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ) )