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Metamath Proof Explorer

Theorem prstcoc

Description: Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024)

Ref Expression
Hypotheses prstcnid.c ⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) )
prstcnid.k ⊢ ( 𝜑 → 𝐾 ∈ Proset )
prstcoc.oc ⊢ ( 𝜑 → ⊥ = ( oc ‘ 𝐾 ) )
Assertion prstcoc ( 𝜑 → ( ⊥ ‘ 𝑋 ) = ( ( oc ‘ 𝐶 ) ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 prstcnid.c ⊢ ( 𝜑 → 𝐶 = ( ProsetToCat ‘ 𝐾 ) )
2 prstcnid.k ⊢ ( 𝜑 → 𝐾 ∈ Proset )
3 prstcoc.oc ⊢ ( 𝜑 → ⊥ = ( oc ‘ 𝐾 ) )
4 1 2 3 prstcocval ⊢ ( 𝜑 → ⊥ = ( oc ‘ 𝐶 ) )
5 4 fveq1d ⊢ ( 𝜑 → ( ⊥ ‘ 𝑋 ) = ( ( oc ‘ 𝐶 ) ‘ 𝑋 ) )