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Metamath Proof Explorer

Theorem chdmj1

Description: De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chdmj1 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 chdmm4 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∨ℋ 𝐵 ) )
2 1 fveq2d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) )
3 choccl ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
4 choccl ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
5 chincl ⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ )
6 3 4 5 syl2an ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ )
7 ococ ⊢ ( ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) )
8 6 7 syl ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) )
9 2 8 eqtr3d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) )