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Metamath Proof Explorer

Theorem chdmm1

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

Ref Expression
Assertion chdmm1 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ineq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) )
2 1 fveq2d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ ( 𝐴 ∩ 𝐵 ) ) = ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ) )
3 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
4 3 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
5 2 4 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ ( 𝐴 ∩ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ↔ ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) )
6 ineq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
7 6 fveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ) = ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
8 fveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ⊥ ‘ 𝐵 ) = ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
9 8 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
10 7 9 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ 𝐵 ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ↔ ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) ) )
11 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
12 ifchhv ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
13 11 12 chdmm1i ⊢ ( ⊥ ‘ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∩ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) = ( ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ∨ℋ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
14 5 10 13 dedth2h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )