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 ) ) )