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Metamath Proof Explorer

Theorem chjass

Description: Associative law for Hilbert lattice join. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chjass ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) )
2 1 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) )
3 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) )
4 2 3 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) ) )
5 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
6 5 oveq1d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ 𝐶 ) )
7 oveq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 ∨ℋ 𝐶 ) = ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) )
8 7 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) ) )
9 6 8 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ 𝐶 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) ) ) )
10 oveq2 ⊢ ( 𝐶 = if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ 𝐶 ) = ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) )
11 oveq2 ⊢ ( 𝐶 = if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) → ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) = ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) )
12 11 oveq2d ⊢ ( 𝐶 = if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) ) )
13 10 12 eqeq12d ⊢ ( 𝐶 = if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ 𝐶 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ 𝐶 ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) ) ) )
14 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
15 ifchhv ⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
16 ifchhv ⊢ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ∈ Cℋ
17 14 15 16 chjassi ⊢ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∨ℋ if ( 𝐶 ∈ Cℋ , 𝐶 , ℋ ) ) )
18 4 9 13 17 dedth3h ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) )