Theorem chjassi

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
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
	chjass.3	⊢ 𝐶 ∈ Cℋ
Assertion	chjassi	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3		chjass.3	⊢ 𝐶 ∈ Cℋ
4		inass	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) ) )
5	1 2	chdmj1i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) )
6	5	ineq1i	⊢ ( ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) ) = ( ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) )
7	2 3	chdmj1i	⊢ ( ⊥ ‘ ( 𝐵 ∨ℋ 𝐶 ) ) = ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) )
8	7	ineq2i	⊢ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( 𝐵 ∨ℋ 𝐶 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ( ⊥ ‘ 𝐵 ) ∩ ( ⊥ ‘ 𝐶 ) ) )
9	4 6 8	3eqtr4i	⊢ ( ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( 𝐵 ∨ℋ 𝐶 ) ) )
10	9	fveq2i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) ) ) = ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( 𝐵 ∨ℋ 𝐶 ) ) ) )
11	1 2	chjcli	⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ
12	11 3	chdmm4i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ∩ ( ⊥ ‘ 𝐶 ) ) ) = ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 )
13	2 3	chjcli	⊢ ( 𝐵 ∨ℋ 𝐶 ) ∈ Cℋ
14	1 13	chdmm4i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( 𝐵 ∨ℋ 𝐶 ) ) ) ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )
15	10 12 14	3eqtr3i	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )