Description: A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | chj12.1 | ⊢ 𝐴 ∈ Cℋ | |
chj12.2 | ⊢ 𝐵 ∈ Cℋ | ||
chj12.3 | ⊢ 𝐶 ∈ Cℋ | ||
Assertion | chj12i | ⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chj12.1 | ⊢ 𝐴 ∈ Cℋ | |
2 | chj12.2 | ⊢ 𝐵 ∈ Cℋ | |
3 | chj12.3 | ⊢ 𝐶 ∈ Cℋ | |
4 | 1 2 | chjcomi | ⊢ ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) |
5 | 4 | oveq1i | ⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) |
6 | 1 2 3 | chjassi | ⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) |
7 | 2 1 3 | chjassi | ⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) |
8 | 5 6 7 | 3eqtr3i | ⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) ) |