Metamath Proof Explorer


Theorem chjjdiri

Description: Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006) (New usage is discouraged.)

Ref Expression
Hypotheses chj12.1 𝐴C
chj12.2 𝐵C
chj12.3 𝐶C
Assertion chjjdiri ( ( 𝐴 𝐵 ) ∨ 𝐶 ) = ( ( 𝐴 𝐶 ) ∨ ( 𝐵 𝐶 ) )

Proof

Step Hyp Ref Expression
1 chj12.1 𝐴C
2 chj12.2 𝐵C
3 chj12.3 𝐶C
4 3 chjidmi ( 𝐶 𝐶 ) = 𝐶
5 4 oveq2i ( ( 𝐴 𝐵 ) ∨ ( 𝐶 𝐶 ) ) = ( ( 𝐴 𝐵 ) ∨ 𝐶 )
6 1 2 3 3 chj4i ( ( 𝐴 𝐵 ) ∨ ( 𝐶 𝐶 ) ) = ( ( 𝐴 𝐶 ) ∨ ( 𝐵 𝐶 ) )
7 5 6 eqtr3i ( ( 𝐴 𝐵 ) ∨ 𝐶 ) = ( ( 𝐴 𝐶 ) ∨ ( 𝐵 𝐶 ) )