I am having trouble with the following question

Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$

My attempt:

Let the lattice be distributive. Hence $$(a \lor b) \land c=(a \land c) \lor (b \land c) ~~~...\mathbf i)$$

We also have $$a \land c \leq a ~and~ b \land c \leq b \lor c $$ Combing the two we get, $$(a \land c) \lor (b \land c) \leq a \lor (b \lor c)$$ Using $\mathbf i)$ we can prove that $$(a\lor b)\land c \leq a \lor(b\lor c)$$ Now I do not know how to prove it the other way around. If anyone has a better way to prove it then please suggest it to me. Thanks in advance.

I am having trouble with the following question

Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$

My attempt:

Let the lattice be distributive. Hence $$(a \lor b) \land c=(a \land c) \lor (b \land c) ~~~...\mathbf i)$$

We also have $$a \land c \leq a ~and~ b \land c \leq b \lor c $$ Combing the two we get, $$(a \land c) \lor (b \land c) \leq a \lor (b \lor c)$$ Using $\mathbf i)$ we can prove that $$(a\lor b)\land c \leq a \lor(b\lor c)$$ Now I do not know how to prove it the other way around. If anyone has a better way to prove it then please suggest it to me. Thanks in advance.

EDIT: Okay I found the question in a text book. It seems the question is $$(a\lor b)\land c \leq a \lor(b\land c) $$. I will try the problem once again and post if I face any difficulty.

I am having trouble with the following question

Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$

My attempt:

Let the lattice be distributive.Hence $$(a \lor b) \land c=(a \land c) \lor (b \land c) ~~~...\mathbf i)$$

We also have $$a \land c \leq a ~and~ b \land c \leq b \lor c $$ Combing the two we get, $$(a \land c) \lor (b \land c) \leq a \lor (b \lor c)$$ Using $\mathbf i)$ we can prove that $$(a\lor b)\land c \leq a \lor(b\lor c)$$ Now I do not know how to prove it the other way around. If anyone has a better way to prove it then please suggest it to me. Thanks in advance.

I am having trouble with the following question

Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$

My attempt:

Let the lattice be distributive. Hence $$(a \lor b) \land c=(a \land c) \lor (b \land c) ~~~...\mathbf i)$$

We also have $$a \land c \leq a ~and~ b \land c \leq b \lor c $$ Combing the two we get, $$(a \land c) \lor (b \land c) \leq a \lor (b \lor c)$$ Using $\mathbf i)$ we can prove that $$(a\lor b)\land c \leq a \lor(b\lor c)$$ Now I do not know how to prove it the other way around. If anyone has a better way to prove it then please suggest it to me. Thanks in advance.