I am reading the proof of Birkhoff's theorem (from T.S. Blyth - Lattices and Ordered Algebraic Structures-Springer (2005), page -$68$) which states that
A lattice $L$ is distributive if and only if it has no sublattice isomorphic to either of $M_3$ and $N_5$. Equivalently, $L$ is distributive iff the cancellation laws hold in $L$.
Here to proof the converse part, the writer defined for given any
$a,b,c\in L,\quad a^*=(b\vee c)\wedge a,\quad b^*=(c\vee a)\wedge b,\quad c^*=(a\vee b)\wedge c$.
Then he claimed that it is obvious to have
$a^*\wedge c^*=a\wedge c, \quad b^*\wedge c^*=b\wedge c,\quad a^*\wedge b^*=a\wedge b\tag{1}$
Then he considered $d=(a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a)$ and showed using modularity twice, \begin{align} a^* ∨ c^* &= a^* ∨ [(a ∨ b) ∧ c]\\ &= (a^* ∨ c) ∧ (a ∨ b)\quad [\text{since}\quad a^*\le a ∨ b]\tag{2}\\ &=\left([(b ∨ c) ∧ a] ∨ c\right)∧ (a ∨ b)\\ &= (b ∨ c) ∧ (a ∨ c) ∧ (a ∨ b) \quad [\text{since}\quad c \le b ∨ c]\tag{3}\\ &= d \end{align}
Now I'm facing problems to understand why $(1),(2),(3)$ are happening. Any help for the calculations for $(1)$ and the reasons for $(2)$ and $(3)$ is highly solicited.
(1) holds by absorption: \begin{align} a^* \wedge c^* &= (b \vee c) \wedge a \wedge (a \vee b) \wedge c\\ &= ((b \vee c) \wedge c) \wedge (a \wedge (a \vee b))\\ &= c \wedge a, \end{align} and analogously for the other two identities.
While I don't know exactly the context of this proof, it seems likely that it is already assumed that the lattice is modular, and so (2) and (3) follow from modularity.