Cancellation Laws/Corollary 1/Proof 2

From ProofWiki
Jump to navigation Jump to search

Corollary to Cancellation Laws

$g h = g \implies h = e$


Proof

\(\displaystyle g h\) \(=\) \(\displaystyle g\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle g^{-1} \paren {g h}\) \(=\) \(\displaystyle g^{-1} g\) Group Axiom $\text G 2$: Existence of Identity Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {g^{-1} g} h\) \(=\) \(\displaystyle g^{-1} g\) Group Axiom $\text G 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle e h\) \(=\) \(\displaystyle e\) Group Axiom $\text G 3$: Existence of Inverse Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h\) \(=\) \(\displaystyle e\) Group Axiom $\text G 2$: Existence of Identity Element

$\blacksquare$


Sources