Theorem sqdivid

Description: The square of a nonzero complex number divided by itself equals that number. (Contributed by AV, 19-Jul-2021)

	Ref	Expression
Assertion	sqdivid	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^ 2 ) / A ) = A )

Proof

Step	Hyp	Ref	Expression
1		sqval	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
2	1	adantr	|- ( ( A e. CC /\ A =/= 0 ) -> ( A ^ 2 ) = ( A x. A ) )
3	2	oveq1d	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^ 2 ) / A ) = ( ( A x. A ) / A ) )
4		divcan3	|- ( ( A e. CC /\ A e. CC /\ A =/= 0 ) -> ( ( A x. A ) / A ) = A )
5	4	3anidm12	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( A x. A ) / A ) = A )
6	3 5	eqtrd	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( A ^ 2 ) / A ) = A )