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Metamath Proof Explorer

Theorem diveq1ad

Description: The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 . Generalization of diveq1d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	$⊢ φ \to A \in ℂ$
	divcld.2	$⊢ φ \to B \in ℂ$
	divcld.3	$⊢ φ \to B \neq 0$
Assertion	diveq1ad	$⊢ φ \to (\frac{A}{B} = 1 \leftrightarrow A = B)$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$⊢ φ \to A \in ℂ$
2		divcld.2	$⊢ φ \to B \in ℂ$
3		divcld.3	$⊢ φ \to B \neq 0$
4		diveq1	$⊢ (A \in ℂ \land B \in ℂ \land B \neq 0) \to (\frac{A}{B} = 1 \leftrightarrow A = B)$
5	1 2 3 4	syl3anc	$⊢ φ \to (\frac{A}{B} = 1 \leftrightarrow A = B)$