iscyggen

Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	iscyg.1	$⊢ B = {Base}_{G}$
	iscyg.2	$⊢ \underset{˙}{\cdot} = \cdot_{G}$
	iscyg3.e	$⊢ E = \{x \in B \| ran (n \in ℤ ⟼ n \underset{˙}{\cdot} x) = B\}$
Assertion	iscyggen	$⊢ X \in E \leftrightarrow (X \in B \land ran (n \in ℤ ⟼ n \underset{˙}{\cdot} X) = B)$

Step	Hyp	Ref	Expression
1		iscyg.1	$⊢ B = {Base}_{G}$
2		iscyg.2	$⊢ \underset{˙}{\cdot} = \cdot_{G}$
3		iscyg3.e	$⊢ E = \{x \in B \| ran (n \in ℤ ⟼ n \underset{˙}{\cdot} x) = B\}$
4		simpl	$⊢ (x = X \land n \in ℤ) \to x = X$
5	4	oveq2d	$⊢ (x = X \land n \in ℤ) \to n \underset{˙}{\cdot} x = n \underset{˙}{\cdot} X$
6	5	mpteq2dva	$⊢ x = X \to (n \in ℤ ⟼ n \underset{˙}{\cdot} x) = (n \in ℤ ⟼ n \underset{˙}{\cdot} X)$
7	6	rneqd	$⊢ x = X \to ran (n \in ℤ ⟼ n \underset{˙}{\cdot} x) = ran (n \in ℤ ⟼ n \underset{˙}{\cdot} X)$
8	7	eqeq1d	$⊢ x = X \to (ran (n \in ℤ ⟼ n \underset{˙}{\cdot} x) = B \leftrightarrow ran (n \in ℤ ⟼ n \underset{˙}{\cdot} X) = B)$
9	8 3	elrab2	$⊢ X \in E \leftrightarrow (X \in B \land ran (n \in ℤ ⟼ n \underset{˙}{\cdot} X) = B)$

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