Theorem nmzbi

Description: Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015)

	Ref	Expression
Hypothesis	elnmz.1	$$⊢N=\left\{x\in X|\forall y\in X\left(x\underset{\dot{}}{+}y\in S\leftrightarrow y\underset{\dot{}}{+}x\in S\right)\right\}$$
Assertion	nmzbi	$$⊢\left(A\in N\wedge B\in X\right)\to \left(A\underset{\dot{}}{+}B\in S\leftrightarrow B\underset{\dot{}}{+}A\in S\right)$$

Proof

Step	Hyp	Ref	Expression
1		elnmz.1	$$⊢N=\left\{x\in X|\forall y\in X\left(x\underset{\dot{}}{+}y\in S\leftrightarrow y\underset{\dot{}}{+}x\in S\right)\right\}$$
2	1	elnmz	$$⊢A\in N\leftrightarrow \left(A\in X\wedge \forall z\in X\left(A\underset{\dot{}}{+}z\in S\leftrightarrow z\underset{\dot{}}{+}A\in S\right)\right)$$
3	2	simprbi	$$⊢A\in N\to \forall z\in X\left(A\underset{\dot{}}{+}z\in S\leftrightarrow z\underset{\dot{}}{+}A\in S\right)$$
4		oveq2	$$⊢z=B\to A\underset{\dot{}}{+}z=A\underset{\dot{}}{+}B$$
5	4	eleq1d	$$⊢z=B\to \left(A\underset{\dot{}}{+}z\in S\leftrightarrow A\underset{\dot{}}{+}B\in S\right)$$
6		oveq1	$$⊢z=B\to z\underset{\dot{}}{+}A=B\underset{\dot{}}{+}A$$
7	6	eleq1d	$$⊢z=B\to \left(z\underset{\dot{}}{+}A\in S\leftrightarrow B\underset{\dot{}}{+}A\in S\right)$$
8	5 7	bibi12d	$$⊢z=B\to \left(\left(A\underset{\dot{}}{+}z\in S\leftrightarrow z\underset{\dot{}}{+}A\in S\right)\leftrightarrow \left(A\underset{\dot{}}{+}B\in S\leftrightarrow B\underset{\dot{}}{+}A\in S\right)\right)$$
9	8	rspccva	$$⊢\left(\forall z\in X\left(A\underset{\dot{}}{+}z\in S\leftrightarrow z\underset{\dot{}}{+}A\in S\right)\wedge B\in X\right)\to \left(A\underset{\dot{}}{+}B\in S\leftrightarrow B\underset{\dot{}}{+}A\in S\right)$$
10	3 9	sylan	$$⊢\left(A\in N\wedge B\in X\right)\to \left(A\underset{\dot{}}{+}B\in S\leftrightarrow B\underset{\dot{}}{+}A\in S\right)$$