Metamath Proof Explorer

Theorem selvval2lem1

Description: T is an associative algebra. For simplicity, I stands for ( I \ J ) and we have J e. W instead of J C_ I . (Contributed by SN, 15-Dec-2023)

		Ref	Expression
	Hypotheses	selvval2lem1.u	$⊢ U = I mPoly R$
		selvval2lem1.t	$⊢ T = J mPoly U$
		selvval2lem1.i	$⊢ φ \to I \in V$
		selvval2lem1.j	$⊢ φ \to J \in W$
		selvval2lem1.r	$⊢ φ \to R \in CRing$
	Assertion	selvval2lem1	$⊢ φ \to T \in AssAlg$

Proof

Step	Hyp	Ref	Expression
1		selvval2lem1.u	$⊢ U = I mPoly R$
2		selvval2lem1.t	$⊢ T = J mPoly U$
3		selvval2lem1.i	$⊢ φ \to I \in V$
4		selvval2lem1.j	$⊢ φ \to J \in W$
5		selvval2lem1.r	$⊢ φ \to R \in CRing$
6	1	mplcrng	$⊢ (I \in V \land R \in CRing) \to U \in CRing$
7	3 5 6	syl2anc	$⊢ φ \to U \in CRing$
8	2	mplassa	$⊢ (J \in W \land U \in CRing) \to T \in AssAlg$
9	4 7 8	syl2anc	$⊢ φ \to T \in AssAlg$