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Metamath Proof Explorer

Theorem selvcllem1

Description: T is an associative algebra. For simplicity, I stands for ( I \ J ) and we have J e. W instead of J C_ I . TODO-SN: In practice, this "simplification" makes the lemmas harder to use. (Contributed by SN, 15-Dec-2023)

	Ref	Expression
Hypotheses	selvcllem1.u	$⊢ U = I mPoly R$
	selvcllem1.t	$⊢ T = J mPoly U$
	selvcllem1.i	$⊢ φ \to I \in V$
	selvcllem1.j	$⊢ φ \to J \in W$
	selvcllem1.r	$⊢ φ \to R \in CRing$
Assertion	selvcllem1	$⊢ φ \to T \in AssAlg$

Proof

Step	Hyp	Ref	Expression
1		selvcllem1.u	$⊢ U = I mPoly R$
2		selvcllem1.t	$⊢ T = J mPoly U$
3		selvcllem1.i	$⊢ φ \to I \in V$
4		selvcllem1.j	$⊢ φ \to J \in W$
5		selvcllem1.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	$⊢ φ \to T \in AssAlg$