Theorem assaassd

Description: Left-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026)

	Ref	Expression
Hypotheses	assaassd.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
	assaassd.2	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	assaassd.3	⊢ 𝐵 = ( Base ‘ 𝐹 )
	assaassd.4	⊢ · = ( ·𝑠 ‘ 𝑊 )
	assaassd.5	⊢ × = ( .r ‘ 𝑊 )
	assaassd.6	⊢ ( 𝜑 → 𝑊 ∈ AssAlg )
	assaassd.7	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	assaassd.8	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	assaassd.9	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	assaassd	⊢ ( 𝜑 → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		assaassd.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		assaassd.2	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3		assaassd.3	⊢ 𝐵 = ( Base ‘ 𝐹 )
4		assaassd.4	⊢ · = ( ·𝑠 ‘ 𝑊 )
5		assaassd.5	⊢ × = ( .r ‘ 𝑊 )
6		assaassd.6	⊢ ( 𝜑 → 𝑊 ∈ AssAlg )
7		assaassd.7	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
8		assaassd.8	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
9		assaassd.9	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
10	1 2 3 4 5	assaass	⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) )
11	6 7 8 9 10	syl13anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝑋 ) × 𝑌 ) = ( 𝐴 · ( 𝑋 × 𝑌 ) ) )