Theorem isassad

Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014) (Revised by SN, 2-Mar-2025)

	Ref	Expression
Hypotheses	isassad.v	⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑊 ) )
	isassad.f	⊢ ( 𝜑 → 𝐹 = ( Scalar ‘ 𝑊 ) )
	isassad.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐹 ) )
	isassad.s	⊢ ( 𝜑 → · = ( ·𝑠 ‘ 𝑊 ) )
	isassad.t	⊢ ( 𝜑 → × = ( .r ‘ 𝑊 ) )
	isassad.1	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	isassad.2	⊢ ( 𝜑 → 𝑊 ∈ Ring )
	isassad.4	⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
	isassad.5	⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
Assertion	isassad	⊢ ( 𝜑 → 𝑊 ∈ AssAlg )

Proof

Step	Hyp	Ref	Expression
1		isassad.v	⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑊 ) )
2		isassad.f	⊢ ( 𝜑 → 𝐹 = ( Scalar ‘ 𝑊 ) )
3		isassad.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐹 ) )
4		isassad.s	⊢ ( 𝜑 → · = ( ·𝑠 ‘ 𝑊 ) )
5		isassad.t	⊢ ( 𝜑 → × = ( .r ‘ 𝑊 ) )
6		isassad.1	⊢ ( 𝜑 → 𝑊 ∈ LMod )
7		isassad.2	⊢ ( 𝜑 → 𝑊 ∈ Ring )
8		isassad.4	⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
9		isassad.5	⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
10	6 7	jca	⊢ ( 𝜑 → ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ) )
11	8 9	jca	⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) )
12	11	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) )
13	2	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐹 ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
14	3 13	eqtrd	⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
15	4	oveqd	⊢ ( 𝜑 → ( 𝑟 · 𝑥 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) )
16		eqidd	⊢ ( 𝜑 → 𝑦 = 𝑦 )
17	5 15 16	oveq123d	⊢ ( 𝜑 → ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) )
18		eqidd	⊢ ( 𝜑 → 𝑟 = 𝑟 )
19	5	oveqd	⊢ ( 𝜑 → ( 𝑥 × 𝑦 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) )
20	4 18 19	oveq123d	⊢ ( 𝜑 → ( 𝑟 · ( 𝑥 × 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) )
21	17 20	eqeq12d	⊢ ( 𝜑 → ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ↔ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) )
22		eqidd	⊢ ( 𝜑 → 𝑥 = 𝑥 )
23	4	oveqd	⊢ ( 𝜑 → ( 𝑟 · 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) )
24	5 22 23	oveq123d	⊢ ( 𝜑 → ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) )
25	24 20	eqeq12d	⊢ ( 𝜑 → ( ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ↔ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) )
26	21 25	anbi12d	⊢ ( 𝜑 → ( ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ↔ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) ) )
27	1 26	raleqbidv	⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) ) )
28	1 27	raleqbidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) ) )
29	14 28	raleqbidv	⊢ ( 𝜑 → ( ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ↔ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) ) )
30	12 29	mpbid	⊢ ( 𝜑 → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) )
31		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
32		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
33		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
34		eqid	⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
35		eqid	⊢ ( .r ‘ 𝑊 ) = ( .r ‘ 𝑊 )
36	31 32 33 34 35	isassa	⊢ ( 𝑊 ∈ AssAlg ↔ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ) ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ∧ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) ) )
37	10 30 36	sylanbrc	⊢ ( 𝜑 → 𝑊 ∈ AssAlg )