Metamath Proof Explorer

Theorem isassad

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

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.3 ⊢ ( 𝜑 → 𝐹 ∈ CRing )
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.3 ⊢ ( 𝜑 → 𝐹 ∈ CRing )
9 isassad.4 ⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
10 isassad.5 ⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) )
11 2 8 eqeltrrd ⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ CRing )
12 6 7 11 3jca ⊢ ( 𝜑 → ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ ( Scalar ‘ 𝑊 ) ∈ CRing ) )
13 9 10 jca ⊢ ( ( 𝜑 ∧ ( 𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) )
14 13 ralrimivvva ⊢ ( 𝜑 → ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) )
15 2 fveq2d ⊢ ( 𝜑 → ( Base ‘ 𝐹 ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
16 3 15 eqtrd ⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
17 4 oveqd ⊢ ( 𝜑 → ( 𝑟 · 𝑥 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) )
18 eqidd ⊢ ( 𝜑 → 𝑦 = 𝑦 )
19 5 17 18 oveq123d ⊢ ( 𝜑 → ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) )
20 eqidd ⊢ ( 𝜑 → 𝑟 = 𝑟 )
21 5 oveqd ⊢ ( 𝜑 → ( 𝑥 × 𝑦 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) )
22 4 20 21 oveq123d ⊢ ( 𝜑 → ( 𝑟 · ( 𝑥 × 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) )
23 19 22 eqeq12d ⊢ ( 𝜑 → ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ↔ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ( .r ‘ 𝑊 ) 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) )
24 eqidd ⊢ ( 𝜑 → 𝑥 = 𝑥 )
25 4 oveqd ⊢ ( 𝜑 → ( 𝑟 · 𝑦 ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) )
26 5 24 25 oveq123d ⊢ ( 𝜑 → ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) )
27 26 22 eqeq12d ⊢ ( 𝜑 → ( ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ↔ ( 𝑥 ( .r ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) = ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) ) )
28 23 27 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 ) ) ) ) )
29 1 28 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 ) ) ) ) )
30 1 29 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 ) ) ) ) )
31 16 30 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 ) ) ) ) )
32 14 31 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 ) ) ) )
33 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
34 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
35 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
36 eqid ⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
37 eqid ⊢ ( .r ‘ 𝑊 ) = ( .r ‘ 𝑊 )
38 33 34 35 36 37 isassa ⊢ ( 𝑊 ∈ AssAlg ↔ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ ( Scalar ‘ 𝑊 ) ∈ CRing ) ∧ ∀ 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 ) ) ) ) )
39 12 32 38 sylanbrc ⊢ ( 𝜑 → 𝑊 ∈ AssAlg )