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Metamath Proof Explorer

Definition df-assa

Description: Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014) (Revised by SN, 2-Mar-2025)

Ref Expression
Assertion df-assa AssAlg = { 𝑤 ∈ ( LMod ∩ Ring ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 casa ⊢ AssAlg
1 vw ⊢ 𝑤
2 clmod ⊢ LMod
3 crg ⊢ Ring
4 2 3 cin ⊢ ( LMod ∩ Ring )
5 csca ⊢ Scalar
6 1 cv ⊢ 𝑤
7 6 5 cfv ⊢ ( Scalar ‘ 𝑤 )
8 vf ⊢ 𝑓
9 vr ⊢ 𝑟
10 cbs ⊢ Base
11 8 cv ⊢ 𝑓
12 11 10 cfv ⊢ ( Base ‘ 𝑓 )
13 vx ⊢ 𝑥
14 6 10 cfv ⊢ ( Base ‘ 𝑤 )
15 vy ⊢ 𝑦
16 cvsca ⊢ ·𝑠
17 6 16 cfv ⊢ ( ·𝑠 ‘ 𝑤 )
18 vs ⊢ 𝑠
19 cmulr ⊢ .r
20 6 19 cfv ⊢ ( .r ‘ 𝑤 )
21 vt ⊢ 𝑡
22 9 cv ⊢ 𝑟
23 18 cv ⊢ 𝑠
24 13 cv ⊢ 𝑥
25 22 24 23 co ⊢ ( 𝑟 𝑠 𝑥 )
26 21 cv ⊢ 𝑡
27 15 cv ⊢ 𝑦
28 25 27 26 co ⊢ ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 )
29 24 27 26 co ⊢ ( 𝑥 𝑡 𝑦 )
30 22 29 23 co ⊢ ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) )
31 28 30 wceq ⊢ ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) )
32 22 27 23 co ⊢ ( 𝑟 𝑠 𝑦 )
33 24 32 26 co ⊢ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) )
34 33 30 wceq ⊢ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) )
35 31 34 wa ⊢ ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
36 35 21 20 wsbc ⊢ [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
37 36 18 17 wsbc ⊢ [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
38 37 15 14 wral ⊢ ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
39 38 13 14 wral ⊢ ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
40 39 9 12 wral ⊢ ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
41 40 8 7 wsbc ⊢ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) )
42 41 1 4 crab ⊢ { 𝑤 ∈ ( LMod ∩ Ring ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) }
43 0 42 wceq ⊢ AssAlg = { 𝑤 ∈ ( LMod ∩ Ring ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) }