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Definition df-assa 21341
Description: Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) 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.)
Assertion
Ref Expression
df-assa AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
Distinct variable group:   𝑓,𝑟,𝑠,𝑡,𝑤,𝑥,𝑦

Detailed syntax breakdown of Definition df-assa
StepHypRef Expression
1 casa 21338 . 2 class AssAlg
2 vr . . . . . . . . . . . . . 14 setvar 𝑟
32cv 1540 . . . . . . . . . . . . 13 class 𝑟
4 vx . . . . . . . . . . . . . 14 setvar 𝑥
54cv 1540 . . . . . . . . . . . . 13 class 𝑥
6 vs . . . . . . . . . . . . . 14 setvar 𝑠
76cv 1540 . . . . . . . . . . . . 13 class 𝑠
83, 5, 7co 7393 . . . . . . . . . . . 12 class (𝑟𝑠𝑥)
9 vy . . . . . . . . . . . . 13 setvar 𝑦
109cv 1540 . . . . . . . . . . . 12 class 𝑦
11 vt . . . . . . . . . . . . 13 setvar 𝑡
1211cv 1540 . . . . . . . . . . . 12 class 𝑡
138, 10, 12co 7393 . . . . . . . . . . 11 class ((𝑟𝑠𝑥)𝑡𝑦)
145, 10, 12co 7393 . . . . . . . . . . . 12 class (𝑥𝑡𝑦)
153, 14, 7co 7393 . . . . . . . . . . 11 class (𝑟𝑠(𝑥𝑡𝑦))
1613, 15wceq 1541 . . . . . . . . . 10 wff ((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦))
173, 10, 7co 7393 . . . . . . . . . . . 12 class (𝑟𝑠𝑦)
185, 17, 12co 7393 . . . . . . . . . . 11 class (𝑥𝑡(𝑟𝑠𝑦))
1918, 15wceq 1541 . . . . . . . . . 10 wff (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))
2016, 19wa 396 . . . . . . . . 9 wff (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
21 vw . . . . . . . . . . 11 setvar 𝑤
2221cv 1540 . . . . . . . . . 10 class 𝑤
23 cmulr 17180 . . . . . . . . . 10 class .r
2422, 23cfv 6532 . . . . . . . . 9 class (.r𝑤)
2520, 11, 24wsbc 3773 . . . . . . . 8 wff [(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
26 cvsca 17183 . . . . . . . . 9 class ·𝑠
2722, 26cfv 6532 . . . . . . . 8 class ( ·𝑠𝑤)
2825, 6, 27wsbc 3773 . . . . . . 7 wff [( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
29 cbs 17126 . . . . . . . 8 class Base
3022, 29cfv 6532 . . . . . . 7 class (Base‘𝑤)
3128, 9, 30wral 3060 . . . . . 6 wff 𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
3231, 4, 30wral 3060 . . . . 5 wff 𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
33 vf . . . . . . 7 setvar 𝑓
3433cv 1540 . . . . . 6 class 𝑓
3534, 29cfv 6532 . . . . 5 class (Base‘𝑓)
3632, 2, 35wral 3060 . . . 4 wff 𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
37 csca 17182 . . . . 5 class Scalar
3822, 37cfv 6532 . . . 4 class (Scalar‘𝑤)
3936, 33, 38wsbc 3773 . . 3 wff [(Scalar‘𝑤) / 𝑓]𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))
40 clmod 20420 . . . 4 class LMod
41 crg 20014 . . . 4 class Ring
4240, 41cin 3943 . . 3 class (LMod ∩ Ring)
4339, 21, 42crab 3431 . 2 class {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
441, 43wceq 1541 1 wff AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
Colors of variables: wff setvar class
This definition is referenced by:  isassa  21344
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