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Definition df-assa 16372
 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.)
Assertion
Ref Expression
df-assa AssAlg Scalar
Distinct variable group:   ,,,,,,

Detailed syntax breakdown of Definition df-assa
StepHypRef Expression
1 casa 16369 . 2 AssAlg
2 vf . . . . . . 7
32cv 1651 . . . . . 6
4 ccrg 15661 . . . . . 6
53, 4wcel 1725 . . . . 5
6 vr . . . . . . . . . . . . . . 15
76cv 1651 . . . . . . . . . . . . . 14
8 vx . . . . . . . . . . . . . . 15
98cv 1651 . . . . . . . . . . . . . 14
10 vs . . . . . . . . . . . . . . 15
1110cv 1651 . . . . . . . . . . . . . 14
127, 9, 11co 6081 . . . . . . . . . . . . 13
13 vy . . . . . . . . . . . . . 14
1413cv 1651 . . . . . . . . . . . . 13
15 vt . . . . . . . . . . . . . 14
1615cv 1651 . . . . . . . . . . . . 13
1712, 14, 16co 6081 . . . . . . . . . . . 12
189, 14, 16co 6081 . . . . . . . . . . . . 13
197, 18, 11co 6081 . . . . . . . . . . . 12
2017, 19wceq 1652 . . . . . . . . . . 11
217, 14, 11co 6081 . . . . . . . . . . . . 13
229, 21, 16co 6081 . . . . . . . . . . . 12
2322, 19wceq 1652 . . . . . . . . . . 11
2420, 23wa 359 . . . . . . . . . 10
25 vw . . . . . . . . . . . 12
2625cv 1651 . . . . . . . . . . 11
27 cmulr 13530 . . . . . . . . . . 11
2826, 27cfv 5454 . . . . . . . . . 10
2924, 15, 28wsbc 3161 . . . . . . . . 9
30 cvsca 13533 . . . . . . . . . 10
3126, 30cfv 5454 . . . . . . . . 9
3229, 10, 31wsbc 3161 . . . . . . . 8
33 cbs 13469 . . . . . . . . 9
3426, 33cfv 5454 . . . . . . . 8
3532, 13, 34wral 2705 . . . . . . 7
3635, 8, 34wral 2705 . . . . . 6
373, 33cfv 5454 . . . . . 6
3836, 6, 37wral 2705 . . . . 5
395, 38wa 359 . . . 4
40 csca 13532 . . . . 5 Scalar
4126, 40cfv 5454 . . . 4 Scalar
4239, 2, 41wsbc 3161 . . 3 Scalar
43 clmod 15950 . . . 4
44 crg 15660 . . . 4
4543, 44cin 3319 . . 3
4642, 25, 45crab 2709 . 2 Scalar
471, 46wceq 1652 1 AssAlg Scalar
 Colors of variables: wff set class This definition is referenced by:  isassa  16375
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