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Mirrors > Home > ILE Home > Th. List > df-mnd | Unicode version |
Description: A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 12713), whose operation is associative (see mndass 12714) and has a two-sided neutral element (see mndid 12715), see also ismnd 12709. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
Ref | Expression |
---|---|
df-mnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmnd 12706 |
. 2
![]() ![]() | |
2 | ve |
. . . . . . . . . . 11
![]() ![]() | |
3 | 2 | cv 1352 |
. . . . . . . . . 10
![]() ![]() |
4 | vx |
. . . . . . . . . . 11
![]() ![]() | |
5 | 4 | cv 1352 |
. . . . . . . . . 10
![]() ![]() |
6 | vp |
. . . . . . . . . . 11
![]() ![]() | |
7 | 6 | cv 1352 |
. . . . . . . . . 10
![]() ![]() |
8 | 3, 5, 7 | co 5869 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8, 5 | wceq 1353 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 5, 3, 7 | co 5869 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10, 5 | wceq 1353 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 9, 11 | wa 104 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | vb |
. . . . . . . 8
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14 | 13 | cv 1352 |
. . . . . . 7
![]() ![]() |
15 | 12, 4, 14 | wral 2455 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15, 2, 14 | wrex 2456 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | vg |
. . . . . . 7
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18 | 17 | cv 1352 |
. . . . . 6
![]() ![]() |
19 | cplusg 12515 |
. . . . . 6
![]() ![]() | |
20 | 18, 19 | cfv 5212 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
21 | 16, 6, 20 | wsbc 2962 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | cbs 12442 |
. . . . 5
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23 | 18, 22 | cfv 5212 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 13, 23 | wsbc 2962 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | csgrp 12696 |
. . 3
![]() | |
26 | 24, 17, 25 | crab 2459 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 1, 26 | wceq 1353 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
This definition is referenced by: ismnddef 12708 |
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