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Mirrors > Home > ILE Home > Th. List > df-sgrp | Unicode version |
Description: A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 12664), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
Ref | Expression |
---|---|
df-sgrp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csgrp 12696 |
. 2
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2 | vx |
. . . . . . . . . . . 12
![]() ![]() | |
3 | 2 | cv 1352 |
. . . . . . . . . . 11
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4 | vy |
. . . . . . . . . . . 12
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5 | 4 | cv 1352 |
. . . . . . . . . . 11
![]() ![]() |
6 | vo |
. . . . . . . . . . . 12
![]() ![]() | |
7 | 6 | cv 1352 |
. . . . . . . . . . 11
![]() ![]() |
8 | 3, 5, 7 | co 5869 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() |
9 | vz |
. . . . . . . . . . 11
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10 | 9 | cv 1352 |
. . . . . . . . . 10
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11 | 8, 10, 7 | co 5869 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 5, 10, 7 | co 5869 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() |
13 | 3, 12, 7 | co 5869 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 11, 13 | wceq 1353 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | vb |
. . . . . . . . 9
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16 | 15 | cv 1352 |
. . . . . . . 8
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17 | 14, 9, 16 | wral 2455 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17, 4, 16 | wral 2455 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18, 2, 16 | wral 2455 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | vg |
. . . . . . 7
![]() ![]() | |
21 | 20 | cv 1352 |
. . . . . 6
![]() ![]() |
22 | cplusg 12515 |
. . . . . 6
![]() ![]() | |
23 | 21, 22 | cfv 5212 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
24 | 19, 6, 23 | wsbc 2962 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | cbs 12442 |
. . . . 5
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26 | 21, 25 | cfv 5212 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
27 | 24, 15, 26 | wsbc 2962 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | cmgm 12662 |
. . 3
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29 | 27, 20, 28 | crab 2459 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 1, 29 | wceq 1353 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
This definition is referenced by: issgrp 12698 |
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