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Mirrors > Home > ILE Home > Th. List > df-submnd | Unicode version |
Description: A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
df-submnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csubmnd 12740 |
. 2
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2 | vs |
. . 3
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3 | cmnd 12709 |
. . 3
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4 | 2 | cv 1352 |
. . . . . . 7
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5 | c0g 12653 |
. . . . . . 7
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6 | 4, 5 | cfv 5212 |
. . . . . 6
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7 | vt |
. . . . . . 7
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8 | 7 | cv 1352 |
. . . . . 6
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9 | 6, 8 | wcel 2148 |
. . . . 5
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10 | vx |
. . . . . . . . . 10
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11 | 10 | cv 1352 |
. . . . . . . . 9
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12 | vy |
. . . . . . . . . 10
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13 | 12 | cv 1352 |
. . . . . . . . 9
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14 | cplusg 12518 |
. . . . . . . . . 10
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15 | 4, 14 | cfv 5212 |
. . . . . . . . 9
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16 | 11, 13, 15 | co 5869 |
. . . . . . . 8
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17 | 16, 8 | wcel 2148 |
. . . . . . 7
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18 | 17, 12, 8 | wral 2455 |
. . . . . 6
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19 | 18, 10, 8 | wral 2455 |
. . . . 5
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20 | 9, 19 | wa 104 |
. . . 4
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21 | cbs 12445 |
. . . . . 6
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22 | 4, 21 | cfv 5212 |
. . . . 5
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23 | 22 | cpw 3574 |
. . . 4
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24 | 20, 7, 23 | crab 2459 |
. . 3
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25 | 2, 3, 24 | cmpt 4061 |
. 2
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26 | 1, 25 | wceq 1353 |
1
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Colors of variables: wff set class |
This definition is referenced by: submrcl 12752 issubm 12753 |
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