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Mirrors > Home > ILE Home > Th. List > df-subg | Unicode version |
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13002), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 12997), contains the neutral element of the group (see subg0 12993) and contains the inverses for all of its elements (see subginvcl 12996). (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
df-subg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csubg 12980 |
. 2
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2 | vw |
. . 3
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3 | cgrp 12831 |
. . 3
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4 | 2 | cv 1352 |
. . . . . 6
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5 | vs |
. . . . . . 7
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6 | 5 | cv 1352 |
. . . . . 6
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7 | cress 12457 |
. . . . . 6
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8 | 4, 6, 7 | co 5874 |
. . . . 5
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9 | 8, 3 | wcel 2148 |
. . . 4
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10 | cbs 12456 |
. . . . . 6
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11 | 4, 10 | cfv 5216 |
. . . . 5
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12 | 11 | cpw 3575 |
. . . 4
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13 | 9, 5, 12 | crab 2459 |
. . 3
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14 | 2, 3, 13 | cmpt 4064 |
. 2
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15 | 1, 14 | wceq 1353 |
1
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Colors of variables: wff set class |
This definition is referenced by: issubg 12986 subgex 12989 |
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