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Mirrors > Home > ILE Home > Th. List > df-mgp | Unicode version |
Description: Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13143). (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
df-mgp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmgp 13130 |
. 2
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2 | vw |
. . 3
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3 | cvv 2738 |
. . 3
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4 | 2 | cv 1352 |
. . . 4
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5 | cnx 12459 |
. . . . . 6
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6 | cplusg 12536 |
. . . . . 6
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7 | 5, 6 | cfv 5217 |
. . . . 5
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8 | cmulr 12537 |
. . . . . 6
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9 | 4, 8 | cfv 5217 |
. . . . 5
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10 | 7, 9 | cop 3596 |
. . . 4
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11 | csts 12460 |
. . . 4
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12 | 4, 10, 11 | co 5875 |
. . 3
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13 | 2, 3, 12 | cmpt 4065 |
. 2
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14 | 1, 13 | wceq 1353 |
1
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Colors of variables: wff set class |
This definition is referenced by: fnmgp 13132 mgpvalg 13133 |
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