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Mirrors > Home > MPE Home > Th. List > df-mgp | Structured version Visualization version 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 unitgrp 17895 shows that 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" (ringmgp 17786) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 9486). (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
df-mgp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmgp 17723 |
. 2
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2 | vw |
. . 3
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3 | cvv 3045 |
. . 3
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4 | 2 | cv 1443 |
. . . 4
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5 | cnx 15118 |
. . . . . 6
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6 | cplusg 15190 |
. . . . . 6
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7 | 5, 6 | cfv 5582 |
. . . . 5
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8 | cmulr 15191 |
. . . . . 6
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9 | 4, 8 | cfv 5582 |
. . . . 5
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10 | 7, 9 | cop 3974 |
. . . 4
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11 | csts 15119 |
. . . 4
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12 | 4, 10, 11 | co 6290 |
. . 3
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13 | 2, 3, 12 | cmpt 4461 |
. 2
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14 | 1, 13 | wceq 1444 |
1
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Colors of variables: wff setvar class |
This definition is referenced by: fnmgp 17725 mgpval 17726 |
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