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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisrhm 15501 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmmhm 15502 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmghm 15503 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremrhmf 15504 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmmul 15505 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom

Theoremisrhm2d 15506* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
RingHom

Theoremisrhmd 15507* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhm1 15508 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmco 15509 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom RingHom RingHom

Theorempwsco1rhm 15510* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                                           RingHom

Theorempwsco2rhm 15511* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                      RingHom        RingHom

10.5  Division rings and fields

10.5.1  Definition and basic properties

Syntaxcdr 15512 Extend class notation with class of all division rings.

Syntaxcfield 15513 Class of fields.
Field

Definitiondf-drng 15514 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
Unit

Definitiondf-field 15515 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng 15516 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngunit 15517 Elementhood in the set of units when is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngui 15518 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngrng 15519 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)

Theoremdrnggrp 15520 A division ring is a group. (Contributed by NM, 8-Sep-2011.)

Theoremisfld 15521 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng2 15522 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
mulGrps

Theoremdrngprop 15523 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)

Theoremdrngmgp 15524 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
mulGrps

Theoremdrngmcl 15525 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)

Theoremdrngid 15526 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
mulGrps

Theoremdrngunz 15527 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)

Theoremdrngid2 15528 Properties showing that an element is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremdrnginvrcl 15529 Closure of the multiplicative inverse in a division ring. (reccl 9431 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrn0 15530 The multiplicative inverse in a division ring is nonzero. (recne0 9437 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrl 15531 Property of the multiplicative inverse in a division ring. (recid2 9439 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrr 15532 Property of the multiplicative inverse in a division ring. (recid 9438 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrngmul0or 15533 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)

Theoremdrngmulne0 15534 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)

Theoremdrngmuleq0 15535 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)

Theoremopprdrng 15536 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
oppr

Theoremisdrngd 15537* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." (Contributed by NM, 2-Aug-2013.)

Theoremisdrngrd 15538* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 15537 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)

Theoremdrngpropd 15539* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)

Theoremfldpropd 15540* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Field Field

10.5.2  Subrings of a ring

Syntaxcsubrg 15541 Extend class notation with all subrings of a ring.
SubRing

Syntaxcrgspn 15542 Extend class notation with span of a set of elements over a ring.
RingSpan

Definitiondf-subrg 15543* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset of (where multiplication is component-wise) contains the false identity which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

SubRing s

Definitiondf-rgspn 15544* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan SubRing

Theoremissubrg 15545 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing s

Theoremsubrgss 15546 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgid 15547 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing

Theoremsubrgrng 15548 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrgcrng 15549 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        SubRing

Theoremsubrgrcl 15550 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing

Theoremsubrgsubg 15551 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremsubrg0 15552 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrg1cl 15553 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgbas 15554 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrg1 15555 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrgacl 15556 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgmcl 15557 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgsubm 15558 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
mulGrp       SubRing SubMnd

Theoremsubrgdvds 15559 If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        r       r       SubRing

Theoremsubrguss 15560 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       SubRing

Theoremsubrginv 15561 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s               Unit              SubRing

Theoremsubrgdv 15562 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        /r       Unit       /r       SubRing

Theoremsubrgunit 15563 An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit              SubRing

Theoremsubrgugrp 15564 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       mulGrps        SubRing SubGrp

Theoremissubrg2 15565* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremopprsubrg 15566 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15567 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing

Theoremsubrgin 15568 The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing SubRing

Theoremsubrgmre 15569 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15570* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15571 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15572 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Theoremissubrg3 15573 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15574 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15575 The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
RingHom RingHom SubRing

Theoremrhmima 15576 The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
RingHom SubRing SubRing

Theoremcntzsubr 15577 Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
mulGrp       Cntz       SubRing

Theorempwsdiagrhm 15578* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
s                      RingHom

Theoremsubrgpropd 15579* If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
SubRing SubRing

Theoremrhmpropd 15580* Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom RingHom

10.5.3  Absolute value (abstract algebra)

Syntaxcabv 15581 The set of absolute values on a ring.
AbsVal

Definitiondf-abv 15582* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 11721 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfval 15583* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabv 15584* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabvd 15585* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
AbsVal

Theoremabvrcl 15586 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfge0 15587 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvf 15588 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvcl 15589 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvge0 15590 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabveq0 15591 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvne0 15592 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvgt0 15593 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvmul 15594 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvtri 15595 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv0 15596 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1z 15597 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1 15598 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvneg 15599 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvsubtri 15600 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal

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