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Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopprsubrg 15401 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15402 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 15403 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 15404 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15405* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15406 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15407 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Theoremissubrg3 15408 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15409 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15410 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 15411 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 15412 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 15413* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
s                      RingHom

Theoremsubrgpropd 15414* 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 15415* 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 15416 The set of absolute values on a ring.
AbsVal

Definitiondf-abv 15417* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 11598 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfval 15418* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabv 15419* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabvd 15420* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
AbsVal

Theoremabvrcl 15421 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfge0 15422 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvf 15423 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvcl 15424 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvge0 15425 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabveq0 15426 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvne0 15427 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvgt0 15428 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvmul 15429 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvtri 15430 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv0 15431 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1z 15432 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1 15433 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvneg 15434 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvsubtri 15435 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal

Theoremabvrec 15436 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvdiv 15437 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal                     /r

Theoremabvdom 15438 Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvres 15439 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
AbsVal       s        AbsVal       SubRing

Theoremabvtrivd 15440* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvtriv 15441* The trivial absolute value. (This theorem is true as long as is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 15438 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvpropd 15442* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal AbsVal

10.5.4  Star rings

Syntaxcstf 15443 Extend class notation with the functionalization of the *-ring involution.

Syntaxcsr 15444 Extend class notation with class of all *-rings.

Definitiondf-staf 15445* Define the functionalization of the involution in a star ring. This is not strictly necessary but by having as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)

Definitiondf-srng 15446* Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
RingHom oppr

Theoremstaffval 15447* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstafval 15448 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstaffn 15449 The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremissrng 15450 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrhm 15451 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrng 15452 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcnv 15453 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngf1o 15454 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcl 15455 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngnvl 15456 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngadd 15457 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngmul 15458 The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng1 15459 The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 15461.) (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng0 15460 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremissrngd 15461* Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)

10.6  Left Modules

10.6.1  Definition and basic properties

Syntaxclmod 15462 Extend class notation with class of all left modules.

Syntaxcscaf 15463 The functionalization of the scalar multiplication operation.

Definitiondf-lmod 15464* Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
Scalar

Definitiondf-scaf 15465* Define the functionalization of the operator. This restricts the value of to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremislmod 15466* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodlema 15467 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremislmodd 15468* Properties that determine a left module. See note in isgrpd2 14340 regarding the on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremlmodgrp 15469 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodrng 15470 The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodfgrp 15471 The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodbn0 15472 The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodacl 15473 Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodmcl 15474 Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodsn0 15475 The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvacl 15476 Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodass 15477 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodlcan 15478 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvscl 15479 Closure of scalar product for a left module. (hvmulcl 21423 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremscaffval 15480* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafval 15481 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafeq 15482 If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscaffn 15483 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodscaf 15484 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodvsdi 15485 Distributive law for scalar product. (ax-hvdistr1 21418 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsdi1OLD 15486 Obsolete version of lmodvsdi 15485 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Scalar

Theoremlmodvsdir 15487 Distributive law for scalar product. (ax-hvdistr1 21418 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsdi2OLD 15488 Obsolete version of lmodvsdir 15487 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Scalar

Theoremlmodvsass 15489 Associative law for scalar product. (ax-hvmulass 21417 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

TheoremlmodvsassOLD 15490 Obsolete version of lmodvsass 15489 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Scalar

Theoremlmod0cl 15491 The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmod1cl 15492 The ring unit in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvs1 15493 Scalar product with ring unit. (ax-hvmulid 21416 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmod0vcl 15494 The zero vector is a vector. (ax-hv0cl 21413 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vlid 15495 Left identity law for the zero vector. (hvaddid2 21432 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vrid 15496 Right identity law for the zero vector. (ax-hvaddid 21414 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vid 15497 Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vs 15498 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21420 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvs0 15499 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 21433 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvnegcl 15500 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

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