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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabvrec 15601 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvdiv 15602 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal                     /r

Theoremabvdom 15603 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 15604 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 15605* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvtriv 15606* 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 15603 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvpropd 15607* 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 15608 Extend class notation with the functionalization of the *-ring involution.

Syntaxcsr 15609 Extend class notation with class of all *-rings.

Definitiondf-staf 15610* 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 15611* 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 15612* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstafval 15613 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstaffn 15614 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 15615 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrhm 15616 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrng 15617 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcnv 15618 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngf1o 15619 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcl 15620 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngnvl 15621 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngadd 15622 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngmul 15623 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 15624 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 15626.) (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng0 15625 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremissrngd 15626* 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 15627 Extend class notation with class of all left modules.

Syntaxcscaf 15628 The functionalization of the scalar multiplication operation.

Definitiondf-lmod 15629* 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 15630* 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 15631* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodlema 15632 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremislmodd 15633* Properties that determine a left module. See note in isgrpd2 14505 regarding the on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremlmodgrp 15634 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodrng 15635 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 15636 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 15637 The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodacl 15638 Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodmcl 15639 Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodsn0 15640 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 15641 Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodass 15642 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodlcan 15643 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvscl 15644 Closure of scalar product for a left module. (hvmulcl 21593 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremscaffval 15645* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafval 15646 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafeq 15647 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 15648 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodscaf 15649 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodvsdi 15650 Distributive law for scalar product. (ax-hvdistr1 21588 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsdi1OLD 15651 Obsolete version of lmodvsdi 15650 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 15652 Distributive law for scalar product. (ax-hvdistr1 21588 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsdi2OLD 15653 Obsolete version of lmodvsdir 15652 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 15654 Associative law for scalar product. (ax-hvmulass 21587 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

TheoremlmodvsassOLD 15655 Obsolete version of lmodvsass 15654 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 15656 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 15657 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 15658 Scalar product with ring unit. (ax-hvmulid 21586 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmod0vcl 15659 The zero vector is a vector. (ax-hv0cl 21583 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vlid 15660 Left identity law for the zero vector. (hvaddid2 21602 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vrid 15661 Right identity law for the zero vector. (ax-hvaddid 21584 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vid 15662 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 15663 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21590 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvs0 15664 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 21603 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvnegcl 15665 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvnegid 15666 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvneg1 15667 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

TheoremlmodvsnegOLD 15668 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Scalar

Theoremlmodvsneg 15669 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsubcl 15670 Closure of vector subtraction. (hvsubcl 21597 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodcom 15671 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)

Theoremlmodabl 15672 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodcmn 15673 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
CMnd

Theoremlmodnegadd 15674 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlmod4 15675 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsubadd 15676 Relationship between vector subtraction and addition. (hvsubadd 21656 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvaddsub4 15677 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvpncan 15678 Addition/subtraction cancellation law for vectors. (hvpncan 21618 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvnpcan 15679 Cancellation law for vector subtraction (npcan 9060 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsubval2 15680 Value of vector subtraction in terms of addition. (hvsubval 21596 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodsubvs 15681 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlmodsubdi 15682 Scalar multiplication distributive law for subtraction. (hvsubdistr1 21628 analog, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubdir 15683 Scalar multiplication distributive law for subtraction. (hvsubdistr2 21629 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubeq0 15684 If the difference between two vectors is zero, they are equal. (hvsubeq0 21647 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodsubid 15685 Subtraction of a vector from itself. (hvsubid 21605 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsghm 15686* Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
Scalar

Theoremlmodprop2d 15687* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15688 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlmodpropd 15688* If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

10.6.2  Subspaces and spans in a left module

Syntaxclss 15689 Extend class notation with linear subspaces of a left module or left vector space.

Definitiondf-lss 15690* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
Scalar

Theoremlssset 15691* The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
Scalar

Theoremislss 15692* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremislssd 15693* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremlssss 15694 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlssel 15695 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlss1 15696 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssuni 15697 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)

Theoremlssn0 15698 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theorem00lss 15699 The empty structure has no subspaces (for use with fvco4i 5597). (Contributed by Stefan O'Rear, 31-Mar-2015.)

Theoremlsscl 15700 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

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