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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexpmhm 16701* Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
flds        mulGrpfld       MndHom

Theoremexpghm 16702* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
flds        mulGrpfld       s

10.11.2  Algebraic constructions based on the complexes

Syntaxczrh 16703 Map the rationals into a field, or the integers into a ring.
RHom

Syntaxczlm 16704 Augment an abelian group with vector space operations to turn it into a -module.
Mod

Syntaxcchr 16705 Syntax for ring characteristic.
chr

Syntaxczn 16706 The ring of integers modulo .
ℤ/n

Definitiondf-zrh 16707 Define the unique homomorphism from the integers into a ring. This encodes the usual notation of for integers (see also df-mulg 14744). (Contributed by Mario Carneiro, 13-Jun-2015.)
RHom flds RingHom

Definitiondf-zlm 16708 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod sSet Scalar flds sSet .g

Definitiondf-chr 16709 The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr

Definitiondf-zn 16710* Define the ring of integers . This is literally the quotient ring of by the ideal , but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.)
ℤ/n flds s ~QG RSpan sSet RHom ..^

Theoremmulgghm2 16711* The powers of a group element give a homomorphism from to a group. (Contributed by Mario Carneiro, 13-Jun-2015.)
flds        .g

Theoremmulgrhm 16712* The powers of the element give a ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        .g                     RingHom

Theoremmulgrhm2 16713* The powers of the element give the unique ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        .g                     RingHom

Theoremzrhval 16714 Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.)
flds        RHom       RingHom

Theoremzrhval2 16715* Alternate value of the RHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
flds        RHom       .g

Theoremzrhmulg 16716 Value of the RHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RHom       .g

Theoremzrhrhmb 16717 The RHom homomorphism is the unique ring homomorphism from . (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RHom       RingHom

Theoremzrhrhm 16718 The RHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
flds        RHom       RingHom

Theoremzrh1 16719 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RHom

Theoremzrh0 16720 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RHom

Theoremzrhpropd 16721* The ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
RHom RHom

Theoremzlmval 16722 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       flds        .g       sSet Scalar sSet

Theoremzlmlem 16723 Lemma for zlmbas 16724 and zlmplusg 16725. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       Slot

Theoremzlmbas 16724 Base set of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmplusg 16725 Group operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmmulr 16726 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmsca 16727 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       flds        Scalar

Theoremzlmvsca 16728 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       .g

Theoremzlmlmod 16729 The -module operation turns an arbitrary abelian group into a left module over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmassa 16730 The -module operation turns a ring into an associative algebra over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       AssAlg

Theoremchrval 16731 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr

Theoremchrcl 16732 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr

Theoremchrid 16733 The canonical ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrdvds 16734 The ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrcong 16735 If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.)
chr       RHom

Theoremchrnzr 16736 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
NzRing chr

Theoremchrrhm 16737 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RingHom chr chr

Theoremdomnchr 16738 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Domn chr chr

Theoremznlidl 16739 The set is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       LIdeal

Theoremzncrng2 16740 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 12-Jun-2015.)
flds        RSpan       s ~QG

Theoremznval 16741 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
flds        RSpan       s ~QG        ℤ/n       RHom        ..^              sSet

Theoremznle 16742 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n       RHom        ..^

Theoremznval2 16743 Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n              sSet

Theoremznbaslem 16744 Lemma for znbas 16749. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
flds        RSpan       s ~QG        ℤ/n       Slot

Theoremznbas2 16745 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n

Theoremznadd 16746 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n

Theoremznmul 16747 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n

Theoremznzrh 16748 The ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n       RHom RHom

Theoremznbas 16749 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ℤ/n       ~QG

Theoremzncrng 16750 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n

Theoremznzrh2 16751* The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ~QG        ℤ/n       RHom

Theoremznzrhval 16752 The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ~QG        ℤ/n       RHom

Theoremznzrhfo 16753 The ring homomorphism is a surjection onto . (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n              RHom

Theoremzncyg 16754 The group is cyclic for all (including ). (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       CycGrp

Theoremzndvds 16755 Express equality of equivalence classes in in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremzndvds0 16756 Special case of zndvds 16755 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremznf1o 16757 The function enumerates all equivalence classes in ℤ/nℤ for each . When , so we let ; otherwise enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
flds        ℤ/n              RHom        ..^

Theoremzzngim 16758 The ring homomorphism is an isomorphism for . (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       RHom       flds GrpIso

Theoremznle2 16759 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremznleval 16760 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremznleval2 16761 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremzntoslem 16762 Lemma for zntos 16763. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^                     Toset

Theoremzntos 16763 The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case when it coincides with the ordering on .) (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Toset

Theoremznhash 16764 The ℤ/nℤ structure has elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n

Theoremznfi 16765 The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n

Theoremznfld 16766 The ℤ/nℤ structure is a finite field when is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Field

Theoremznidomb 16767 The ℤ/nℤ structure is a domain (and hence a field) precisely when is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       IDomn

Theoremznchr 16768 Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
ℤ/n       chr

Theoremznunit 16769 The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
ℤ/n       Unit       RHom

Theoremznunithash 16770 The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
ℤ/n       Unit

Theoremznrrg 16771 The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for , where all nonzero integers are regular, but only are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
ℤ/n       Unit       RLReg

Theoremcygznlem1 16772* Lemma for cygzn 16776. (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem2a 16773* Lemma for cygzn 16776. (Contributed by Mario Carneiro, 23-Dec-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem2 16774* Lemma for cygzn 16776. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem3 16775* A cyclic group with elements is isomorphic to . (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       .g       RHom              CycGrp                     𝑔

Theoremcygzn 16776 A cyclic group with elements is isomorphic to , and an infinite cyclic group is isomorphic to . (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       CycGrp 𝑔

Theoremcygth 16777* The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups , for (where is the infinite cyclic group ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp 𝑔 ℤ/n

Theoremcyggic 16778 Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp 𝑔

Theoremfrgpcyg 16779 A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       CycGrp

10.12  Hilbert spaces

10.12.1  Definition and basic properties

Syntaxcphl 16780 Extend class notation with class all pre-Hilbert spaces.

Syntaxcipf 16781 Extend class notation with inner product function.

Definitiondf-phl 16782* Define class all generalized pre-Hilbert (inner product) spaces. (Contributed by NM, 22-Sep-2011.)
Scalar LMHom ringLMod

Definitiondf-ipf 16783* Define group addition function. Usually we will use directly instead of , and they have the same behavior in most cases. The main advantage of is that it is a guaranteed function (mndplusf 14635), while only has closure (mndcl 14624). (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremisphl 16784* The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar                                   LMHom ringLMod

Theoremphllvec 16785 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremphllmod 16786 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremphlsrng 16787 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremphllmhm 16788* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar                            LMHom ringLMod

Theoremipcl 16789 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipcj 16790 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremiporthcom 16791 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip0l 16792 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip0r 16793 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipeq0 16794 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipdir 16795 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipdi 16796 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip2di 16797 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipsubdir 16798 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipsubdi 16799 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip2subdi 16800 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

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