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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzlmmulr 16801 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmsca 16802 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       flds        Scalar

Theoremzlmvsca 16803 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       .g

Theoremzlmlmod 16804 The -module operation turns an arbitrary abelian group into a left module over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmassa 16805 The -module operation turns a ring into an associative algebra over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       AssAlg

Theoremchrval 16806 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr

Theoremchrcl 16807 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr

Theoremchrid 16808 The canonical ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrdvds 16809 The ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrcong 16810 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 16811 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
NzRing chr

Theoremchrrhm 16812 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RingHom chr chr

Theoremdomnchr 16813 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Domn chr chr

Theoremznlidl 16814 The set is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       LIdeal

Theoremzncrng2 16815 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 16816 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 16817 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 16818 Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.)
flds        RSpan       s ~QG        ℤ/n              sSet

Theoremznbaslem 16819 Lemma for znbas 16824. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
flds        RSpan       s ~QG        ℤ/n       Slot

Theoremznbas2 16820 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 16821 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 16822 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 16823 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 16824 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ℤ/n       ~QG

Theoremzncrng 16825 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n

Theoremznzrh2 16826* The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ~QG        ℤ/n       RHom

Theoremznzrhval 16827 The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        RSpan       ~QG        ℤ/n       RHom

Theoremznzrhfo 16828 The ring homomorphism is a surjection onto . (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n              RHom

Theoremzncyg 16829 The group is cyclic for all (including ). (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       CycGrp

Theoremzndvds 16830 Express equality of equivalence classes in in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremzndvds0 16831 Special case of zndvds 16830 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremznf1o 16832 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 16833 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 16834 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremznleval 16835 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremznleval2 16836 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^

Theoremzntoslem 16837 Lemma for zntos 16838. (Contributed by Mario Carneiro, 15-Jun-2015.)
flds        ℤ/n       RHom        ..^                     Toset

Theoremzntos 16838 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 16839 The ℤ/nℤ structure has elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n

Theoremznfi 16840 The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n

Theoremznfld 16841 The ℤ/nℤ structure is a finite field when is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Field

Theoremznidomb 16842 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 16843 Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
ℤ/n       chr

Theoremznunit 16844 The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
ℤ/n       Unit       RHom

Theoremznunithash 16845 The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
ℤ/n       Unit

Theoremznrrg 16846 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 16847* Lemma for cygzn 16851. (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem2a 16848* Lemma for cygzn 16851. (Contributed by Mario Carneiro, 23-Dec-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem2 16849* Lemma for cygzn 16851. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
ℤ/n       .g       RHom              CycGrp

Theoremcygznlem3 16850* A cyclic group with elements is isomorphic to . (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       .g       RHom              CycGrp                     𝑔

Theoremcygzn 16851 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 16852* 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 16853 Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp 𝑔

Theoremfrgpcyg 16854 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 16855 Extend class notation with class all pre-Hilbert spaces.

Syntaxcipf 16856 Extend class notation with inner product function.

Definitiondf-phl 16857* Define class all generalized pre-Hilbert (inner product) spaces. (Contributed by NM, 22-Sep-2011.)
Scalar LMHom ringLMod

Definitiondf-ipf 16858* 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 14706), while only has closure (mndcl 14695). (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremisphl 16859* 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 16860 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremphllmod 16861 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremphlsrng 16862 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremphllmhm 16863* 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 16864 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipcj 16865 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremiporthcom 16866 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip0l 16867 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 16868 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipeq0 16869 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 16870 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 16871 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip2di 16872 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipsubdir 16873 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipsubdi 16874 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremip2subdi 16875 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremipass 16876 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipassr 16877 "Associative" law for second argument of inner product (compare ipass 16876). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipassr2 16878 "Associative" law for inner product. Conjugate version of ipassr 16877. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremipffval 16879* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)

Theoremipfval 16880 The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremipfeq 16881 If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremipffn 16882 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Theoremphlipf 16883 The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Scalar

Theoremip2eq 16884* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)

Theoremisphld 16885* Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremphlpropd 16886* If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar       Scalar

10.12.2  Orthocomplements and closed subspaces

Syntaxcocv 16887 Extend class notation with orthocomplement of a subspace.

Syntaxccss 16888 Extend class notation with set of closed subspaces.

Syntaxcthl 16889 Extend class notation with the Hilbert lattice.
toHL

Definitiondf-ocv 16890* Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.)
Scalar

Definitiondf-css 16891* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)

Definitiondf-thl 16892 Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL toInc sSet

Theoremocvfval 16893* The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Scalar

Theoremocvval 16894* Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Scalar

Theoremelocv 16895* Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Scalar

Theoremocvi 16896 Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
Scalar

Theoremocvss 16897 The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremocvocv 16898 A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremocvlss 16899 The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremocv2ss 16900 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)

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