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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempsr1bas2 16101 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
PwSer1              mPwSer

Theorempsr1bas 16102 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
PwSer1

Theoremvr1val 16103 The value of the generator of the power series algebra (the in ). Since all univariate polynomial rings over a fixed base ring are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
var1       mVar

Theoremvr1cl2 16104 The variable is a member of the power series algebra . (Contributed by Mario Carneiro, 8-Feb-2015.)
var1       PwSer1

Theoremply1val 16105 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1       s mPoly

Theoremply1bas 16106 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              mPoly

Theoremply1lss 16107 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1

Theoremply1subrg 16108 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       PwSer1              SubRing

Theoremply1crng 16109 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1

Theoremply1assa 16110 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1       AssAlg

Theorempsr1rclOLD 16111 Obsolete version of elbasfv 13065 as of 5-Apr-2016. Reverse closure for ring existence from the univariate power series base set. (Contributed by Stefan O'Rear, 25-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
PwSer1

Theorempsr1bascl 16112 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1              mPwSer

Theorempsr1basf 16113 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1

Theoremply1rclOLD 16114 Obsolete version of elbasfv 13065 as of 5-Apr-2016. Reverse closure for ring existence from the univariate polynomial base set. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Poly1

Theoremply1basf 16115 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1

Theoremply1bascl 16116 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              PwSer1

Theoremply1bascl2 16117 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1              mPoly

Theoremcoe1fval 16118* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fv 16119 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremfvcoe1 16120 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1

Theoremcoe1fval3 16121* Univariate power series coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1f2 16122 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
coe1              PwSer1

Theoremcoe1fval2 16123* Univariate polynomial coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1f 16124 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremcoe1sfi 16125 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1              Poly1

Theoremvr1cl 16126 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
var1       Poly1

Theoremopsr0 16127 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer

Theoremopsr1 16128 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPwSer        ordPwSer

Theoremmplplusg 16129 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

TheoremmplvscafvalOLD 16130 Obsolete version of mplvsca2 16022 as of 5-Apr-2016. Value of scalar multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
mPoly        mPwSer

Theoremmplmulr 16131 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
mPoly        mPwSer

Theorempsr1plusg 16132 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theorempsr1vsca 16133 Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theorempsr1mulr 16134 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
PwSer1       mPwSer

Theoremply1plusg 16135 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremply1vsca 16136 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremply1mulr 16137 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
Poly1       mPoly

Theoremressply1bas2 16138 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       PwSer1

Theoremressply1bas 16139 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1add 16140 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1mul 16141 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremressply1vsca 16142 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing       s

Theoremsubrgply1 16143 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
Poly1       s        Poly1              SubRing SubRing

Theorempsrbaspropd 16144 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer mPwSer

Theorempsrplusgpropd 16145* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPwSer mPwSer

Theoremmplbaspropd 16146* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPoly mPoly

Theoremstrov2rcl 16147 Reverse closure for polynomial-resembling things. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theorempsropprmul 16148 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
mPwSer        oppr       mPwSer

Theoremply1opprmul 16149 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       oppr       Poly1

Theorem00ply1bas 16150 Lemma for ply1basfvi 16151 and deg1fvi 19303. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1

Theoremply1basfvi 16151 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgfvi 16152 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1baspropd 16153* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremply1plusgpropd 16154* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1 Poly1

Theoremopsrrng 16155 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
ordPwSer

Theoremopsrlmod 16156 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
ordPwSer

Theorempsr1rng 16157 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
PwSer1

Theoremply1rng 16158 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Poly1

Theorempsr1lmod 16159 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
PwSer1

Theorempsr1sca 16160 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
PwSer1       Scalar

Theorempsr1sca2 16161 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
PwSer1       Scalar

Theoremply1lmod 16162 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1

Theoremply1sca 16163 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       Scalar

Theoremply1sca2 16164 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Poly1       Scalar

Theoremply1mpl0 16165 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPoly        Poly1

Theoremply1mpl1 16166 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
mPoly        Poly1

Theoremply1ascl 16167 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
Poly1       algSc       algSc mPoly

Theoremsubrg1ascl 16168 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
Poly1       algSc       s        Poly1       SubRing       algSc

Theoremsubrg1asclcl 16169 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
Poly1       algSc       s        Poly1       SubRing

Theoremsubrgvr1 16170 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
var1       SubRing       s        var1

Theoremsubrgvr1cl 16171 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
var1       SubRing       s        Poly1

Theoremcoe1z 16172 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                     coe1

Theoremcoe1add 16173 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
Poly1                            coe1 coe1 coe1

Theoremcoe1addfv 16174 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                            coe1 coe1 coe1

Theoremcoe1subfv 16175 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Poly1                            coe1 coe1coe1

Theoremcoe1mul2lem1 16176 An equivalence for coe1mul2 16178. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremcoe1mul2lem2 16177* An equivalence for coe1mul2 16178. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Theoremcoe1mul2 16178* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
PwSer1                            coe1 g coe1 coe1

Theoremcoe1mul 16179* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Poly1                            coe1 g coe1 coe1

Theoremply1tmcl 16180 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g

Theoremcoe1tm 16181* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g       coe1

Theoremcoe1tmfv1 16182 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g       coe1

Theoremcoe1tmfv2 16183 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                          coe1

Theoremcoe1tmmul2 16184* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                                        coe1 coe1

Theoremcoe1tmmul 16185* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       var1              mulGrp       .g                                                        coe1 coe1

Theoremcoe1tmmul2fv 16186 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       var1              mulGrp       .g                                                               coe1 coe1

Theoremcoe1pwmul 16187* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       var1       mulGrp       .g                                          coe1 coe1

Theoremcoe1pwmulfv 16188 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       var1       mulGrp       .g                                                 coe1 coe1

Theoremply1scltm 16189 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       var1              mulGrp       .g       algSc

Theoremcoe1sclmul 16190 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     algSc                     coe1 coe1

Theoremcoe1sclmulfv 16191 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1                     algSc                     coe1 coe1

Theoremcoe1sclmul2 16192 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     algSc                     coe1 coe1

Theoremply1sclf 16193 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc

Theoremcoe1scl 16194* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc                     coe1

Theoremply1sclid 16195 Recover the base scalar from a scalar polynomial.. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc              coe1

Theoremply1sclf1 16196 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       algSc

Theoremply1scl0 16197 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       algSc

Theoremply1scln0 16198 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       algSc

Theoremply1scl1 16199 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       algSc

Theoremply1coe 16200* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Poly1       var1                     mulGrp       .g       coe1              g

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