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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembrtpid2 25101 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)

Theorembrtpid3 25102 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)

Theoremceqsrexv2 25103* Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)

Theoremiota5f 25104* A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)

Theoremdfid4 25105 The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)

19.7.5  Properties of reals and complexes

Theoremsqdivzi 25106 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremdivelunit 25107 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theorempm2.61iine 25108 Equality version of pm2.61ii 159. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremdedekind 25109* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 9015 with appropriate adjustments, states that, if completely preceeds , then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremdedekindle 25110* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)

Theoremmulcan1g 25111 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremmulcan2g 25112 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremmulge0b 25113 A condition for multiplication to be non-negative. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulle0b 25114 A condition for multiplication to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulsuble0b 25115 A condition for multiplication of subtraction to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremrelin01 25116 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)

Theoremsubdivcomb1 25117 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)

Theoremsubdivcomb2 25118 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)

Theoremsubeqrev 25119 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)

Theoremfznatpl1 25120 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremsupfz 25121 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theoreminffz 25122 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theorembcnm1 25123 The binomial coefficent of is . (Contributed by Scott Fenton, 16-May-2014.)

Theoremfz0n 25124 The sequence is empty iff is zero. (Contributed by Scott Fenton, 16-May-2014.)

Theorem4bc3eq4 25125 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)

Theorem4bc2eq6 25126 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremhalfthird 25127 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)

Theorem5recm6rec 25128 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
;

Theorempnpncand 25129 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremshftvalg 25130 Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)

Theorempossumd 25131 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfzp1nel 25132 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremdivcnvshft 25133* Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremdivcnvlin 25134* Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)

Theoremmuls1d 25135 Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.)

Theoremclimlec3 25136* Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)

19.7.6  Product sequences

Theoremprodf 25137* An infinite product of complex terms is a function from an upper set of integers to . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremclim2prod 25138* The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremclim2div 25139* The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremprodfmul 25140* The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremprodf1 25141 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodf1f 25142 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfclim1 25143 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfn0 25144* No term of a non-zero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)

Theoremprodfrec 25145* The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremprodfdiv 25146* The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)

19.7.7  Non-trivial convergence

Theoremntrivcvg 25147* A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremntrivcvgn0 25148* A product that converges to a non-zero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremntrivcvgfvn0 25149* Any value of a product sequence that converges to a non-zero value is itself non-zero. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremntrivcvgtail 25150* A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremntrivcvgmullem 25151* Lemma for ntrivcvgmul 25152. (Contributed by Scott Fenton, 19-Dec-2017.)

Theoremntrivcvgmul 25152* The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)

19.7.8  Complex products

Syntaxcprod 25153 Extend class notation to include complex products.

Definitiondf-prod 25154* Define the product of a series with an index set of integers . This definition takes most of the aspects of df-sum 12421 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a non-zero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodex 25155 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1f 25156 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremprodeq1 25157* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremnfcprod1 25158* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremnfcprod 25159* Bound-variable hypothesis builder for product: if is (effectively) not free in and , it is not free in . (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremprodeq2w 25160* Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2ii 25161* Equality theorem for product, with the class expressions and guarded by to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2 25162* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprod 25163* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodv 25164* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodi 25165* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1i 25166* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2i 25167* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12i 25168* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1d 25169* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2d 25170* Equality deduction for sum. Note that unlike prodeq2dv 25171, may occur in . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2dv 25171* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2sdv 25172* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theorem2cprodeq2dv 25173* Equality deduction for double sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12dv 25174* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12rdv 25175* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprod2id 25176* The second class argument to a sum can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrblem 25177* Lemma for prodrb 25180. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremfprodcvg 25178* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrblem2 25179* Lemma for prodrb 25180. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrb 25180* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem3 25181* Lemma for prodmo 25184. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2a 25182* Lemma for prodmo 25184. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2 25183* Lemma for prodmo 25184. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmo 25184* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremzprod 25185* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremiprod 25186* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremzprodn0 25187* Non-zero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremiprodn0 25188* Non-zero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)

19.7.9  Finite products

Theoremfprod 25189* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremfprodntriv 25190* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremprod0 25191 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprod1 25192* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodfc 25193* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremfprodf1o 25194* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodss 25195* Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.)

Theoremfprodss 25196* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprodser 25197* A finite product expressed in terms of a partial product of an infinite sequence. The recursive definition of a finite product follows from here. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcl2lem 25198* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcllem 25199* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcl 25200* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

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