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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaov31 5901* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov13 5902* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov4 5903* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov411 5904* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov42 5905* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaovdir 5906* Reverse distributive law. (Contributed by NM, 26-Aug-1995.)

Theoremcaovdilem 5907* Lemma used by real number construction. (Contributed by NM, 26-Aug-1995.)

Theoremcaovlem2 5908* Lemma used in real number construction. (Contributed by NM, 26-Aug-1995.)

Theoremcaovmo 5909* Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)

Theoremgrprinvlem 5910* Lemma for grprinvd 5911. (Contributed by NM, 9-Aug-2013.)

Theoremgrprinvd 5911* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpridd 5912* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

2.4.9  "Maps to" notation

Theoremelmpt2cl 5913* If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremelmpt2cl1 5914* If a two-parameter class is not empty, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremelmpt2cl2 5915* If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremelovmpt2 5916* Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 14518, islmhm 15619. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremrelmptopab 5917* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremf1ocnvd 5918* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremf1od 5919* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremf1ocnv2d 5920* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremf1o2d 5921* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

TheoremxpexgALT 5922 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4707 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.)

Theoremf1opw2 5923* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5924 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)

Theoremf1opw 5924* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremsuppss2 5925* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremsuppssfv 5926* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremsuppssov1 5927* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.10  Function operation

Syntaxcof 5928 Extend class notation to include mapping of an operation to a function operation.

Syntaxcofr 5929 Extend class notation to include mapping of an binary relation to a function relation.

Definitiondf-of 5930* Define the function operation map. The definition is designed so that if is a binary operation, then is the analogous operation on functions which corresponds to applying pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Definitiondf-ofr 5931* Define the function relation map. The definition is designed so that if is a binary relation, then is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofeq 5932 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofreq 5933 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofexg 5934 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)

Theoremnfof 5935* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremnfofr 5936* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffval 5937* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval 5938* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofval 5939 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrval 5940 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffn 5941 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremfnfvof 5942 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)

Theoremoffval3 5943* General value of with no assumptions on functionality of and . (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoffres 5944 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoff 5945* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofres 5946 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremoffval2 5947* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval2 5948* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofco 5949 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremoffveq 5950* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremoffveqb 5951* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremofc1 5952 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofc2 5953 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)

Theoremofc12 5954 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofref 5955* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofinvl 5956* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Theoremcaofid0l 5957* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid0r 5958* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid1 5959* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofid2 5960* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofcom 5961* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofrss 5962* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofass 5963* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaoftrn 5964* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofdi 5965* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofdir 5966* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)

Theoremcaonncan 5967* Transfer nncan 8956-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theoremofmres 5968* Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 5970, allowing it to be used as a function or structure argument. By ofmresval 5969, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Theoremofmresval 5969 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremofmresex 5970 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremsuppssof1 5971* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.11  First and second members of an ordered pair

Syntaxc1st 5972 Extend the definition of a class to include the first member an ordered pair function.

Syntaxc2nd 5973 Extend the definition of a class to include the second member an ordered pair function.

Definitiondf-1st 5974 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5980 proves that it does this. For example, . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5060 and op1stb 4460). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Definitiondf-2nd 5975 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5981 proves that it does this. For example, . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5063 and op2ndb 5062). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Theorem1stval 5976 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem2ndval 5977 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem1st0 5978 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theorem2nd0 5979 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theoremop1st 5980 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop2nd 5981 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop1std 5982 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop2ndd 5983 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop1stg 5984 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremop2ndg 5985 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremot1stg 5986 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5986, ot2ndg 5987, ot3rdg 5988.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot2ndg 5987 Extract the second member of an ordered triple. (See ot1stg 5986 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot3rdg 5988 Extract the third member of an ordered triple. (See ot1stg 5986 comment.) (Contributed by NM, 3-Apr-2015.)

Theorem1stval2 5989 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theorem2ndval2 5990 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theoremfo1st 5991 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo2nd 5992 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf1stres 5993 Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf2ndres 5994 Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo1stres 5995 Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theoremfo2ndres 5996 Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theorem1st2val 5997* Value of an alternate definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem2nd2val 5998* Value of an alternate definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem1stcof 5999 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Theorem2ndcof 6000 Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)

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