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Type | Label | Description |
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Statement | ||
Theorem | mnd12g 16601 | Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | mnd4g 16602 | Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | mndidcl 16603 | The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
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Theorem | mndplusf 16604 | The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.) |
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Theorem | mndlrid 16605 | A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | mndlid 16606 | The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | mndrid 16607 | The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | ismndd 16608* | Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | mndpfo 16609 | The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.) |
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Theorem | mndfo 16610 | The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.) |
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Theorem | mndpropd 16611* | If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | mndprop 16612 | If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
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Theorem | issubmnd 16613* | Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Theorem | ress0g 16614 |
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Theorem | submnd0 16615 | The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element.) (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Theorem | prdsplusgcl 16616 | Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
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Theorem | prdsidlem 16617* | Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Theorem | prdsmndd 16618 | The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
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Theorem | prds0g 16619 | Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
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Theorem | pwsmnd 16620 | The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.) |
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Theorem | pws0g 16621 | Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.) |
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Theorem | imasmnd2 16622* | The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | imasmnd 16623* | The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | imasmndf1 16624 | The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.) |
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Theorem | xpsmnd 16625 | The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.) |
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Theorem | mnd1 16626 | The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.) |
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Theorem | mnd1OLD 16627 | The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) Obsolete version of mnd1 16626 as of 11-Feb-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | mnd1id 16628 | The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.) |
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Syntax | cmhm 16629 | Hom-set generator class for monoids. |
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Syntax | csubmnd 16630 | Class function taking a monoid to its lattice of submonoids. |
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Definition | df-mhm 16631* | A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Definition | df-submnd 16632* | A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | ismhm 16633* | Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmrcl1 16634 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmrcl2 16635 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmf 16636 | A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmpropd 16637* | Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.) |
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Theorem | mhmlin 16638 | A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhm0 16639 | A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | idmhm 16640 | The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.) |
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Theorem | mhmf1o 16641 | A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.) |
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Theorem | submrcl 16642 | Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubm 16643* | Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubm2 16644 | Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubmd 16645* | Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | submss 16646 | Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | submid 16647 | Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | subm0cl 16648 | Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | submcl 16649 | Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
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Theorem | submmnd 16650 | Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | submbas 16651 | The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.) |
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Theorem | subm0 16652 | Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | subsubm 16653 | A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.) |
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Theorem | 0mhm 16654 | The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | resmhm 16655 | Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.) |
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Theorem | resmhm2 16656 | One direction of resmhm2b 16657. (Contributed by Mario Carneiro, 18-Jun-2015.) |
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Theorem | resmhm2b 16657 | Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.) |
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Theorem | mhmco 16658 | The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
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Theorem | mhmima 16659 | The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) |
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Theorem | mhmeql 16660 | The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
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Theorem | submacs 16661 | Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | mrcmndind 16662* | (( From SO's determinants branch )). TODO: Appropriate description to be added! (Contributed by SO, 14-Jul-2018.) |
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Theorem | prdspjmhm 16663* | A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
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Theorem | pwspjmhm 16664* | A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.) |
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Theorem | pwsdiagmhm 16665* | Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.) |
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Theorem | pwsco1mhm 16666* | Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
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Theorem | pwsco2mhm 16667* | Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
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One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 15390. If order is not significant, it is simpler to use families instead. | ||
Theorem | gsumvallem2 16668* |
Lemma for properties of the set of identities of ![]() |
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Theorem | gsumsubm 16669 | Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.) |
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Theorem | gsumz 16670* | Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
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Theorem | gsumwsubmcl 16671 | Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) |
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Theorem | gsumws1 16672 | A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
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Theorem | gsumwcl 16673 |
Closure of the composite of a word in a structure ![]() |
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Theorem | gsumccat 16674 | Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) |
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Theorem | gsumws2 16675 | Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | gsumccatsn 16676 | Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.) |
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Theorem | gsumspl 16677 | The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
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Theorem | gsumwmhm 16678 | Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
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Theorem | gsumwspan 16679* | The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Syntax | cfrmd 16680 | Extend class definition with the free monoid construction. |
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Syntax | cvrmd 16681 | Extend class notation with free monoid injection. |
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Definition | df-frmd 16682 |
Define a free monoid over a set ![]() ![]() |
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Definition | df-vrmd 16683* |
Define a free monoid over a set ![]() ![]() |
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Theorem | frmdval 16684 | Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | frmdbas 16685 | The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmdelbas 16686 | An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmdplusg 16687 | The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmdadd 16688 | Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | vrmdfval 16689* |
The canonical injection from the generating set ![]() |
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Theorem | vrmdval 16690 | The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
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Theorem | vrmdf 16691 | The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmdmnd 16692 | A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmd0 16693 | The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | frmdsssubm 16694 | The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | frmdgsum 16695 | Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | frmdss2 16696 |
A subset of generators is contained in a submonoid iff the set of words
on the generators is in the submonoid. This can be viewed as an
elementary way of saying "the monoidal closure of ![]() ![]() |
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Theorem | frmdup1 16697* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | frmdup2 16698* | The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.) |
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Theorem | frmdup3lem 16699* | Lemma for frmdup3 16700. (Contributed by Mario Carneiro, 18-Jul-2016.) |
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Theorem | frmdup3 16700* | Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.) |
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