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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremringcinv 38801 An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
RingCat                                   Inv       RingIso

Theoremringciso 38802 An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)
RingCat                                          RingIso

Theoremringcbasbas 38803 An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.)
RingCat              WUni

Theoremfuncringcsetc 38804* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
RingCat                     WUni              RingHom

TheoremfuncringcsetcALTV2lem1 38805* Lemma 1 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni

TheoremfuncringcsetcALTV2lem2 38806* Lemma 2 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni

TheoremfuncringcsetcALTV2lem3 38807* Lemma 3 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni

TheoremfuncringcsetcALTV2lem4 38808* Lemma 4 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom

TheoremfuncringcsetcALTV2lem5 38809* Lemma 5 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom        RingHom

TheoremfuncringcsetcALTV2lem6 38810* Lemma 6 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom        RingHom

TheoremfuncringcsetcALTV2lem7 38811* Lemma 7 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom

TheoremfuncringcsetcALTV2lem8 38812* Lemma 8 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom

TheoremfuncringcsetcALTV2lem9 38813* Lemma 9 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom        comp comp

TheoremfuncringcsetcALTV2 38814* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
RingCat                            WUni              RingHom

TheoremringcbasALTV 38815 Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremringchomfvalALTV 38816* Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            RingHom

TheoremringchomALTV 38817 Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                                          RingHom

TheoremelringchomALTV 38818 A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremringccofvalALTV 38819* Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                     comp       RingHom RingHom

TheoremringccoALTV 38820 Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                     comp                            RingHom        RingHom

TheoremringccatidALTV 38821* Lemma for ringccatALTV 38822. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremringccatALTV 38822 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremringcidALTV 38823 The identity arrow in the category of rings is the identity function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremringcsectALTV 38824 A section in the category of rings, written out. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                                          Sect       RingHom RingHom

TheoremringcinvALTV 38825 An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                                   Inv       RingIso

TheoremringcisoALTV 38826 An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
RingCatALTV                                          RingIso

TheoremringcbasbasALTV 38827 An element of the base set of the base set of the category of rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV              WUni

Theoremfuncringcsetclem1ALTV 38828* Lemma 1 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni

Theoremfuncringcsetclem2ALTV 38829* Lemma 2 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni

Theoremfuncringcsetclem3ALTV 38830* Lemma 3 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni

Theoremfuncringcsetclem4ALTV 38831* Lemma 4 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom

Theoremfuncringcsetclem5ALTV 38832* Lemma 5 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom        RingHom

Theoremfuncringcsetclem6ALTV 38833* Lemma 6 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom        RingHom

Theoremfuncringcsetclem7ALTV 38834* Lemma 7 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom

Theoremfuncringcsetclem8ALTV 38835* Lemma 8 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom

Theoremfuncringcsetclem9ALTV 38836* Lemma 9 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom        comp comp

TheoremfuncringcsetcALTV 38837* The "natural forgetful functor" from the category of rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
RingCatALTV                            WUni              RingHom

Theoremirinitoringc 38838 The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
ring        RingCat       ring InitO

Theoremzrtermoringc 38839 The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
RingCat       NzRing              TermO

Theoremzrninitoringc 38840* The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
RingCat       NzRing              NzRing       InitO

Theoremnzerooringczr 38841 There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
RingCat       NzRing              ring        ZeroO

21.33.12.10  Subcategories of the category of rings

Theoremsrhmsubclem1 38842* Lemma 1 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)

Theoremsrhmsubclem2 38843* Lemma 2 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)
RingCat

Theoremsrhmsubclem3 38844* Lemma 3 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)
RingHom        RingCat

Theoremsrhmsubc 38845* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
RingHom        SubcatRingCat

Theoremsringcat 38846* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
RingHom        RingCat cat

Theoremcrhmsubc 38847* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
RingHom        SubcatRingCat

Theoremcringcat 38848* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
RingHom        RingCat cat

Theoremdrhmsubc 38849* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
RingHom        SubcatRingCat

Theoremdrngcat 38850* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
RingHom        RingCat cat

Theoremfldcat 38851* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
RingHom        Field       RingHom        RingCat cat

Theoremfldc 38852* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
RingHom        Field       RingHom        RingCat cat cat

Theoremfldhmsubc 38853* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
RingHom        Field       RingHom        SubcatRingCat cat

Theoremrngcrescrhm 38854 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
RngCat              RingHom        cat s sSet

Theoremrhmsubclem1 38855 Lemma 1 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
RngCat              RingHom

Theoremrhmsubclem2 38856 Lemma 2 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
RngCat              RingHom        RingHom

Theoremrhmsubclem3 38857* Lemma 3 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
RngCat              RingHom        RngCat

Theoremrhmsubclem4 38858* Lemma 4 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
RngCat              RingHom        compRngCat

Theoremrhmsubc 38859 According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
RngCat              RingHom        SubcatRngCat

Theoremrhmsubccat 38860 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
RngCat              RingHom        RngCat cat

TheoremsrhmsubcALTVlem1 38861* Lemma 1 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)

TheoremsrhmsubcALTVlem2 38862* Lemma 2 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingCatALTV

TheoremsrhmsubcALTVlem3 38863* Lemma 3 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingHom        RingCatALTV

TheoremsrhmsubcALTV 38864* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingHom        SubcatRingCatALTV

TheoremsringcatALTV 38865* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingHom        RingCatALTV cat

TheoremcrhmsubcALTV 38866* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingHom        SubcatRingCatALTV

TheoremcringcatALTV 38867* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
RingHom        RingCatALTV cat

TheoremdrhmsubcALTV 38868* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
RingHom        SubcatRingCatALTV

TheoremdrngcatALTV 38869* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
RingHom        RingCatALTV cat

TheoremfldcatALTV 38870* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
RingHom        Field       RingHom        RingCatALTV cat

TheoremfldcALTV 38871* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
RingHom        Field       RingHom        RingCatALTV cat cat

TheoremfldhmsubcALTV 38872* According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
RingHom        Field       RingHom        SubcatRingCatALTV cat

TheoremrngcrescrhmALTV 38873 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        cat s sSet

TheoremrhmsubcALTVlem1 38874 Lemma 1 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom

TheoremrhmsubcALTVlem2 38875 Lemma 2 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        RingHom

TheoremrhmsubcALTVlem3 38876* Lemma 3 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        RngCatALTV

TheoremrhmsubcALTVlem4 38877* Lemma 4 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        compRngCatALTV

TheoremrhmsubcALTV 38878 According to df-subc 15668, the subcategories Subcat of a category are subsets of the homomorphisms of ( see subcssc 15696 and subcss2 15699). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        SubcatRngCatALTV

TheoremrhmsubcALTVcat 38879 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
RngCatALTV              RingHom        RngCatALTV cat

21.33.13  Basic algebraic structures (extension)

21.33.13.1  Auxiliary theorems

Theoremrabeqsn 38880* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)

Theoremrabsssn 38881* Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)

Theoremxpprsng 38882 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)

Theoremopeliun2xp 38883 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 4906. (Contributed by AV, 30-Mar-2019.)

Theoremeliunxp2 38884* Membership in a union of Cartesian products over its second component, analogous to eliunxp 4992. (Contributed by AV, 30-Mar-2019.)

Theoremmpt2mptx2 38885* Express a two-argument function as a one-argument function, or vice-versa. In this version is not assumed to be constant w.r.t , analogous to mpt2mptx 6401. (Contributed by AV, 30-Mar-2019.)

Theoremcbvmpt2x2 38886* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6384 allows to be a function of , analogous to cbvmpt2x 6383. (Contributed by AV, 30-Mar-2019.)

Theoremdmmpt2ssx2 38887* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6872. (Contributed by AV, 30-Mar-2019.)

Theoremmpt2exxg2 38888* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpt2exxg 6881. (Contributed by AV, 30-Mar-2019.)

Theoremovmpt2rdxf 38889* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6436. (Contributed by AV, 30-Mar-2019.)

Theoremovmpt2rdx 38890* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6436. (Contributed by AV, 30-Mar-2019.)

Theoremovmpt2x2 38891* The value of an operation class abstraction. Variant of ovmpt2ga 6440 which does not require and to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremfdmdifeqresdif 38892* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)

Theoremoffvalfv 38893* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)

Theoremofaddmndmap 38894 The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)

Theoremmapsnop 38895 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)

Theoremmapprop 38896 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)

Theoremztprmneprm 38897 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)

Theorem2t6m3t4e0 38898 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)

Theoremssnn0ssfz 38899* For any finite subset of , find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28203. (Contributed by AV, 30-Sep-2019.)

Theoremnn0sumltlt 38900 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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