Home Metamath Proof ExplorerTheorem List (p. 159 of 329) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22426) Hilbert Space Explorer (22427-23949) Users' Mathboxes (23950-32836)

Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvrcan1 15801 A cancellation law for division. (divcan1 9692 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit       /r

Theoremdvrcan3 15802 A cancellation law for division. (divcan3 9707 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
Unit       /r

Theoremdvreq1 15803 A cancellation law for division. (diveq1 9713 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
Unit       /r

Theoremrnginvdv 15804 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
Unit       /r

Theoremrngidpropd 15805* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Theoremdvdsrpropd 15806* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
r r

Theoremunitpropd 15807* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Unit Unit

Theoreminvrpropd 15808* The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)

Theoremisirred 15809* An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremisnirred 15810* The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremisirred2 15811* Expand out the set differences from isirred 15809. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremopprirred 15812 Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
oppr       Irred       Irred

Theoremirredn0 15813 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirredcl 15814 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirrednu 15815 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredn1 15816 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirredrmul 15817 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredlmul 15818 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredmul 15819 If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred              Unit

Theoremirredneg 15820 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirrednegb 15821 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

10.4.5  Ring homomorphisms

Syntaxcrh 15822 Ring homomorphisms.
RingHom

Syntaxcrs 15823 Ring isomorphisms.
RingIso

Definitiondf-rnghom 15824* Define the set of ring homomorphisms from to . (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Definitiondf-rngiso 15825* Define the set of ring isomorphisms from to . (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingIso RingHom RingHom

Theoremdfrhm2 15826* The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom mulGrp MndHom mulGrp

Theoremrhmrcl1 15827 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremrhmrcl2 15828 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremisrhm 15829 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmmhm 15830 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmghm 15831 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremrhmf 15832 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmmul 15833 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom

Theoremisrhm2d 15834* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
RingHom

Theoremisrhmd 15835* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhm1 15836 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmco 15837 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom RingHom RingHom

Theorempwsco1rhm 15838* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                                           RingHom

Theorempwsco2rhm 15839* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                      RingHom        RingHom

10.5  Division rings and fields

10.5.1  Definition and basic properties

Syntaxcdr 15840 Extend class notation with class of all division rings.

Syntaxcfield 15841 Class of fields.
Field

Definitiondf-drng 15842 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
Unit

Definitiondf-field 15843 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng 15844 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngunit 15845 Elementhood in the set of units when is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngui 15846 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngrng 15847 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)

Theoremdrnggrp 15848 A division ring is a group. (Contributed by NM, 8-Sep-2011.)

Theoremisfld 15849 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng2 15850 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
mulGrps

Theoremdrngprop 15851 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)

Theoremdrngmgp 15852 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
mulGrps

Theoremdrngmcl 15853 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)

Theoremdrngid 15854 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
mulGrps

Theoremdrngunz 15855 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)

Theoremdrngid2 15856 Properties showing that an element is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremdrnginvrcl 15857 Closure of the multiplicative inverse in a division ring. (reccl 9690 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrn0 15858 The multiplicative inverse in a division ring is nonzero. (recne0 9696 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrl 15859 Property of the multiplicative inverse in a division ring. (recid2 9698 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrr 15860 Property of the multiplicative inverse in a division ring. (recid 9697 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrngmul0or 15861 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)

Theoremdrngmulne0 15862 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)

Theoremdrngmuleq0 15863 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)

Theoremopprdrng 15864 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
oppr

Theoremisdrngd 15865* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." (Contributed by NM, 2-Aug-2013.)

Theoremisdrngrd 15866* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 15865 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)

Theoremdrngpropd 15867* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)

Theoremfldpropd 15868* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Field Field

10.5.2  Subrings of a ring

Syntaxcsubrg 15869 Extend class notation with all subrings of a ring.
SubRing

Syntaxcrgspn 15870 Extend class notation with span of a set of elements over a ring.
RingSpan

Definitiondf-subrg 15871* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset of (where multiplication is component-wise) contains the false identity which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

SubRing s

Definitiondf-rgspn 15872* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan SubRing

Theoremissubrg 15873 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing s

Theoremsubrgss 15874 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgid 15875 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing

Theoremsubrgrng 15876 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrgcrng 15877 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        SubRing

Theoremsubrgrcl 15878 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing

Theoremsubrgsubg 15879 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremsubrg0 15880 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrg1cl 15881 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgbas 15882 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrg1 15883 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrgacl 15884 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgmcl 15885 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgsubm 15886 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
mulGrp       SubRing SubMnd

Theoremsubrgdvds 15887 If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        r       r       SubRing

Theoremsubrguss 15888 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       SubRing

Theoremsubrginv 15889 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s               Unit              SubRing

Theoremsubrgdv 15890 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        /r       Unit       /r       SubRing

Theoremsubrgunit 15891 An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit              SubRing

Theoremsubrgugrp 15892 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       mulGrps        SubRing SubGrp

Theoremissubrg2 15893* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremopprsubrg 15894 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15895 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing

Theoremsubrgin 15896 The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing SubRing

Theoremsubrgmre 15897 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15898* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15899 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15900 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32836
 Copyright terms: Public domain < Previous  Next >