P. 333 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 33201-33300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	paddunN 33201	The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 5876.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	poldmj1N 33202	De Morgan's law for polarity of projective sum. (oldmj1 32496 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pmapj2N 33203	The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapocjN 33204	The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$

Theorem	polatN 33205	The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polatN 33206	Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pnonsingN 33207	The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
$/\$

Syntax	cpscN 33208	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.


Definition	df-psubclN 33209*	Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
$/\$

Theorem	psubclsetN 33210*	The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispsubclN 33211	The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubcliN 33212	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	psubcli2N 33213	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubclsubN 33214	A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubclssatN 33215	A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pmapidclN 33216	Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$

Theorem	0psubclN 33217	The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)


Theorem	1psubclN 33218	The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)


Theorem	atpsubclN 33219	A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pmapsubclN 33220	A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispsubcl2N 33221*	Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)


Theorem	psubclinN 33222	The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	paddatclN 33223	The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pclfinclN 33224	The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 33174 and also pclcmpatN 33175. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	linepsubclN 33225	A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	polsubclN 33226	A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	poml4N 33227	Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	poml5N 33228	Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	poml6N 33229	Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	osumcllem1N 33230	Lemma for osumclN 33241. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	osumcllem2N 33231	Lemma for osumclN 33241. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	osumcllem3N 33232	Lemma for osumclN 33241. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$

Theorem	osumcllem4N 33233	Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem5N 33234	Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	osumcllem6N 33235	Lemma for osumclN 33241. Use atom exchange hlatexch1 32669 to swap and . (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	osumcllem7N 33236*	Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem8N 33237	Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem9N 33238	Lemma for osumclN 33241. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem10N 33239	Lemma for osumclN 33241. Contradict osumcllem9N 33238. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem11N 33240	Lemma for osumclN 33241. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	osumclN 33241	Closure of orthogonal sum. If and are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmapojoinN 33242	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33126 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidN 33243	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33227. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33241. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pexmidlem1N 33244	Lemma for pexmidN 33243. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	pexmidlem2N 33245	Lemma for pexmidN 33243. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem3N 33246	Lemma for pexmidN 33243. Use atom exchange hlatexch1 32669 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem4N 33247*	Lemma for pexmidN 33243. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem5N 33248	Lemma for pexmidN 33243. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem6N 33249	Lemma for pexmidN 33243. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem7N 33250	Lemma for pexmidN 33243. Contradict pexmidlem6N 33249. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem8N 33251	Lemma for pexmidN 33243. The contradiction of pexmidlem6N 33249 and pexmidlem7N 33250 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pexmidALTN 33252	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 33227. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pl42lem1N 33253	Lemma for pl42N 33257. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	pl42lem2N 33254	Lemma for pl42N 33257. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42lem3N 33255	Lemma for pl42N 33257. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pl42lem4N 33256	Lemma for pl42N 33257. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42N 33257	Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Syntax	clh 33258	Extend class notation with set of all co-atoms (lattice hyperplanes).


Syntax	claut 33259	Extend class notation with set of all lattice automorphisms.


Syntax	cwpointsN 33260	Extend class notation with W points.


Syntax	cpautN 33261	Extend class notation with set of all projective automorphisms.


Definition	df-lhyp 33262*	Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)


Definition	df-laut 33263*	Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-watsN 33264*	Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" . These are all atoms not in the polarity of , which is the hyperplane determined by . Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
$\$

Definition	df-pautN 33265*	Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
$/\$

Theorem	watfvalN 33266*	The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$\$

Theorem	watvalN 33267	Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $\$

Theorem	iswatN 33268	The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lhpset 33269*	The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)


Theorem	islhp 33270	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islhp2 33271	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpbase 33272	A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)


Theorem	lhp1cvr 33273	The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhplt 33274	An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	lhp2lt 33275	The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpexlt 33276*	There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhp0lt 33277	A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhpn0 33278	A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexle 33279*	There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexnle 33280*	There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
$/\$

Theorem	lhpexle1lem 33281*	Lemma for lhpexle1 33282 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle1 33282*	There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$

Theorem	lhpexle2lem 33283*	Lemma for lhpexle2 33284. (Contributed by NM, 19-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle2 33284*	There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$

Theorem	lhpexle3lem 33285*	There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant,simp usage can be improved. (Contributed by NM, 9-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle3 33286*	There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	lhpex2leN 33287*	There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	lhpoc 33288	The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpoc2N 33289	The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$/\$

Theorem	lhpocnle 33290	The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
$/\$

Theorem	lhpocat 33291	The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
$/\$

Theorem	lhpocnel 33292	The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
$/\$ $/\$

Theorem	lhpocnel2 33293	The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
$/\$ $/\$

Theorem	lhpjat1 33294	The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpjat2 33295	The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpj1 33296	The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpmcvr 33297	The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmcvr2 33298*	Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr3 33299	Specialization of lhpmcvr2 33298. TODO: Use this to simplify many uses of $.\/$ $./\$ to become . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr4N 33300	Specialization of lhpmcvr2 33298. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$

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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 32801-32900 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 392 39101-39200 393 39201-39300 394 39301-39314

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