P. 153 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 15201-15300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	mstri2 15201	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmstri 15202	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	mstri 15203	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmstri3 15204	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	mstri3 15205	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	msrtri 15206	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmspropd 15207	Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)


Theorem	mspropd 15208	Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)


Theorem	setsmsbasg 15209	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theorem	setsmsdsg 15210	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theorem	setsmstsetg 15211	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
sSet TopSet TopSet

Theorem	mopni 15212*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	mopni2 15213*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	mopni3 15214*	An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	blssopn 15215	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)


Theorem	unimopn 15216	The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$

Theorem	mopnin 15217	The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$ $/\$

Theorem	mopn0 15218	The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)


Theorem	rnblopn 15219	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
$/\$

Theorem	blopn 15220	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	neibl 15221*	The neighborhoods around a point of a metric space are those subsets containing a ball around . Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$ $/\$

Theorem	blnei 15222	A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$

Theorem	blsscls2 15223*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	metss 15224*	Two ways of saying that metric generates a finer topology than metric . (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$

Theorem	metequiv 15225*	Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	metequiv2 15226*	If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
$/\$ $/\$

Theorem	metss2lem 15227*	Lemma for metss2 15228. (Contributed by Mario Carneiro, 14-Sep-2015.)
$/\$ $/\$ $/\$ $/\$

Theorem	metss2 15228*	If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), then generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
$/\$ $/\$

Theorem	comet 15229*	The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	bdmetval 15230*	Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
inf $/\$ $/\$ $/\$ inf

Theorem	bdxmet 15231*	The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
inf $/\$ $/\$

Theorem	bdmet 15232*	The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf $/\$

Theorem	bdbl 15233*	The standard bounded metric corresponding to generates the same balls as for radii less than . (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	bdmopn 15234*	The standard bounded metric corresponding to generates the same topology as . (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf $/\$ $/\$

Theorem	mopnex 15235*	The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	metrest 15236	Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
$/\$ ↾_t

Theorem	xmetxp 15237*	The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)


Theorem	xmetxpbl 15238*	The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point with radius . (Contributed by Jim Kingdon, 22-Oct-2023.)


Theorem	xmettxlem 15239*	Lemma for xmettx 15240. (Contributed by Jim Kingdon, 15-Oct-2023.)


Theorem	xmettx 15240*	The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)


9.2.5 Continuity in metric spaces

Theorem	metcnp3 15241*	Two ways to express that is continuous at for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
$/\$ $/\$ $/\$

Theorem	metcnp 15242*	Two ways to say a mapping from metric to metric is continuous at point . (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
$/\$ $/\$ $/\$

Theorem	metcnp2 15243*	Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 15242 (and Munkres' metcn 15244) for compatibility with df-lm 14920. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	metcn 15244*	Two ways to say a mapping from metric to metric is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" there is a positive "delta" such that a distance less than delta in maps to a distance less than epsilon in . (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
$/\$ $/\$

Theorem	metcnpi 15245*	Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 15242. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	metcnpi2 15246*	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 15243. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	metcnpi3 15247*	Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 15246 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	txmetcnp 15248*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	txmetcn 15249*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
$/\$ $/\$ $/\$ $/\$

Theorem	metcnpd 15250*	Two ways to say a mapping from metric to metric is continuous at point . (Contributed by Jim Kingdon, 14-Jun-2023.)
$/\$

9.2.6 Topology on the reals

Theorem	qtopbasss 15251*	The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
$/\$ $/\$ inf

Theorem	qtopbas 15252	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)


Theorem	retopbas 15253	A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)


Theorem	retop 15254	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)


Theorem	uniretop 15255	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)


Theorem	retopon 15256	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
TopOn

Theorem	retps 15257	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
TopSet

Theorem	iooretopg 15258	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
$/\$

Theorem	cnmetdval 15259	Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
$/\$

Theorem	cnmet 15260	The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)


Theorem	cnxmet 15261	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)


Theorem	cntoptopon 15262	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
TopOn

Theorem	cntoptop 15263	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)


Theorem	cnbl0 15264	Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)


Theorem	cnblcld 15265*	Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)


Theorem	cnfldms 15266	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
ℂ_fld

Theorem	cnfldxms 15267	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
ℂ_fld

Theorem	cnfldtps 15268	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
ℂ_fld

Theorem	cnfldtopn 15269	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
ℂ_fld

Theorem	cnfldtopon 15270	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
ℂ_fld TopOn

Theorem	cnfldtop 15271	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
ℂ_fld

Theorem	unicntopcntop 15272	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)


Theorem	unicntop 15273	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℂ_fld

Theorem	cnopncntop 15274	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)


Theorem	cnopn 15275	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℂ_fld

Theorem	reopnap 15276*	The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
#

Theorem	remetdval 15277	Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
$/\$

Theorem	remet 15278	The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)


Theorem	rexmet 15279	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)


Theorem	bl2ioo 15280	A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
$/\$

Theorem	ioo2bl 15281	An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
$/\$

Theorem	ioo2blex 15282	An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
$/\$

Theorem	blssioo 15283	The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)


Theorem	tgioo 15284	The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)


Theorem	tgqioo 15285	The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)


Theorem	resubmet 15286	The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
↾_t

Theorem	tgioo2cntop 15287	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
↾_t

Theorem	rerestcntop 15288	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
↾_t ↾_t

Theorem	tgioo2 15289	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
ℂ_fld ↾_t

Theorem	rerest 15290	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
ℂ_fld ↾_t ↾_t

Theorem	addcncntoplem 15291*	Lemma for addcncntop 15292, subcncntop 15293, and mulcncntop 15294. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
$/\$ $/\$ $/\$

Theorem	addcncntop 15292	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)


Theorem	subcncntop 15293	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)


Theorem	mulcncntop 15294	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)


Theorem	divcnap 15295*	Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
↾_t # #

Theorem	mpomulcn 15296*	Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
ℂ_fld

Theorem	fsumcncntop 15297*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for normally contains free variables and to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
TopOn $/\$

Theorem	fsumcn 15298*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for normally contains free variables and to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
ℂ_fld TopOn $/\$

Theorem	expcn 15299*	The power function on complex numbers, for fixed exponent , is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 8155. (Revised by GG, 16-Mar-2025.)
ℂ_fld

9.2.7 Topological definitions using the reals

Syntax	ccncf 15300	Extend class notation to include the operation which returns a class of continuous complex functions.

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