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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbiota1 26801 Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremsbaniota 26802 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremeubi 26803 Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotasbcq 26804 Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)

16.17.5  Set Theory

Theoremelnev 26805* Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)

TheoremrusbcALT 26806 A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (Proof modification is discouraged.)

Theoremcompel 26807 Equivalence between two ways of saying "is a member of the complement of ." (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremcompeq 26808* Equality between two ways of saying "the complement of ." (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremcompne 26809 The complement of is not equal to . (Contributed by Andrew Salmon, 15-Jul-2011.)

TheoremcompneOLD 26810 Obsolete proof of compne 26809 as of 28-Jun-2015. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcompab 26811 Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremconss34 26812 Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremconss2 26813 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremconss1 26814 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremralbidar 26815 More general form of ralbida 2521. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremrexbidar 26816 More general form of rexbida 2522. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremdropab1 26817 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremdropab2 26818 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremipo0 26819 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremifr0 26820 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremordpss 26821 ordelpss 4313 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)

Theoremfvsb 26822* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremfveqsb 26823* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

TheoremxrltneNEW 26824 'Less than' implies not equal for extended reals. (Contributed by Andrew Salmon, 11-Nov-2011.)

Theoremxpexb 26825 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

Theoremxpexcnv 26826 A condition where the converse of xpex 4708 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

Theoremtrelpss 26827 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4270, ax-reg 7190 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)

16.17.6  Arithmetic

Theoremaddcomgi 26828 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)

16.17.7  Geometry

Syntaxcplusr 26829 Introduce the operation of vector addition.

Syntaxcminusr 26830 Introduce the operation of vector subtraction.

Syntaxctimesr 26831 Introduce the operation of scalar multiplication.

Syntaxcptdfc 26832 is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.

Syntaxcrr3c 26833 is a class.

Syntaxcline3 26834 is a class.

Definitiondf-addr 26835* Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

Definitiondf-subr 26836* Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)

Definitiondf-mulv 26837* Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremaddrval 26838* Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremsubrval 26839* Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremmulvval 26840* Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremaddrfv 26841 Vector addition at a value. The operation takes each vector and and forms a new vector whose values are the sum of each of the values of and . (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremsubrfv 26842 Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremmulvfv 26843 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremaddrfn 26844 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremsubrfn 26845 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

Theoremmulvfn 26846 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

Definitiondf-ptdf 26848* Define the predicate , which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)

Definitiondf-rr3 26849 Define the set of all points . We define each point as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)

Definitiondf-line3 26850* Define the set of all lines. A line is an infinite subset of that satisfies a property. (Contributed by Andrew Salmon, 15-Jul-2012.)

16.18  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 26851 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.)

Theoremax3h 26852 Recovery of ax-3 9 from hirstL-ax3 26851. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.)

Theoremnotatnand 26853 Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaistia 26854 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 26855 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 26856 Given both a,b are equivalent to T., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorembothfbothsame 26857 Given both a,b are equivalent to F., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaiffbbtat 26858 Given a is equivalent to b, b is equivalent to T. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaisbbisfaisf 26859 Given a is equivalent to b, b is equivalent to F. there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaibnbna 26860 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 26861 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 26862 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremastbstanbst 26863 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaisbnaxb 26864 Given a is equivalent to b, there exists a proof for (not (a xor b)) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 26865 Given a implies b, there exists a proof for a implies not not b (Contributed by Jarvin Udandy, created by Norman Megill 2-Sep-2016.)

Theorematbiffatnnb 26866 If a implies b, is is implied a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 26867 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy 31-Aug-2016.)

Theorematbiffatnnbalt 26868 If a implies b, it is implied a implies not not b (Contributed by Jarvin Udandy, but created by Mario Carnerio, shorter proof 29-Aug-2016.)

Theoremabnotbtaxb 26869 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 26870 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 26871 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 26872 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, but created by Mario Carneiro, shorter proof 29-Aug-2016.)

Theoremaistbisfiaxb 26873 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaisfbistiaxb 26874 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremdandysum 26875 Given the right hypotheses we can prove a dandysum of 2+2=4 Values that when added exceed a 4bit value, are not supported. (Contributed by Jarvin Udandy, 2-Sep-2016.)
jph        jps        jch        jth        jta        jet        jze        jth jta jph jet jps jze jch

16.19  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

16.19.1  Natural deduction

Theorem19.8ad 26876 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1758. (Contributed by DAW, 13-Feb-2017.)

Theoremsbidd 26877 An identity theorem for substitution. See sbid 1895. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

Theoremsbidd-misc 26878 An identity theorem for substitution. See sbid 1895. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

16.19.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

Syntaxcge-real 26879 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 26881.

Syntaxcgt 26880 Extend wff notation to include the 'greater than' relation, see df-gt 26882.

Definitiondf-gte 26881 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 8753.

We do not write this as , and similarly we do not write ` > ` as , because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: and but these are very complicated. This definition of , and the similar one for (df-gt 26882), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 26883 for a more conventional expression of the relationship between and . As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Definitiondf-gt 26882 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 8630. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 26881 for a discussion on why this approach is used for the definition. See gt-lt 26884 and gt-lth 26886 for more conventional expression of the relationship between and .

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremgte-lte 26883 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Theoremgt-lt 26884 Simple relationship between and . (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremgte-lteh 26885 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Theoremgt-lth 26886 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremex-gt 26887 Simple example of , in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)

Theoremex-gte 26888 Simple example of , in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)

16.19.3  Hyperbolic trig functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as . However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

Syntaxcsinh 26889 Extend class notation to include the hyperbolic sine function, see df-sinh 26892.
sinh

Syntaxccosh 26890 Extend class notation to include the hyperbolic cosine function. see df-cosh 26893.
cosh

Syntaxctanh 26891 Extend class notation to include the hyperbolic tangent function, see df-tanh 26894.
tanh

Definitiondf-sinh 26892 Define the hyperbolic sine function (sinh). We define it this way for cmpt 3974, which requires the form . See sinhval-named 26895 for a simple way to evaluate it. We define this function by dividing by , which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 26898 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
sinh

Definitiondf-cosh 26893 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 3974, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.)
cosh

Definitiondf-tanh 26894 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 3974, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.)
tanh cosh

Theoremsinhval-named 26895 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 26892. See sinhval 12308 for a theorem to convert this further. See sinh-conventional 26898 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
sinh

Theoremcoshval-named 26896 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 26893. See coshval 12309 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
cosh

Theoremtanhval-named 26897 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 26894. (Contributed by David A. Wheeler, 10-May-2015.)
cosh tanh

Theoremsinh-conventional 26898 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
sinh

Theoremsinhpcosh 26899 Prove that sinh cosh using the conventional hyperbolic trig functions. (Contributed by David A. Wheeler, 27-May-2015.)
sinh cosh

16.19.4  Reciprocal trig functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

Syntaxcsec 26900 Extend class notation to include the secant function, see df-sec 26903.

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