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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgtned 7901 'Less than' implies not equal. See also gtapd 8424 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 7902 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 7903 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletri3d 7904 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 7905 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 7906 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 7907 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnltled 7908 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremlensymd 7909 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremmulgt0d 7910 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 7911 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 7912 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremlttrd 7913 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

Theorem0lt1 7914 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)

Theoremltntri 7915 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, . Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)

4.2.5  Initial properties of the complex numbers

Theoremmul12 7916 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 7917 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 7918 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 7919 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 7920 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 7921 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 7922 A theorem for complex numbers analogous the second Peano postulate peano2 4517. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 7923 A theorem for reals analogous the second Peano postulate peano2 4517. (Contributed by NM, 5-Jul-2005.)

Theoremaddid2 7926 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremreaddcan 7927 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)

Theorem00id 7928 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid1i 7929 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 7930 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)

Theoremaddcomi 7931 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul12i 7933 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmul32i 7934 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 7935 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremaddid2d 7937 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 7938 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul12d 7939 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 7940 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 7941 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 7942 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmuladd11r 7943 A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)

Theoremcomraddd 7944 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)

4.3  Real and complex numbers - basic operations

Theoremadd12 7945 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 7946 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd32r 7947 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)

Theoremadd4 7948 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 7949 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 7950 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 7951 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 7952 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 7953 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 7954 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 7955 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 7956 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 7957 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

4.3.2  Subtraction

Syntaxcmin 7958 Extend class notation to include subtraction.

Syntaxcneg 7959 Extend class notation to include unary minus. The symbol is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus () and subtraction cmin 7958 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5782, if we used the same symbol then " " could mean either " " minus "", or it could represent the (meaningless) operation of classes " " and " " connected with "operation" "". On the other hand, " " is unambiguous.

Definitiondf-sub 7960* Define subtraction. Theorem subval 7979 shows its value (and describes how this definition works), theorem subaddi 8074 relates it to addition, and theorems subcli 8063 and resubcli 8050 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 7961 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 7959 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theoremcnegexlem1 7962 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7965. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegexlem2 7963 Existence of a real number which produces a real number when multiplied by . (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7965. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegexlem3 7964* Existence of real number difference. Lemma for cnegex 7965. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegex 7965* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)

Theoremcnegex2 7966* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcan 7967 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremaddcan2 7968 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 7969 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcan2i 7970 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcand 7971 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 7972 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 7973 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7971. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 7974 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7972. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 7975 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7973. Consequence of addcand 7971. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 7976 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7974. Consequence of addcan2d 7972. (Contributed by David Moews, 28-Feb-2017.)

Theorem0cnALT 7977 Alternate proof of 0cn 7783. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegeu 7978* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubval 7979* Value of subtraction, which is the (unique) element such that . (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremnegeq 7980 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)

Theoremnegeqi 7981 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)

Theoremnegeqd 7982 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)

Theoremnfnegd 7983 Deduction version of nfneg 7984. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfneg 7984 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbnegg 7985 Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubcl 7986 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremnegcl 7987 Closure law for negative. (Contributed by NM, 6-Aug-2003.)

Theoremnegicn 7988 is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)

Theoremsubf 7989 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremsubadd 7990 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2 7991 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubsub23 7992 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)

Theorempncan 7993 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempncan2 7994 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

Theorempncan3 7995 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)

Theoremnpcan 7996 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsubass 7997 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsub 7998 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubadd23 7999 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)

Theoremaddsub12 8000 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)

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