Home Intuitionistic Logic ExplorerTheorem List (p. 72 of 102) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremletr 7101 Transitive law. (Contributed by NM, 12-Nov-1999.)

Theoremleid 7102 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltne 7103 'Less than' implies not equal. See also ltap 7622 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)

Theoremltnsym 7104 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)

Theoremltle 7105 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)

Theoremlelttr 7106 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)

Theoremltletr 7107 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)

Theoremltnsym2 7108 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremeqle 7109 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)

Theoremltnri 7110 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremeqlei 7111 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)

Theoremeqlei2 7112 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)

Theoremgtneii 7113 'Less than' implies not equal. See also gtapii 7623 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)

Theoremltneii 7114 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremlttri3i 7115 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremletri3i 7116 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremltnsymi 7117 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)

Theoremlenlti 7118 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)

Theoremltlei 7119 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)

Theoremltleii 7120 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)

Theoremltnei 7121 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)

Theoremlttri 7122 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 7123 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 7124 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 7125 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 7126 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremmulgt0i 7127 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 7128 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 7129 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 7130 'Less than' implies not equal. See also gtapd 7626 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 7131 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 7132 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletri3d 7133 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 7134 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 7135 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 7136 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulgt0d 7137 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 7138 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 7139 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremlttrd 7140 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

Theorem0lt1 7141 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.)

3.2.5  Initial properties of the complex numbers

Theoremmul12 7142 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 7143 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 7144 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 7145 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 7146 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 7147 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 7148 A theorem for complex numbers analogous the second Peano postulate peano2 4318. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 7149 A theorem for reals analogous the second Peano postulate peano2 4318. (Contributed by NM, 5-Jul-2005.)

Theoremaddid2 7152 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremreaddcan 7153 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)

Theorem00id 7154 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid1i 7155 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 7156 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)

Theoremaddcomi 7157 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul12i 7159 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 7160 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 7161 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremaddid2d 7163 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 7164 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul12d 7165 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 7166 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 7167 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 7168 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

3.3  Real and complex numbers - basic operations

Theoremadd12 7169 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 7170 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd32r 7171 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)

Theoremadd4 7172 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 7173 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 7174 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 7175 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 7176 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 7177 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 7178 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 7179 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 7180 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 7181 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

3.3.2  Subtraction

Syntaxcmin 7182 Extend class notation to include subtraction.

Syntaxcneg 7183 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 7182 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5512, 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 7184* Define subtraction. Theorem subval 7203 shows its value (and describes how this definition works), theorem subaddi 7298 relates it to addition, and theorems subcli 7287 and resubcli 7274 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 7185 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 7183 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theoremcnegexlem1 7186 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7189. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegexlem2 7187 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 7189. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegexlem3 7188* Existence of real number difference. Lemma for cnegex 7189. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremcnegex 7189* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)

Theoremcnegex2 7190* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcan 7191 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 7192 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 7193 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 7194 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 7195 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 7196 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 7197 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7195. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 7198 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7196. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 7199 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7197. Consequence of addcand 7195. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 7200 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7198. Consequence of addcan2d 7196. (Contributed by David Moews, 28-Feb-2017.)

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 10101-10124
 Copyright terms: Public domain < Previous  Next >