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Intuitionistic Logic Explorer Theorem List (p. 91 of 156) | < Previous Next > |
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Type | Label | Description |
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Statement | ||
Theorem | nn1suc 9001* | If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | nnaddcl 9002 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
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Theorem | nnmulcl 9003 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
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Theorem | nnmulcli 9004 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
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Theorem | nnge1 9005 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnle1eq1 9006 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
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Theorem | nngt0 9007 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
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Theorem | nnnlt1 9008 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | 0nnn 9009 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnne0 9010 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
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Theorem | nnap0 9011 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
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Theorem | nngt0i 9012 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
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Theorem | nnap0i 9013 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
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Theorem | nnne0i 9014 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
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Theorem | nn2ge 9015* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
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Theorem | nn1gt1 9016 |
A positive integer is either one or greater than one. This is for
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Theorem | nngt1ne1 9017 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
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Theorem | nndivre 9018 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
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Theorem | nnrecre 9019 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
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Theorem | nnrecgt0 9020 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnsub 9021 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | nnsubi 9022 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
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Theorem | nndiv 9023* |
Two ways to express "![]() ![]() |
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Theorem | nndivtr 9024 |
Transitive property of divisibility: if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nnge1d 9025 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nngt0d 9026 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnne0d 9027 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnap0d 9028 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
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Theorem | nnrecred 9029 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnaddcld 9030 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnmulcld 9031 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nndivred 9032 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
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The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7879 through df-9 9048), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7879 and df-1 7880).
Integers can also be exhibited as sums of powers of 10 (e.g., the number 103
can be expressed as Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12. | ||
Syntax | c2 9033 | Extend class notation to include the number 2. |
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Syntax | c3 9034 | Extend class notation to include the number 3. |
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Syntax | c4 9035 | Extend class notation to include the number 4. |
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Syntax | c5 9036 | Extend class notation to include the number 5. |
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Syntax | c6 9037 | Extend class notation to include the number 6. |
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Syntax | c7 9038 | Extend class notation to include the number 7. |
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Syntax | c8 9039 | Extend class notation to include the number 8. |
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Syntax | c9 9040 | Extend class notation to include the number 9. |
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Definition | df-2 9041 | Define the number 2. (Contributed by NM, 27-May-1999.) |
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Definition | df-3 9042 | Define the number 3. (Contributed by NM, 27-May-1999.) |
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Definition | df-4 9043 | Define the number 4. (Contributed by NM, 27-May-1999.) |
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Definition | df-5 9044 | Define the number 5. (Contributed by NM, 27-May-1999.) |
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Definition | df-6 9045 | Define the number 6. (Contributed by NM, 27-May-1999.) |
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Definition | df-7 9046 | Define the number 7. (Contributed by NM, 27-May-1999.) |
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Definition | df-8 9047 | Define the number 8. (Contributed by NM, 27-May-1999.) |
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Definition | df-9 9048 | Define the number 9. (Contributed by NM, 27-May-1999.) |
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Theorem | 0ne1 9049 |
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Theorem | 1ne0 9050 |
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Theorem | 1m1e0 9051 |
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Theorem | 2re 9052 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 2cn 9053 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) |
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Theorem | 2ex 9054 | 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 2cnd 9055 | 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 3re 9056 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 3cn 9057 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) |
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Theorem | 3ex 9058 | 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 4re 9059 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 4cn 9060 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
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Theorem | 5re 9061 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 5cn 9062 | The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 6re 9063 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 6cn 9064 | The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 7re 9065 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 7cn 9066 | The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 8re 9067 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 8cn 9068 | The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 9re 9069 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
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Theorem | 9cn 9070 | The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 0le0 9071 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
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Theorem | 0le2 9072 | 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
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Theorem | 2pos 9073 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 2ne0 9074 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
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Theorem | 2ap0 9075 | The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
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Theorem | 3pos 9076 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 3ne0 9077 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
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Theorem | 3ap0 9078 | The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
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Theorem | 4pos 9079 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 4ne0 9080 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
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Theorem | 4ap0 9081 | The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
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Theorem | 5pos 9082 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 6pos 9083 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 7pos 9084 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 8pos 9085 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
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Theorem | 9pos 9086 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
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This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10. | ||
Theorem | neg1cn 9087 | -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
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Theorem | neg1rr 9088 | -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
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Theorem | neg1ne0 9089 | -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | neg1lt0 9090 | -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | neg1ap0 9091 | -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) |
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Theorem | negneg1e1 9092 |
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Theorem | 1pneg1e0 9093 |
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Theorem | 0m0e0 9094 | 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 1m0e1 9095 | 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 0p1e1 9096 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
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Theorem | fv0p1e1 9097 |
Function value at ![]() ![]() ![]() ![]() ![]() |
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Theorem | 1p0e1 9098 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | 1p1e2 9099 | 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) |
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Theorem | 2m1e1 9100 | 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9127. (Contributed by David A. Wheeler, 4-Jan-2017.) |
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