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Type | Label | Description |
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Statement | ||
Theorem | msq11 8301 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | ledivp1 8302 | Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.) |
Theorem | squeeze0 8303* | If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.) |
Theorem | ltp1i 8304 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
Theorem | recgt0i 8305 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
Theorem | recgt0ii 8306 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
Theorem | prodgt0i 8307 | Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.) |
Theorem | prodge0i 8308 | Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) |
Theorem | divgt0i 8309 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
Theorem | divge0i 8310 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
Theorem | ltreci 8311 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
Theorem | lereci 8312 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
Theorem | lt2msqi 8313 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
Theorem | le2msqi 8314 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
Theorem | msq11i 8315 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
Theorem | divgt0i2i 8316 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
Theorem | ltrecii 8317 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
Theorem | divgt0ii 8318 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
Theorem | ltmul1i 8319 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
Theorem | ltdiv1i 8320 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
Theorem | ltmuldivi 8321 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
Theorem | ltmul2i 8322 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
Theorem | lemul1i 8323 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.) |
Theorem | lemul2i 8324 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.) |
Theorem | ltdiv23i 8325 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
Theorem | ltdiv23ii 8326 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
Theorem | ltmul1ii 8327 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.) |
Theorem | ltdiv1ii 8328 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
Theorem | ltp1d 8329 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lep1d 8330 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | ltm1d 8331 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lem1d 8332 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | recgt0d 8333 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | divgt0d 8334 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | mulgt1d 8335 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemulge11d 8336 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemulge12d 8337 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul1ad 8338 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul2ad 8339 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | ltmul12ad 8340 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul12ad 8341 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul12bd 8342 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | mulle0r 8343 | Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) |
Theorem | lbreu 8344* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
Theorem | lbcl 8345* | If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | lble 8346* | If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | lbinf 8347* | If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | lbinfcl 8348* | If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | lbinfle 8349* | If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | suprubex 8350* | A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.) |
Theorem | suprlubex 8351* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprnubex 8352* | An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprleubex 8353* | The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Theorem | negiso 8354 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | dfinfre 8355* | The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | crap0 8356 | The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.) |
# # # | ||
Theorem | creur 8357* | The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | creui 8358* | The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | cju 8359* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Syntax | cn 8360 | Extend class notation to include the class of positive integers. |
Definition | df-inn 8361* | Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8362 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.) |
Theorem | dfnn2 8362* | Definition of the set of positive integers. Another name for df-inn 8361. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
Theorem | peano5nni 8363* | Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnssre 8364 | The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnsscn 8365 | The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | nnex 8366 | The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnre 8367 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncn 8368 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nnrei 8369 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncni 8370 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | 1nn 8371 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Theorem | peano2nn 8372 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnred 8373 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nncnd 8374 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | peano2nnd 8375 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnind 8376* | Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8380 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnindALT 8377* |
Principle of Mathematical Induction (inference schema). The last four
hypotheses give us the substitution instances we need; the first two are
the induction step and the basis.
This ALT version of nnind 8376 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | nn1m1nn 8378 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Theorem | nn1suc 8379* | 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.) |
Theorem | nnaddcl 8380 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcl 8381 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcli 8382 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nnge1 8383 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnle1eq1 8384 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Theorem | nngt0 8385 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Theorem | nnnlt1 8386 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | 0nnn 8387 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnne0 8388 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Theorem | nnap0 8389 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
# | ||
Theorem | nngt0i 8390 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
Theorem | nnne0i 8391 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Theorem | nn2ge 8392* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Theorem | nn1gt1 8393 | A positive integer is either one or greater than one. This is for ; 0elnn 4407 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Theorem | nngt1ne1 8394 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Theorem | nndivre 8395 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Theorem | nnrecre 8396 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Theorem | nnrecgt0 8397 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnsub 8398 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nnsubi 8399 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
Theorem | nndiv 8400* | Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
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