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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ltaddsubi 7101 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.) |
Theorem | lt2addi 7102 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Theorem | le2addi 7103 | Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.) |
Theorem | gt0ne0d 7104 | Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt0ne0d 7105 | Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.) |
Theorem | leidd 7106 | 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt0neg1d 7107 | Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt0neg2d 7108 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le0neg1d 7109 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le0neg2d 7110 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addgegt0d 7111 | Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addgt0d 7112 | Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge0d 7113 | Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegd 7114 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegd 7115 | Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegcon1d 7116 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegcon2d 7117 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegcon1d 7118 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegcon2d 7119 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddposd 7120 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddpos2d 7121 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubposd 7122 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | posdifd 7123 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge01d 7124 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge02d 7125 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subge0d 7126 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | suble0d 7127 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subge02d 7128 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1d 7129 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leadd1d 7130 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leadd2d 7131 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubaddd 7132 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubaddd 7133 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubadd2d 7134 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubadd2d 7135 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsubd 7136 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsub2d 7137 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Theorem | leaddsub2d 7138 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subled 7139 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubd 7140 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub23d 7141 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub13d 7142 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub1d 7143 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub2d 7144 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub1d 7145 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub2d 7146 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1dd 7147 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub1dd 7148 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub2dd 7149 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd1dd 7150 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd2dd 7151 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub1dd 7152 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub2dd 7153 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | le2addd 7154 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le2subd 7155 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltleaddd 7156 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leltaddd 7157 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2addd 7158 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2subd 7159 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsublt 7160 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
Theorem | 1le1 7161 | . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
Theorem | gt0add 7162 | A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Syntax | creap 7163 | Class of real apartness relation. |
#_{ℝ} | ||
Definition | df-reap 7164* | Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapval 7165 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7177 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapirr 7166 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT]], p. (varies). Beyond the development of # itself, proofs should use apirr 7189 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
#_{ℝ} | ||
Theorem | recexre 7167* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapti 7168 | Real apartness is tight. (Contributed by Jim Kingdon, 30-Jan-2020.) |
#_{ℝ} | ||
Theorem | ixi 7169 | times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | recexgt0 7170* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Syntax | cap 7171 | Class of complex apartness relation. |
# | ||
Definition | df-ap 7172* | Define complex apartness. Definition 6.1 of Skeleton for the Proof development leading to the. Fundamental Theorem of Algebra, Herman Geuvers, Randy Pollack, Freek Wiedijk, Jan Zwanenburg, October 2, 2000. (Contributed by Jim Kingdon, 26-Jan-2020.) |
# #_{ℝ} #_{ℝ} | ||
Theorem | inelr 7173 | The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.) |
Theorem | rimul 7174 | A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | rereim 7175 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
Theorem | apreap 7176 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
# #_{ℝ} | ||
Theorem | reaplt 7177 | Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT]], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) |
# | ||
Theorem | ltmul1a 7178 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | ltmul1 7179 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT]], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | lemul1 7180 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Theorem | reapmul1lem 7181 | Lemma for reapmul1 7182. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # | ||
Theorem | reapmul1 7182 | Multiplication of both sides of real apartness by a real number apart from zero. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # # | ||
Theorem | reapadd1 7183 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapneg 7184 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapcotr 7185 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | apsqgt0 7186 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
# | ||
Theorem | cru 7187 | The representation of complex numbers in terms of real and imaginary parts 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 | apreim 7188 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
# # # | ||
Theorem | apirr 7189 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# | ||
Theorem | apsym 7190 | Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # | ||
Theorem | apcotr 7191 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | apadd1 7192 | Addition respects apartness. Analogue of addcan 6793 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | apadd2 7193 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # | ||
Theorem | extadd 7194 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5441. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
# # # | ||
Theorem | apneg 7195 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
# # | ||
Theorem | mathbox 7196 |
(This theorem is a dummy placeholder for these guidelines. The name of
this theorem, "mathbox", is hard-coded into the Metamath program
to
identify the start of the mathbox section for web page generation.)
A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm. Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it. Guidelines: 1. If at all possible, please use only 0-ary class constants for new definitions. 2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website. 3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm. 4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed. Notes: 1. We may decide to move some theorems to the main part of set.mm for general use. 2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on. 3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.) |
Theorem | ax1hfs 7197 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) |
Theorem | nnexmid 7198 | Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.) |
Theorem | nndc 7199 | Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.) |
DECID | ||
Theorem | dcdc 7200 | Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.) |
DECID DECID DECID |
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