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Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ltaddsubi 7501 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.) |
Theorem | lt2addi 7502 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Theorem | le2addi 7503 | Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.) |
Theorem | gt0ne0d 7504 | Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt0ne0d 7505 | Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.) |
Theorem | leidd 7506 | 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt0neg1d 7507 | 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 7508 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le0neg1d 7509 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le0neg2d 7510 | Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addgegt0d 7511 | Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addgt0d 7512 | Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge0d 7513 | Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegd 7514 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegd 7515 | Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegcon1d 7516 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnegcon2d 7517 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegcon1d 7518 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenegcon2d 7519 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddposd 7520 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddpos2d 7521 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubposd 7522 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | posdifd 7523 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge01d 7524 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addge02d 7525 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subge0d 7526 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | suble0d 7527 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subge02d 7528 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1d 7529 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leadd1d 7530 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leadd2d 7531 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubaddd 7532 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubaddd 7533 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubadd2d 7534 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubadd2d 7535 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsubd 7536 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsub2d 7537 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Theorem | leaddsub2d 7538 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subled 7539 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubd 7540 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub23d 7541 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub13d 7542 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub1d 7543 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub2d 7544 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub1d 7545 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub2d 7546 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1dd 7547 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub1dd 7548 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub2dd 7549 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd1dd 7550 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd2dd 7551 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub1dd 7552 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub2dd 7553 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | le2addd 7554 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le2subd 7555 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltleaddd 7556 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leltaddd 7557 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2addd 7558 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2subd 7559 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | possumd 7560 | Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | sublt0d 7561 | When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | ltaddsublt 7562 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
Theorem | 1le1 7563 | . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
Theorem | gt0add 7564 | 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 7565 | Class of real apartness relation. |
#ℝ | ||
Definition | df-reap 7566* | Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 7573 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 7578). (Contributed by Jim Kingdon, 26-Jan-2020.) |
#ℝ | ||
Theorem | reapval 7567 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7579 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
#ℝ | ||
Theorem | reapirr 7568 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7596 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
#ℝ | ||
Theorem | recexre 7569* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
#ℝ | ||
Theorem | reapti 7570 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7613. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
#ℝ | ||
Theorem | recexgt0 7571* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Syntax | cap 7572 | Class of complex apartness relation. |
# | ||
Definition | df-ap 7573* |
Define complex apartness. Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 7658 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 7596), symmetry (apsym 7597), and cotransitivity (apcotr 7598). Apartness implies negated equality, as seen at apne 7614, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 7613). (Contributed by Jim Kingdon, 26-Jan-2020.) |
# #ℝ #ℝ | ||
Theorem | ixi 7574 | times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | inelr 7575 | The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.) |
Theorem | rimul 7576 | 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 7577 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
Theorem | apreap 7578 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
# #ℝ | ||
Theorem | reaplt 7579 | 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 | reapltxor 7580 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
# | ||
Theorem | 1ap0 7581 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# | ||
Theorem | ltmul1a 7582 | 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 7583 | 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 7584 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Theorem | reapmul1lem 7585 | Lemma for reapmul1 7586. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # | ||
Theorem | reapmul1 7586 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7764. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # # | ||
Theorem | reapadd1 7587 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapneg 7588 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapcotr 7589 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | remulext1 7590 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
# # | ||
Theorem | remulext2 7591 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | apsqgt0 7592 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
# | ||
Theorem | cru 7593 | 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 7594 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
# # # | ||
Theorem | mulreim 7595 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Theorem | apirr 7596 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# | ||
Theorem | apsym 7597 | 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 7598 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | apadd1 7599 | Addition respects apartness. Analogue of addcan 7191 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | apadd2 7600 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # |
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