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Type | Label | Description |
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Statement | ||
Theorem | recid2d 10401 | Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | recrecd 10402 | A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | dividd 10403 | A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div0d 10404 | Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcld 10405 | Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan1d 10406 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan2d 10407 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divrecd 10408 | Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divrec2d 10409 | Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan3d 10410 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan4d 10411 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | diveq0d 10412 | A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | diveq1d 10413 | Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | diveq1ad 10414 | The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 10323. Generalization of diveq1d 10413. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | diveq0ad 10415 | A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 10302. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | divne1d 10416 | If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 10413. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | divne0bd 10417 | A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divnegd 10418 | Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divneg2d 10419 | Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div2negd 10420 | Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divne0d 10421 | The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | recdivd 10422 | The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | recdiv2d 10423 | Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan6d 10424 | Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ddcand 10425 | Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | rec11d 10426 | Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmuld 10427 | Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div32d 10428 | A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div13d 10429 | A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divdiv32d 10430 | Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan5d 10431 | Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divcan5rd 10432 | Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.) |
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Theorem | divcan7d 10433 | Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | dmdcand 10434 | Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | dmdcan2d 10435 | Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | divdiv1d 10436 | Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divdiv2d 10437 | Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmul2d 10438 | Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmul3d 10439 | Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divassd 10440 | An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div12d 10441 | A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div23d 10442 | A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divdird 10443 | Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divsubdird 10444 | Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | div11d 10445 | One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmuldivd 10446 | Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmul13d 10447 | Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmul24d 10448 | Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divadddivd 10449 | Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divsubdivd 10450 | Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divmuleqd 10451 | Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | divdivdivd 10452 | Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | diveq1bd 10453 | If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 10323. Converse of diveq1d 10413. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | div2sub 10454 | Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.) |
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Theorem | div2subd 10455 | Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 10454. (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | rereccld 10456 | Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | redivcld 10457 | Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subrec 10458 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.) |
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Theorem | subreci 10459 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.) |
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Theorem | subrecd 10460 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.) |
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Theorem | mvllmuld 10461 | Move LHS left multiplication to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
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Theorem | mvllmuli 10462 | Move LHS left multiplication to RHS. Uses divcan4i 10376. (Contributed by David A. Wheeler, 11-Oct-2018.) |
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Theorem | elimgt0 10463 |
Hypothesis for weak deduction theorem to eliminate ![]() ![]() ![]() |
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Theorem | elimge0 10464 |
Hypothesis for weak deduction theorem to eliminate ![]() ![]() ![]() |
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Theorem | ltp1 10465 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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Theorem | lep1 10466 | A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.) |
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Theorem | ltm1 10467 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
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Theorem | lem1 10468 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | letrp1 10469 | A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.) |
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Theorem | p1le 10470 | A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
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Theorem | recgt0 10471 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | prodgt0 10472 | Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | prodgt02 10473 | Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.) |
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Theorem | prodge0 10474 | Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | prodge02 10475 | Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.) |
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Theorem | ltmul1a 10476 | Lemma for ltmul1 10477. 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.) |
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Theorem | ltmul1 10477 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | ltmul2 10478 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) |
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Theorem | lemul1 10479 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
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Theorem | lemul2 10480 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.) |
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Theorem | lemul1a 10481 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.) |
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Theorem | lemul2a 10482 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) |
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Theorem | ltmul12a 10483 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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Theorem | lemul12b 10484 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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Theorem | lemul12a 10485 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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Theorem | mulgt1 10486 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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Theorem | ltmulgt11 10487 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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Theorem | ltmulgt12 10488 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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Theorem | lemulge11 10489 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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Theorem | lemulge12 10490 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | ltdiv1 10491 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | lediv1 10492 | Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.) |
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Theorem | gt0div 10493 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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Theorem | ge0div 10494 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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Theorem | divgt0 10495 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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Theorem | divge0 10496 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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Theorem | mulge0b 10497 | A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.) |
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Theorem | mulle0b 10498 | A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
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Theorem | mulsuble0b 10499 | A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
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Theorem | ltmuldiv 10500 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
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