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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecid2d 10401 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecrecd 10402 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdividd 10403 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv0d 10404 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcld 10405 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan1d 10406 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan2d 10407 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrecd 10408 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrec2d 10409 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan3d 10410 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan4d 10411 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq0d 10412 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1d 10413 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1ad 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.)

Theoremdiveq0ad 10415 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 10302. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne1d 10416 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 10413. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne0bd 10417 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivnegd 10418 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivneg2d 10419 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv2negd 10420 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivne0d 10421 The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdivd 10422 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdiv2d 10423 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan6d 10424 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremddcand 10425 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrec11d 10426 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuld 10427 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv32d 10428 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13d 10429 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv32d 10430 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5d 10431 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5rd 10432 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdivcan7d 10433 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcand 10434 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcan2d 10435 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivdiv1d 10436 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv2d 10437 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul2d 10438 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul3d 10439 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivassd 10440 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv12d 10441 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv23d 10442 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdird 10443 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdird 10444 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv11d 10445 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuldivd 10446 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul13d 10447 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul24d 10448 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivadddivd 10449 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdivd 10450 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuleqd 10451 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdivdivd 10452 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1bd 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.)

Theoremdiv2sub 10454 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremdiv2subd 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.)

Theoremrereccld 10456 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremredivcld 10457 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubrec 10458 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)

Theoremsubreci 10459 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremsubrecd 10460 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremmvllmuld 10461 Move LHS left multiplication to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)

Theoremmvllmuli 10462 Move LHS left multiplication to RHS. Uses divcan4i 10376. (Contributed by David A. Wheeler, 11-Oct-2018.)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 10463 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremelimge0 10464 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 30-Jul-1999.)

Theoremltp1 10465 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 10466 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 10467 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 10468 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 10469 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 10470 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 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.)

Theoremprodgt0 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.)

Theoremprodgt02 10473 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 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.)

Theoremprodge02 10475 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul1a 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.)

Theoremltmul1 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.)

Theoremltmul2 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.)

Theoremlemul1 10479 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2 10480 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 10481 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 10482 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 10483 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 10484 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 10485 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 10486 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 10487 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 10488 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 10489 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 10490 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 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.)

Theoremlediv1 10492 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 10493 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 10494 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 10495 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 10496 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremmulge0b 10497 A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulle0b 10498 A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremmulsuble0b 10499 A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)

Theoremltmuldiv 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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