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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlediv1 9501 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 9502 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 9503 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 9504 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 9505 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 9506 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 9507 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 9508 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 9509 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

TheoremledivmulOLD 9510 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltdivmul2 9511 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 9512 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 9513 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremledivmul2OLD 9514 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlemuldiv 9515 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 9516 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 9517 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 9518 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2msq1 9519 Lemma for lt2msq 9520. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 9520 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltdiv2 9521 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltdiv2OLD 9522 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltrec1 9523 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 9524 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 9525 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 9526 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 9527 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 9528 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 9529 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 9530 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 9531 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 9532 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 9533 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 9534 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 9535 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 9536 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 9537 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.)

Theoremledivp1 9538 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.)

Theoremsqueeze0 9539* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 9540 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 9541 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 9542 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 9543 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 9544 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 9545 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 9546 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 9547 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 9548 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 9549 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 9550 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 9551 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 9552 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 9553 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 9554 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltmul1i 9555 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremltdiv1i 9556 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltmuldivi 9557 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)

Theoremltmul2i 9558 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremlemul1i 9559 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)

Theoremlemul2i 9560 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)

Theoremltdiv23i 9561 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremledivp1i 9562 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdivp1i 9563 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdiv23ii 9564 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltmul1ii 9565 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.)

Theoremltdiv1ii 9566 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltp1d 9567 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlep1d 9568 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltm1d 9569 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlem1d 9570 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt0d 9571 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivgt0d 9572 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulgt1d 9573 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge11d 9574 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge12d 9575 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1ad 9576 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2ad 9577 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul12ad 9578 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12ad 9579 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12bd 9580 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

5.3.8  Completeness Axiom and Suprema

Theoremfimaxre 9581* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimaxre2 9582* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)

Theoremfimaxre3 9583* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremlbreu 9584* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)

Theoremlbcl 9585* 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.)

Theoremlble 9586* 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.)

Theoremlbinfm 9587* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremlbinfmcl 9588* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.)

Theoremlbinfmle 9589* If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.)

Theoremsup2 9590* A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)

Theoremsup3 9591* A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)

Theoreminfm3lem 9592* Lemma for infm3 9593. (Contributed by NM, 14-Jun-2005.)

Theoreminfm3 9593* The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has a infimum. (This theorem is the dual of sup3 9591.) (Contributed by NM, 14-Jun-2005.)

Theoremsuprcl 9594* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Oct-2004.)

Theoremsuprub 9595* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)

Theoremsuprlub 9596* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprnub 9597* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprleub 9598* The supremum of a non-empty 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.)

Theoremsupmul1 9599* The supremum function distributes over multiplication, in the sense that , where is shorthand for and is defined as below. This is the simple version, with only one set argument; see supmul 9602 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmullem1 9600* Lemma for supmul 9602. (Contributed by Mario Carneiro, 5-Jul-2013.)

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