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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivcanap5rd 8601 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap7d 8602 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanapd 8603 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanap2d 8604 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap1d 8605 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap2d 8606 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivmulap2d 8607 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivmulap3d 8608 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivassapd 8609 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv12apd 8610 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv23apd 8611 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivdirapd 8612 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivsubdirapd 8613 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv11apd 8614 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivmuldivapd 8615 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
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Theoremrerecclapd 8616 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremredivclapd 8617 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdiveqap1bd 8618 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8488. Converse of diveqap1d 8581. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
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Theoremdiv2subap 8619 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
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Theoremdiv2subapd 8620 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8619. (Contributed by David Moews, 28-Feb-2017.)
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Theoremsubrecap 8621 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
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Theoremsubrecapi 8622 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
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Theoremsubrecapd 8623 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
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Theoremmvllmulapd 8624 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
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4.3.9  Ordering on reals (cont.)

Theoremltp1 8625 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 8626 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 8627 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 8628 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 8629 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 8630 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 8631 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.)

Theoremprodgt0gt0 8632 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8633 for the same theorem with replaced by the weaker condition . (Contributed by Jim Kingdon, 29-Feb-2020.)

Theoremprodgt0 8633 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 8634 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 8635 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 8636 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul2 8637 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)

Theoremlemul2 8638 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 8639 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 8640 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 8641 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 8642 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 8643 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 8644 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 8645 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 8646 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 8647 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 8648 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 8649 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 8650 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 8651 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 8652 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 8653 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 8654 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 8655 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 8656 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 8657 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 8658 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremltdivmul2 8659 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 8660 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 8661 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremlemuldiv 8662 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 8663 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 8664 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 8665 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 8666 Lemma for lt2msq 8667. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 8667 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 8668 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltrec1 8669 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 8670 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 8671 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 8672 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 8673 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 8674 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 8675 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 8676 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 8677 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 8678 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 8679 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 8680 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 8681 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 8682 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 8683 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 8684 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 8685* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 8686 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 8687 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 8688 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 8689 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 8690 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 8691 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 8692 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 8693 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 8694 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 8695 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 8696 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 8697 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 8698 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 8699 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 8700 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

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