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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaddext 7601 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5521. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
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Theoremapneg 7602 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
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Theoremmulext1 7603 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremmulext2 7604 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremmulext 7605 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5521. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
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Theoremmulap0r 7606 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremmsqge0 7607 A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmsqge0i 7608 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmsqge0d 7609 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulge0 7610 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulge0i 7611 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)

Theoremmulge0d 7612 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremapti 7613 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremapne 7614 Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremleltap 7615 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
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Theoremgt0ap0 7616 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremgt0ap0i 7617 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremgt0ap0ii 7618 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremgt0ap0d 7619 Positive implies apart from zero. Because of the way we define #, must be an element of , not just . (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremnegap0 7620 A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremltleap 7621 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremltap 7622 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremgtapii 7623 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremltapii 7624 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremltapi 7625 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremgtapd 7626 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremltapd 7627 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
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Theoremleltapd 7628 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
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Theoremap0gt0 7629 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
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Theoremap0gt0d 7630 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
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Theoremsubap0d 7631 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
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3.3.7  Reciprocals

Theoremrecextlem1 7632 Lemma for recexap 7634. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecexaplem2 7633 Lemma for recexap 7634. (Contributed by Jim Kingdon, 20-Feb-2020.)
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Theoremrecexap 7634* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
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Theoremmulap0 7635 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremmulap0b 7636 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremmulap0i 7637 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
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Theoremmulap0bd 7638 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremmulap0d 7639 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
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Theoremmulap0bad 7640 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7639 and consequence of mulap0bd 7638. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremmulap0bbd 7641 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7639 and consequence of mulap0bd 7638. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremmulcanapd 7642 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanap2d 7643 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanapad 7644 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7642. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanap2ad 7645 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7643. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanap 7646 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanap2 7647 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmulcanapi 7648 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremmuleqadd 7649 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Theoremreceuap 7650* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
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3.3.8  Division

Syntaxcdiv 7651 Extend class notation to include division.

Definitiondf-div 7652* Define division. Theorem divmulap 7654 relates it to multiplication, and divclap 7657 and redivclap 7707 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Theoremdivvalap 7653* Value of division: the (unique) element such that . This is meaningful only when is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theoremdivmulap 7654 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivmulap2 7655 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivmulap3 7656 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivclap 7657 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremrecclap 7658 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivcanap2 7659 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivcanap1 7660 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdiveqap0 7661 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivap0b 7662 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremdivap0 7663 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theoremrecap0 7664 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremrecidap 7665 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremrecidap2 7666 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremdivrecap 7667 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
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Theoremdivrecap2 7668 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivassap 7669 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv23ap 7670 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv32ap 7671 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv13ap 7672 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv12ap 7673 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivdirap 7674 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivcanap3 7675 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivcanap4 7676 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv11ap 7677 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdividap 7678 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv0ap 7679 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdiv1 7680 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorem1div1e1 7681 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)

Theoremdiveqap1 7682 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivnegap 7683 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivsubdirap 7684 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
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Theoremrecrecap 7685 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremrec11ap 7686 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremrec11rap 7687 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivmuldivap 7688 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivdivdivap 7689 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivcanap5 7690 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theoremdivmul13ap 7691 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivmul24ap 7692 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivmuleqap 7693 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremrecdivap 7694 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivcanap6 7695 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivdiv32ap 7696 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivcanap7 7697 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdmdcanap 7698 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivdivap1 7699 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremdivdivap2 7700 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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