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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremconjmulap 8501 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremrerecclap 8502 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremredivclap 8503 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theoremeqneg 8504 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqnegd 8505 A complex number equals its negative iff it is zero. Deduction form of eqneg 8504. (Contributed by David Moews, 28-Feb-2017.)

Theoremeqnegad 8506 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8504. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2negap 8507 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivneg2ap 8508 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecclapzi 8509 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecap0apzi 8510 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrecidapzi 8511 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdiv1i 8512 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)

Theoremeqnegi 8513 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)

Theoremrecclapi 8514 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theoremrecidapi 8515 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
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Theoremrecrecapi 8516 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
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Theoremdividapi 8517 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
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Theoremdiv0api 8518 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theoremdivclapzi 8519 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap1zi 8520 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap2zi 8521 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivrecapzi 8522 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap3zi 8523 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremdivcanap4zi 8524 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theoremrec11api 8525 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivclapi 8526 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap2i 8527 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap1i 8528 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivrecapi 8529 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap3i 8530 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivcanap4i 8531 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivap0i 8532 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremrec11apii 8533 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivassapzi 8534 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivmulapzi 8535 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivdirapzi 8536 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivdiv23apzi 8537 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theoremdivmulapi 8538 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivdiv32api 8539 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivassapi 8540 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivdirapi 8541 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv23api 8542 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv11api 8543 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivmuldivapi 8544 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivmul13api 8545 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivadddivapi 8546 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdivdivdivapi 8547 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremrerecclapzi 8548 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremrerecclapi 8549 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremredivclapzi 8550 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremredivclapi 8551 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theoremdiv1d 8552 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecclapd 8553 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecap0d 8554 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecidapd 8555 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecidap2d 8556 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecrecapd 8557 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdividapd 8558 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdiv0apd 8559 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremapmul1 8560 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
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Theoremapmul2 8561 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
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Theoremdivclapd 8562 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap1d 8563 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap2d 8564 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivrecapd 8565 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivrecap2d 8566 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap3d 8567 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdivcanap4d 8568 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremdiveqap0d 8569 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap1d 8570 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap1ad 8571 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8477. Generalization of diveqap1d 8570. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiveqap0ad 8572 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8454. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivap1d 8573 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
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Theoremdivap0bd 8574 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivnegapd 8575 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivneg2apd 8576 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdiv2negapd 8577 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theoremdivap0d 8578 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecdivapd 8579 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrecdivap2d 8580 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdivcanap6d 8581 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremddcanapd 8582 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremrec11apd 8583 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
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Theoremdivmulapd 8584 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremapdivmuld 8585 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
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Theoremdiv32apd 8586 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdiv13apd 8587 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdiv32apd 8588 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap5d 8589 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap5rd 8590 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivcanap7d 8591 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanapd 8592 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdmdcanap2d 8593 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap1d 8594 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivdivap2d 8595 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theoremdivmulap2d 8596 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivmulap3d 8597 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivassapd 8598 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv12apd 8599 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv23apd 8600 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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