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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcanap2 7401 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC #  0  x.
 
Theoremdivcanap1 7402 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC #  0  x.
 
Theoremdiveqap0 7403 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC #  0  0  0
 
Theoremdivap0b 7404 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC #  0 #  0 #  0
 
Theoremdivap0 7405 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC #  0  CC #  0 #  0
 
Theoremrecap0 7406 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 CC #  0  1 #  0
 
Theoremrecidap 7407 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 CC #  0  x. 
 1  1
 
Theoremrecidap2 7408 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 CC #  0  1  x.  1
 
Theoremdivrecap 7409 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 CC  CC #  0  x. 
 1
 
Theoremdivrecap2 7410 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  1  x.
 
Theoremdivassap 7411 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  x.  C  x.  C
 
Theoremdiv23ap 7412 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  x.  C  C  x.
 
Theoremdiv32ap 7413 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  C  CC  x.  C  x.  C
 
Theoremdiv13ap 7414 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  C  CC  x.  C  C  x.
 
Theoremdiv12ap 7415 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  x.  C  x.  C
 
Theoremdivdirap 7416 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  +  C  C  +  C
 
Theoremdivcanap3 7417 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  x.
 
Theoremdivcanap4 7418 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  x.
 
Theoremdiv11ap 7419 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  C  C
 
Theoremdividap 7420 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC #  0  1
 
Theoremdiv0ap 7421 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC #  0  0  0
 
Theoremdiv1 7422 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  1
 
Theorem1div1e1 7423 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 1  1  1
 
Theoremdiveqap1 7424 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  1
 
Theoremdivnegap 7425 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  -u  -u
 
Theoremdivsubdirap 7426 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
 CC  CC  C  CC  C #  0  -  C  C  -  C
 
Theoremrecrecap 7427 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC #  0  1 
 1
 
Theoremrec11ap 7428 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC #  0  CC #  0  1  1
 
Theoremrec11rap 7429 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC #  0  CC #  0  1  1
 
Theoremdivmuldivap 7430 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C  x.  D  x.  C  x.  D
 
Theoremdivdivdivap 7431 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  D  CC  D #  0  C  D  x.  D  x.  C
 
Theoremdivcanap5 7432 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  C  x.  C  x.
 
Theoremdivmul13ap 7433 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C  x.  D  C  x.  D
 
Theoremdivmul24ap 7434 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C  x.  D  D  x.  C
 
Theoremdivmuleqap 7435 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C  D  x.  D  x.  C
 
Theoremrecdivap 7436 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC #  0  CC #  0 
 1
 
Theoremdivcanap6 7437 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC #  0  CC #  0  x.  1
 
Theoremdivdiv32ap 7438 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  C  C
 
Theoremdivcanap7 7439 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  C  C
 
Theoremdmdcanap 7440 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC #  0  CC #  0  C  CC  x.  C  C
 
Theoremdivdivap1 7441 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  C  x.  C
 
Theoremdivdivap2 7442 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC #  0  C  CC  C #  0  C  x.  C
 
Theoremrecdivap2 7443 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC #  0  CC #  0  1  1  x.
 
Theoremddcanap 7444 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC #  0  CC #  0
 
Theoremdivadddivap 7445 Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C  +  D  x.  D  +  x.  C  C  x.  D
 
Theoremdivsubdivap 7446 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 CC  CC  C  CC  C #  0  D  CC  D #  0  C 
 -  D  x.  D  -  x.  C  C  x.  D
 
Theoremconjmulap 7447 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
 P  CC  P #  0  Q  CC  Q #  0  1  P  +  1  Q  1  P  -  1  x.  Q  -  1  1
 
Theoremrerecclap 7448 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 RR #  0  1  RR
 
Theoremredivclap 7449 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
 RR  RR #  0 
 RR
 
Theoremeqneg 7450 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  -u  0
 
Theoremeqnegd 7451 A complex number equals its negative iff it is zero. Deduction form of eqneg 7450. (Contributed by David Moews, 28-Feb-2017.)
 CC   =>     -u  0
 
Theoremeqnegad 7452 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7450. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     -u   =>     0
 
Theoremdiv2negap 7453 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC  CC #  0  -u  -u
 
Theoremdivneg2ap 7454 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC  CC #  0  -u  -u
 
Theoremrecclapzi 7455 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   =>    #  0  1  CC
 
Theoremrecap0apzi 7456 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   =>    #  0  1 #  0
 
Theoremrecidapzi 7457 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   =>    #  0  x. 
 1  1
 
Theoremdiv1i 7458 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 CC   =>     1
 
Theoremeqnegi 7459 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 CC   =>     -u  0
 
Theoremrecclapi 7460 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 CC   &    #  0   =>     1  CC
 
Theoremrecidapi 7461 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     x. 
 1  1
 
Theoremrecrecapi 7462 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     1 
 1
 
Theoremdividapi 7463 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     1
 
Theoremdiv0api 7464 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 CC   &    #  0   =>     0  0
 
Theoremdivclapzi 7465 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  CC
 
Theoremdivcanap1zi 7466 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivcanap2zi 7467 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivrecapzi 7468 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.  1
 
Theoremdivcanap3zi 7469 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivcanap4zi 7470 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremrec11api 7471 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   =>    #  0 #  0  1  1
 
Theoremdivclapi 7472 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     CC
 
Theoremdivcanap2i 7473 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap1i 7474 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivrecapi 7475 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.  1
 
Theoremdivcanap3i 7476 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap4i 7477 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivap0i 7478 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    #  0
 
Theoremrec11apii 7479 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     1 
 1
 
Theoremdivassapzi 7480 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>     C #  0  x.  C  x.  C
 
Theoremdivmulapzi 7481 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>    #  0  C  x.  C
 
Theoremdivdirapzi 7482 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>     C #  0  +  C  C  +  C
 
Theoremdivdiv23apzi 7483 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>    #  0  C #  0  C  C
 
Theoremdivmulapi 7484 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     C  x.  C
 
Theoremdivdiv32api 7485 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdivassapi 7486 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdivdirapi 7487 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     +  C  C  +  C
 
Theoremdiv23api 7488 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  C  x.
 
Theoremdiv11api 7489 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  C
 
Theoremdivmuldivapi 7490 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     x.  C  D  x.  C  x.  D
 
Theoremdivmul13api 7491 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     x.  C  D  C  x.  D
 
Theoremdivadddivapi 7492 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     +  C  D  x.  D  +  C  x.  x.  D
 
Theoremdivdivdivapi 7493 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   &     C #  0   =>     C  D  x.  D  x.  C
 
Theoremrerecclapzi 7494 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   =>    #  0  1  RR
 
Theoremrerecclapi 7495 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &    #  0   =>     1  RR
 
Theoremredivclapzi 7496 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &     RR   =>    #  0  RR
 
Theoremredivclapi 7497 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &     RR   &    #  0   =>     RR
 
Theoremdiv1d 7498 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     1
 
Theoremrecclapd 7499 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1  CC
 
Theoremrecap0d 7500 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1 #  0
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