HomeHome Intuitionistic Logic Explorer
Theorem List (p. 76 of 94)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecidapd 7501 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     x.  1  1
 
Theoremrecidap2d 7502 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     1  x.  1
 
Theoremrecrecapd 7503 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1  1
 
Theoremdividapd 7504 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     1
 
Theoremdiv0apd 7505 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 0  0
 
Theoremapmul1 7506 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
 CC  CC  C  CC  C #  0 #  x.  C #  x.  C
 
Theoremdivclapd 7507 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>    
 CC
 
Theoremdivcanap1d 7508 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap2d 7509 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivrecapd 7510 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x. 
 1
 
Theoremdivrecap2d 7511 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     1  x.
 
Theoremdivcanap3d 7512 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap4d 7513 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdiveqap0d 7514 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   &     0   =>     0
 
Theoremdiveqap1d 7515 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   &     1   =>   
 
Theoremdiveqap1ad 7516 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7424. Generalization of diveqap1d 7515. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     1
 
Theoremdiveqap0ad 7517 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7403. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     0  0
 
Theoremdivap1d 7518 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
 CC   &     CC   &    #  0   &    #    =>    #  1
 
Theoremdivap0bd 7519 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>    #  0 #  0
 
Theoremdivnegapd 7520 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdivneg2apd 7521 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdiv2negapd 7522 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdivap0d 7523 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    #  0
 
Theoremrecdivapd 7524 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    
 1
 
Theoremrecdivap2d 7525 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     1  1  x.
 
Theoremdivcanap6d 7526 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     x.  1
 
Theoremddcanapd 7527 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>   
 
Theoremrec11apd 7528 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   &    
 1  1    =>   
 
Theoremdivmulapd 7529 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     C  x.  C
 
Theoremdiv32apd 7530 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     x.  C  x.  C
 
Theoremdiv13apd 7531 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     x.  C  C  x.
 
Theoremdivdiv32apd 7532 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdivcanap5d 7533 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C  x.
 
Theoremdivcanap5rd 7534 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     x.  C  x.  C
 
Theoremdivcanap7d 7535 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdmdcanapd 7536 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdmdcanap2d 7537 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     x.  C  C
 
Theoremdivdivap1d 7538 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdivdivap2d 7539 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdivmulap2d 7540 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  C  x.
 
Theoremdivmulap3d 7541 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  x.  C
 
Theoremdivassapd 7542 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdiv12apd 7543 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdiv23apd 7544 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  C  x.
 
Theoremdivdirapd 7545 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     +  C  C  +  C
 
Theoremdivsubdirapd 7546 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     -  C  C  -  C
 
Theoremdiv11apd 7547 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   &     C  C   =>   
 
Theoremrerecclapd 7548 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR   &    #  0   =>    
 1  RR
 
Theoremredivclapd 7549 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR   &     RR   &    #  0   =>    
 RR
 
Theoremmvllmulapd 7550 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC   &     CC   &    #  0   &     x.  C   =>     C
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7551 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 RR  <  +  1
 
Theoremlep1 7552 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 RR  <_  +  1
 
Theoremltm1 7553 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 RR  -  1  <
 
Theoremlem1 7554 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
 RR  -  1  <_
 
Theoremletrp1 7555 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
 RR  RR  <_  <_  +  1
 
Theoremp1le 7556 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
 RR  RR  +  1  <_  <_
 
Theoremrecgt0 7557 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.)
 RR  0  <  0  < 
 1
 
Theoremprodgt0gt0 7558 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7559 for the same theorem with  0  < replaced by the weaker condition 
0  <_ . (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR  RR 
 0  <  0  <  x.  0  <
 
Theoremprodgt0 7559 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.)
 RR  RR 
 0  <_  0  <  x.  0  <
 
Theoremprodgt02 7560 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
 RR  RR 
 0  <_  0  <  x.  0  <
 
Theoremprodge0 7561 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.)
 RR  RR 
 0  <  0  <_  x.  0  <_
 
Theoremprodge02 7562 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 RR  RR 
 0  <  0  <_  x.  0  <_
 
Theoremltmul2 7563 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
 RR  RR  C  RR  0  <  C 
 <  C  x.  <  C  x.
 
Theoremlemul2 7564 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
 RR  RR  C  RR  0  <  C 
 <_  C  x.  <_  C  x.
 
Theoremlemul1a 7565 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.)
 RR  RR  C  RR  0  <_  C  <_  x.  C  <_  x.  C
 
Theoremlemul2a 7566 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 RR  RR  C  RR  0  <_  C  <_  C  x.  <_  C  x.
 
Theoremltmul12a 7567 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
 RR  RR  0  <_  <  C  RR  D  RR  0  <_  C  C  <  D  x.  C  <  x.  D
 
Theoremlemul12b 7568 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 RR  0  <_  RR  C  RR  D  RR  0  <_  D  <_  C  <_  D  x.  C  <_  x.  D
 
Theoremlemul12a 7569 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 RR  0  <_  RR  C  RR  0  <_  C  D  RR 
 <_  C  <_  D  x.  C  <_  x.  D
 
Theoremmulgt1 7570 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
 RR  RR 
 1  <  1  <  1  <  x.
 
Theoremltmulgt11 7571 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 RR  RR  0  <  1  <  <  x.
 
Theoremltmulgt12 7572 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 RR  RR  0  <  1  <  <  x.
 
Theoremlemulge11 7573 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
 RR  RR 
 0  <_  1  <_  <_  x.
 
Theoremlemulge12 7574 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
 RR  RR 
 0  <_  1  <_  <_  x.
 
Theoremltdiv1 7575 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  0  <  C 
 <  C  <  C
 
Theoremlediv1 7576 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C 
 <_  C  <_  C
 
Theoremgt0div 7577 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
 RR  RR  0  <  0  <  0  <
 
Theoremge0div 7578 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
 RR  RR  0  <  0  <_  0  <_
 
Theoremdivgt0 7579 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
 RR  0  <  RR  0  <  0  <
 
Theoremdivge0 7580 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
 RR  0  <_  RR  0  <  0  <_
 
Theoremltmuldiv 7581 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  0  <  C  x.  C 
 <  <  C
 
Theoremltmuldiv2 7582 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C  C  x. 
 <  <  C
 
Theoremltdivmul 7583 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C  C 
 <  <  C  x.
 
Theoremledivmul 7584 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 RR  RR  C  RR  0  <  C  C 
 <_  <_  C  x.
 
Theoremltdivmul2 7585 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
 RR  RR  C  RR  0  <  C  C 
 <  <  x.  C
 
Theoremlt2mul2div 7586 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
 RR  RR  0  <  C  RR  D  RR  0  <  D  x.  <  C  x.  D  D  <  C
 
Theoremledivmul2 7587 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 RR  RR  C  RR  0  <  C  C 
 <_  <_  x.  C
 
Theoremlemuldiv 7588 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 RR  RR  C  RR  0  <  C  x.  C 
 <_  <_  C
 
Theoremlemuldiv2 7589 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 RR  RR  C  RR  0  <  C  C  x. 
 <_  <_  C
 
Theoremltrec 7590 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <  RR  0  <  <  1  < 
 1
 
Theoremlerec 7591 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.)
 RR  0  <  RR  0  <  <_  1  <_ 
 1
 
Theoremlt2msq1 7592 Lemma for lt2msq 7593. (Contributed by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  <  x.  <  x.
 
Theoremlt2msq 7593 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  0  <_  <  x.  <  x.
 
Theoremltdiv2 7594 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
 RR  0  <  RR  0  <  C  RR  0  <  C  <  C  <  C
 
Theoremltrec1 7595 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
 RR  0  <  RR  0  <  1  <  1  <
 
Theoremlerec2 7596 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
 RR  0  <  RR  0  <  <_  1  <_  1
 
Theoremledivdiv 7597 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
 RR  0  <  RR  0  <  C  RR  0  <  C  D  RR  0  <  D 
 <_  C  D  D  C  <_
 
Theoremlediv2 7598 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
 RR  0  <  RR  0  <  C  RR  0  <  C  <_  C  <_  C
 
Theoremltdiv23 7599 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
 RR  RR  0  <  C  RR  0  <  C 
 <  C  C  <
 
Theoremlediv23 7600 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
 RR  RR  0  <  C  RR  0  <  C 
 <_  C  C  <_
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9381
  Copyright terms: Public domain < Previous  Next >