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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnumadd 8401 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 8402 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 8403 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 8404 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 8405 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                            ;

Theoremdecmac 8406 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecma2c 8407 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecadd 8408 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddc 8409 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;       ;

Theoremdecaddc2 8410 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddi 8411 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 8412 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 8413 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;

Theoremdecmul1c 8414 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theoremdecmul2c 8415 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theorem6p5lem 8416 Lemma for 6p5e11 8417 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem6p5e11 8417 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6p6e12 8418 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p4e11 8419 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p5e12 8420 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p6e13 8421 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p7e14 8422 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p3e11 8423 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p4e12 8424 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p5e13 8425 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p6e14 8426 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p7e15 8427 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p8e16 8428 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p2e11 8429 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p3e12 8430 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p4e13 8431 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p5e14 8432 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p6e15 8433 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p7e16 8434 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p8e17 8435 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p9e18 8436 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem10p10e20 8437 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem4t3lem 8438 Lemma for 4t3e12 8439 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem4t3e12 8439 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem4t4e16 8440 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t3e15 8441 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t4e20 8442 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t5e25 8443 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t2e12 8444 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t3e18 8445 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t4e24 8446 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t5e30 8447 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t6e36 8448 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t2e14 8449 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t3e21 8450 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t4e28 8451 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t5e35 8452 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t6e42 8453 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t7e49 8454 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t2e16 8455 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t3e24 8456 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t4e32 8457 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t5e40 8458 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t6e48 8459 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t7e56 8460 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t8e64 8461 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t2e18 8462 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t3e27 8463 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t4e36 8464 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t5e45 8465 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t6e54 8466 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t7e63 8467 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t8e72 8468 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t9e81 8469 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdecbin0 8470 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecbin2 8471 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecbin3 8472 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

3.4.10  Upper sets of integers

Syntaxcuz 8473 Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ."

Definitiondf-uz 8474* Define a function whose value at is the semi-infinite set of contiguous integers starting at , which we will also call the upper integers starting at . Read " " as "the set of integers greater than or equal to ." See uzval 8475 for its value, uzssz 8492 for its relationship to , nnuz 8508 and nn0uz 8507 for its relationships to and , and eluz1 8477 and eluz2 8479 for its membership relations. (Contributed by NM, 5-Sep-2005.)

Theoremuzval 8475* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremuzf 8476 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz1 8477 Membership in the upper set of integers starting at . (Contributed by NM, 5-Sep-2005.)

Theoremeluzel2 8478 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz2 8479 Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz1i 8480 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)

Theoremeluzuzle 8481 An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)

Theoremeluzelz 8482 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)

Theoremeluzelre 8483 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)

Theoremeluzelcn 8484 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremeluzle 8485 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)

Theoremeluz 8486 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)

Theoremuzid 8487 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)

Theoremuzn0 8488 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)

Theoremuztrn 8489 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)

Theoremuztrn2 8490 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremuzneg 8491 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)

Theoremuzssz 8492 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremuzss 8493 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)

Theoremuztric 8494 Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)

Theoremuz11 8495 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)

Theoremeluzp1m1 8496 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1l 8497 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1p1 8498 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)

Theoremeluzaddi 8499 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)

Theoremeluzsubi 8500 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)

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