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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 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = ((𝑇 · 𝐴) + 𝐵)    &   𝑁 = ((𝑇 · 𝐶) + 𝐷)    &   (𝐴 + 𝐶) = 𝐸    &   (𝐵 + 𝐷) = 𝐹       (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theoremnumaddc 8402 Add two decimal integers 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = ((𝑇 · 𝐴) + 𝐵)    &   𝑁 = ((𝑇 · 𝐶) + 𝐷)    &   𝐹 ∈ ℕ0    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   (𝐵 + 𝐷) = ((𝑇 · 1) + 𝐹)       (𝑀 + 𝑁) = ((𝑇 · 𝐸) + 𝐹)

Theoremnummul1c 8403 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = ((𝑇 · 𝐴) + 𝐵)    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐸) = 𝐶    &   (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷)       (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷)

Theoremnummul2c 8404 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = ((𝑇 · 𝐴) + 𝐵)    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝑃 · 𝐴) + 𝐸) = 𝐶    &   (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷)       (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷)

Theoremdecma 8405 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   𝑃 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐶) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = 𝐹       ((𝑀 · 𝑃) + 𝑁) = 𝐸𝐹

Theoremdecmac 8406 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = 𝐺𝐹       ((𝑀 · 𝑃) + 𝑁) = 𝐸𝐹

Theoremdecma2c 8407 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸    &   ((𝑃 · 𝐵) + 𝐷) = 𝐺𝐹       ((𝑃 · 𝑀) + 𝑁) = 𝐸𝐹

Theoremdecadd 8408 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   (𝐴 + 𝐶) = 𝐸    &   (𝐵 + 𝐷) = 𝐹       (𝑀 + 𝑁) = 𝐸𝐹

Theoremdecaddc 8409 Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   𝐹 ∈ ℕ0    &   (𝐵 + 𝐷) = 1𝐹       (𝑀 + 𝑁) = 𝐸𝐹

Theoremdecaddc2 8410 Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   (𝐵 + 𝐷) = 10       (𝑀 + 𝑁) = 𝐸0

Theoremdecaddi 8411 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐵 + 𝑁) = 𝐶       (𝑀 + 𝑁) = 𝐴𝐶

Theoremdecaddci 8412 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   𝐶 ∈ ℕ0    &   (𝐵 + 𝑁) = 1𝐶       (𝑀 + 𝑁) = 𝐷𝐶

Theoremdecaddci2 8413 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   (𝐵 + 𝑁) = 10       (𝑀 + 𝑁) = 𝐷0

Theoremdecmul1c 8414 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐸) = 𝐶    &   (𝐵 · 𝑃) = 𝐸𝐷       (𝑁 · 𝑃) = 𝐶𝐷

Theoremdecmul2c 8415 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝑃 · 𝐴) + 𝐸) = 𝐶    &   (𝑃 · 𝐵) = 𝐸𝐷       (𝑃 · 𝑁) = 𝐶𝐷

Theorem6p5lem 8416 Lemma for 6p5e11 8417 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐵 = (𝐷 + 1)    &   𝐶 = (𝐸 + 1)    &   (𝐴 + 𝐷) = 1𝐸       (𝐴 + 𝐵) = 1𝐶

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem4t3lem 8438 Lemma for 4t3e12 8439 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 = (𝐵 + 1)    &   (𝐴 · 𝐵) = 𝐷    &   (𝐷 + 𝐴) = 𝐸       (𝐴 · 𝐶) = 𝐸

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theoremdecbin0 8470 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       (4 · 𝐴) = (2 · (2 · 𝐴))

Theoremdecbin2 8471 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))

Theoremdecbin3 8472 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)

3.4.10  Upper sets of integers

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

Definitiondf-uz 8474* Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 8475 for its value, uzssz 8492 for its relationship to , nnuz 8508 and nn0uz 8507 for its relationships to and 0, and eluz1 8477 and eluz2 8479 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})

Theoremuzval 8475* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})

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.)
:ℤ⟶𝒫 ℤ

Theoremeluz1 8477 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁)))

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

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 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))

Theoremeluz1i 8480 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))

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.)
((𝐵 ∈ ℤ ∧ 𝐵𝐴) → (𝐶 ∈ (ℤ𝐴) → 𝐶 ∈ (ℤ𝐵)))

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

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

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

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

Theoremeluz 8486 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))

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

Theoremuzn0 8488 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ∈ ran ℤ𝑀 ≠ ∅)

Theoremuztrn 8489 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝑀 ∈ (ℤ𝑁))

Theoremuztrn2 8490 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁𝑍𝑀 ∈ (ℤ𝑁)) → 𝑀𝑍)

Theoremuzneg 8491 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ (ℤ𝑀) → -𝑀 ∈ (ℤ‘-𝑁))

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.)
(ℤ𝑀) ⊆ ℤ

Theoremuzss 8493 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))

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.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ∨ 𝑀 ∈ (ℤ𝑁)))

Theoremuz11 8495 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))

Theoremeluzp1m1 8496 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ𝑀))

Theoremeluzp1l 8497 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)

Theoremeluzp1p1 8498 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))

Theoremeluzaddi 8499 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))

Theoremeluzsubi 8500 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) ∈ (ℤ𝑀))

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