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

Theorem5p4e9 8501 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem6p2e8 8502 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 8503 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem7p2e9 8504 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem1t1e1 8505 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 8506 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

Theorem2t2e4 8507 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4

Theorem3t1e3 8508 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3

Theorem3t2e6 8509 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6

Theorem3t3e9 8510 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9

Theorem4t2e8 8511 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8

Theorem2t0e0 8512 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0

Theorem4d2e2 8513 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2

Theorem2nn 8514 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem3nn 8515 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem4nn 8516 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem5nn 8517 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem6nn 8518 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem7nn 8519 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem8nn 8520 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem9nn 8521 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem1lt2 8522 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2

Theorem2lt3 8523 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3

Theorem1lt3 8524 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3

Theorem3lt4 8525 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4

Theorem2lt4 8526 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4

Theorem1lt4 8527 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 8528 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 8529 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 8530 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 8531 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 8532 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 8533 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 8534 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 8535 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 8536 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 8537 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 8538 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 8539 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 8540 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 8541 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

Theorem1lt7 8542 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7

Theorem7lt8 8543 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8

Theorem6lt8 8544 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8

Theorem5lt8 8545 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8

Theorem4lt8 8546 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8

Theorem3lt8 8547 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8

Theorem2lt8 8548 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8

Theorem1lt8 8549 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8

Theorem8lt9 8550 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9

Theorem7lt9 8551 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9

Theorem6lt9 8552 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9

Theorem5lt9 8553 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9

Theorem4lt9 8554 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9

Theorem3lt9 8555 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9

Theorem2lt9 8556 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9

Theorem1lt9 8557 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9

Theorem0ne2 8558 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2

Theorem1ne2 8559 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2

Theorem1le2 8560 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2

Theorem2cnne0 8561 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)

Theorem2rene0 8562 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)

Theorem1le3 8563 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3

Theoremneg1mulneg1e1 8564 -1 · -1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1

Theoremhalfre 8565 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ

Theoremhalfcn 8566 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ

Theoremhalfgt0 8567 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)

Theoremhalfge0 8568 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)

Theoremhalflt1 8569 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1

Theorem1mhlfehlf 8570 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)

Theorem8th4div3 8571 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)

Theoremhalfpm6th 8572 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
(((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))

Theoremit0e0 8573 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0

Theorem2mulicn 8574 (2 · i) ∈ ℂ (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ

Theoremiap0 8575 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0

Theorem2muliap0 8576 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0

Theorem2muline0 8577 (2 · i) ≠ 0. See also 2muliap0 8576. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0

3.4.5  Simple number properties

Theoremhalfcl 8578 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)

Theoremrehalfcl 8579 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)

Theoremhalf0 8580 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(𝐴 ∈ ℂ → ((𝐴 / 2) = 0 ↔ 𝐴 = 0))

Theorem2halves 8581 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremhalfpos2 8582 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))

Theoremhalfpos 8583 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (𝐴 / 2) < 𝐴))

Theoremhalfnneg2 8584 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))

Theoremhalfaddsubcl 8585 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) / 2) ∈ ℂ ∧ ((𝐴𝐵) / 2) ∈ ℂ))

Theoremhalfaddsub 8586 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐵) / 2) + ((𝐴𝐵) / 2)) = 𝐴 ∧ (((𝐴 + 𝐵) / 2) − ((𝐴𝐵) / 2)) = 𝐵))

Theoremlt2halves 8587 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶))

Theoremaddltmul 8588 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))

Theoremnominpos 8589* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦𝑦 < 𝑥))

Theoremavglt1 8590 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < ((𝐴 + 𝐵) / 2)))

Theoremavglt2 8591 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))

Theoremavgle1 8592 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵𝐴 ≤ ((𝐴 + 𝐵) / 2)))

Theoremavgle2 8593 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵))

Theorem2timesd 8594 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))

Theoremtimes2d 8595 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))

Theoremhalfcld 8596 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 2) ∈ ℂ)

Theorem2halvesd 8597 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremrehalfcld 8598 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 / 2) ∈ ℝ)

Theoremlt2halvesd 8599 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐶 / 2))    &   (𝜑𝐵 < (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) < 𝐶)

Theoremrehalfcli 8600 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       (𝐴 / 2) ∈ ℝ

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