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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnummul2c 11601 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 11602 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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

TheoremdecmaOLD 11603 Obsolete proof of decma 11602 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
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 11604 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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

TheoremdecmacOLD 11605 Obsolete proof of decmac 11604 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
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 11606 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplier 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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

Theoremdecma2cOLD 11607 Obsolete proof of decma2c 11606 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
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 11608 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   (𝐴 + 𝐶) = 𝐸    &   (𝐵 + 𝐷) = 𝐹       (𝑀 + 𝑁) = 𝐸𝐹

TheoremdecaddOLD 11609 Obsolete proof of decadd 11608 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   (𝐴 + 𝐶) = 𝐸    &   (𝐵 + 𝐷) = 𝐹       (𝑀 + 𝑁) = 𝐸𝐹

Theoremdecaddc 11610 Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   𝐹 ∈ ℕ0    &   (𝐵 + 𝐷) = 1𝐹       (𝑀 + 𝑁) = 𝐸𝐹

TheoremdecaddcOLD 11611 Obsolete proof of decaddc 11610 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   𝐹 ∈ ℕ0    &   (𝐵 + 𝐷) = 1𝐹       (𝑀 + 𝑁) = 𝐸𝐹

Theoremdecaddc2OLD 11612 Obsolete version of decaddc2 11613 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   (𝐵 + 𝐷) = 10       (𝑀 + 𝑁) = 𝐸0

Theoremdecaddc2 11613 Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑁 = 𝐶𝐷    &   ((𝐴 + 𝐶) + 1) = 𝐸    &   (𝐵 + 𝐷) = 10       (𝑀 + 𝑁) = 𝐸0

Theoremdecrmanc 11614 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by AV, 16-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑃 ∈ ℕ0    &   (𝐴 · 𝑃) = 𝐸    &   ((𝐵 · 𝑃) + 𝑁) = 𝐹       ((𝑀 · 𝑃) + 𝑁) = 𝐸𝐹

Theoremdecrmac 11615 Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐺) = 𝐸    &   ((𝐵 · 𝑃) + 𝑁) = 𝐺𝐹       ((𝑀 · 𝑃) + 𝑁) = 𝐸𝐹

Theoremdecaddm10 11616 The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (𝐴0 + 𝐵0) = (𝐴 + 𝐵)0

Theoremdecaddi 11617 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 11618 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 11619 Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   (𝐵 + 𝑁) = 10       (𝑀 + 𝑁) = 𝐷0

Theoremdecaddci2OLD 11620 Obsolete version of decaddci2 11619 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   (𝐵 + 𝑁) = 10       (𝑀 + 𝑁) = 𝐷0

Theoremdecsubi 11621 Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   (𝐵𝑁) = 𝐶       (𝑀𝑁) = 𝐴𝐶

TheoremdecsubiOLD 11622 Obsolete proof of decsubi 11621 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   𝑀 = 𝐴𝐵    &   (𝐴 + 1) = 𝐷    &   (𝐵𝑁) = 𝐶       (𝑀𝑁) = 𝐴𝐶

Theoremdecmul1 11623 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝑃) = 𝐶    &   (𝐵 · 𝑃) = 𝐷       (𝑁 · 𝑃) = 𝐶𝐷

Theoremdecmul1OLD 11624 Obsolete proof of decmul1 11623 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝑃) = 𝐶    &   (𝐵 · 𝑃) = 𝐷       (𝑁 · 𝑃) = 𝐶𝐷

Theoremdecmul1c 11625 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐸) = 𝐶    &   (𝐵 · 𝑃) = 𝐸𝐷       (𝑁 · 𝑃) = 𝐶𝐷

Theoremdecmul1cOLD 11626 Obsolete proof of decmul1c 11625 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐸) = 𝐶    &   (𝐵 · 𝑃) = 𝐸𝐷       (𝑁 · 𝑃) = 𝐶𝐷

Theoremdecmul2c 11627 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝑃 · 𝐴) + 𝐸) = 𝐶    &   (𝑃 · 𝐵) = 𝐸𝐷       (𝑃 · 𝑁) = 𝐶𝐷

Theoremdecmul2cOLD 11628 Obsolete proof of decmul2c 11627 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   ((𝑃 · 𝐴) + 𝐸) = 𝐶    &   (𝑃 · 𝐵) = 𝐸𝐷       (𝑃 · 𝑁) = 𝐶𝐷

Theoremdecmulnc 11629 The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (𝑁 · 𝐴𝐵) = (𝑁 · 𝐴)(𝑁 · 𝐵)

Theorem11multnc 11630 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0       (𝑁 · 11) = 𝑁𝑁

Theoremdecmul10add 11631 A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   𝐸 = (𝑀 · 𝐴)    &   𝐹 = (𝑀 · 𝐵)       (𝑀 · 𝐴𝐵) = (𝐸0 + 𝐹)

Theoremdecmul10addOLD 11632 Obsolete proof of decmul10add 11631 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   𝐸 = (𝑀 · 𝐴)    &   𝐹 = (𝑀 · 𝐵)       (𝑀 · 𝐴𝐵) = (𝐸0 + 𝐹)

Theorem6p5lem 11633 Lemma for 6p5e11 11638 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

Theorem5p5e10 11634 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(5 + 5) = 10

Theorem5p5e10bOLD 11635 Obsolete proof of 5p5e10 11634 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 + 5) = 10

Theorem6p4e10 11636 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(6 + 4) = 10

Theorem6p4e10bOLD 11637 Obsolete proof of 6p4e10 11636 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 4) = 10

Theorem6p5e11 11638 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 + 5) = 11

Theorem6p5e11OLD 11639 Obsolete proof of 6p5e11 11638 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 5) = 11

Theorem6p6e12 11640 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 + 6) = 12

Theorem7p3e10 11641 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(7 + 3) = 10

Theorem7p3e10bOLD 11642 Obsolete proof of 7p3e10 11641 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 3) = 10

Theorem7p4e11 11643 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(7 + 4) = 11

Theorem7p4e11OLD 11644 Obsolete proof of 7p4e11 11643 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 4) = 11

Theorem7p5e12 11645 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 5) = 12

Theorem7p6e13 11646 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 6) = 13

Theorem7p7e14 11647 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 7) = 14

Theorem8p2e10 11648 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
(8 + 2) = 10

Theorem8p2e10bOLD 11649 Obsolete proof of 8p2e10 11648 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 2) = 10

Theorem8p3e11 11650 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 + 3) = 11

Theorem8p3e11OLD 11651 Obsolete proof of 8p3e11 11650 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 3) = 11

Theorem8p4e12 11652 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 4) = 12

Theorem8p5e13 11653 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 5) = 13

Theorem8p6e14 11654 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 6) = 14

Theorem8p7e15 11655 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 7) = 15

Theorem8p8e16 11656 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 8) = 16

Theorem9p2e11 11657 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(9 + 2) = 11

Theorem9p2e11OLD 11658 Obsolete proof of 9p2e11 11657 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(9 + 2) = 11

Theorem9p3e12 11659 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 3) = 12

Theorem9p4e13 11660 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 4) = 13

Theorem9p5e14 11661 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 5) = 14

Theorem9p6e15 11662 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 6) = 15

Theorem9p7e16 11663 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 7) = 16

Theorem9p8e17 11664 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 8) = 17

Theorem9p9e18 11665 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 9) = 18

Theorem10p10e20 11666 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 10) = 20

Theorem10p10e20OLD 11667 Obsolete version of 10p10e20 11666 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(10 + 10) = 20

Theorem10m1e9 11668 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
(10 − 1) = 9

Theorem4t3lem 11669 Lemma for 4t3e12 11670 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 11670 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 3) = 12

Theorem4t4e16 11671 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 4) = 16

Theorem5t2e10 11672 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
(5 · 2) = 10

Theorem5t3e15 11673 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 3) = 15

Theorem5t3e15OLD 11674 Obsolete proof of 5t3e15 11673 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 3) = 15

Theorem5t4e20 11675 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 4) = 20

Theorem5t4e20OLD 11676 Obsolete proof of 5t4e20 11675 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 4) = 20

Theorem5t5e25 11677 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 5) = 25

Theorem5t5e25OLD 11678 Obsolete proof of 5t5e25 11677 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 5) = 25

Theorem6t2e12 11679 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 2) = 12

Theorem6t3e18 11680 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18

Theorem6t4e24 11681 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24

Theorem6t5e30 11682 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 5) = 30

Theorem6t5e30OLD 11683 Obsolete proof of 6t5e30 11682 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 · 5) = 30

Theorem6t6e36 11684 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 6) = 36

Theorem6t6e36OLD 11685 Obsolete proof of 6t6e36 11684 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 · 6) = 36

Theorem7t2e14 11686 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14

Theorem7t3e21 11687 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21

Theorem7t4e28 11688 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28

Theorem7t5e35 11689 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35

Theorem7t6e42 11690 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42

Theorem7t7e49 11691 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 7) = 49

Theorem8t2e16 11692 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 2) = 16

Theorem8t3e24 11693 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24

Theorem8t4e32 11694 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32

Theorem8t5e40 11695 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 5) = 40

Theorem8t5e40OLD 11696 Obsolete proof of 8t5e40 11695 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 · 5) = 40

Theorem8t6e48 11697 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 6) = 48

Theorem8t6e48OLD 11698 Obsolete proof of 8t6e48 11697 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 · 6) = 48

Theorem8t7e56 11699 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56

Theorem8t8e64 11700 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64

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