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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnumaddc 11301 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 11302 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 11303 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 11304 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 11305 Obsolete proof of decma 11304 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 11306 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 11307 Obsolete proof of decmac 11306 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 11308 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 11309 Obsolete proof of decma2c 11308 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 11310 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 11311 Obsolete proof of decadd 11310 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 11312 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 11313 Obsolete proof of decaddc 11312 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 11314 Obsolete version of decaddc2 11315 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 11315 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 11316 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 11317 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 11318 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 11319 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 11320 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 11321 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 11322 Obsolete version of decaddci2 11321 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 11323 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 11324 Obsolete proof of decsubi 11323 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 11325 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 11326 Obsolete proof of decmul1 11325 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 11327 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 11328 Obsolete proof of decmul1c 11327 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 11329 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 11330 Obsolete proof of decmul2c 11329 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 11331 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 11332 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0       (𝑁 · 11) = 𝑁𝑁
 
Theoremdecmul10add 11333 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 11334 Obsolete proof of decmul10add 11333 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 11335 Lemma for 6p5e11 11340 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 11336 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 11337 Obsolete proof of 5p5e10 11336 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 + 5) = 10
 
Theorem6p4e10 11338 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 11339 Obsolete proof of 6p4e10 11338 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 4) = 10
 
Theorem6p5e11 11340 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 + 5) = 11
 
Theorem6p5e11OLD 11341 Obsolete proof of 6p5e11 11340 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 5) = 11
 
Theorem6p6e12 11342 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 + 6) = 12
 
Theorem7p3e10 11343 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 11344 Obsolete proof of 7p3e10 11343 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 3) = 10
 
Theorem7p4e11 11345 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(7 + 4) = 11
 
Theorem7p4e11OLD 11346 Obsolete proof of 7p4e11 11345 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 4) = 11
 
Theorem7p5e12 11347 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 5) = 12
 
Theorem7p6e13 11348 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 6) = 13
 
Theorem7p7e14 11349 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 7) = 14
 
Theorem8p2e10 11350 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 11351 Obsolete proof of 8p2e10 11350 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 2) = 10
 
Theorem8p3e11 11352 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 + 3) = 11
 
Theorem8p3e11OLD 11353 Obsolete proof of 8p3e11 11352 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 3) = 11
 
Theorem8p4e12 11354 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 4) = 12
 
Theorem8p5e13 11355 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 5) = 13
 
Theorem8p6e14 11356 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 6) = 14
 
Theorem8p7e15 11357 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 7) = 15
 
Theorem8p8e16 11358 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 8) = 16
 
Theorem9p2e11 11359 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(9 + 2) = 11
 
Theorem9p2e11OLD 11360 Obsolete proof of 9p2e11 11359 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(9 + 2) = 11
 
Theorem9p3e12 11361 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 3) = 12
 
Theorem9p4e13 11362 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 4) = 13
 
Theorem9p5e14 11363 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 5) = 14
 
Theorem9p6e15 11364 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 6) = 15
 
Theorem9p7e16 11365 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 7) = 16
 
Theorem9p8e17 11366 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 8) = 17
 
Theorem9p9e18 11367 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 9) = 18
 
Theorem10p10e20 11368 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 10) = 20
 
Theorem10p10e20OLD 11369 Obsolete version of 10p10e20 11368 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(10 + 10) = 20
 
Theorem10m1e9 11370 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
(10 − 1) = 9
 
Theorem4t3lem 11371 Lemma for 4t3e12 11372 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 11372 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 3) = 12
 
Theorem4t4e16 11373 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 4) = 16
 
Theorem5t2e10 11374 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
(5 · 2) = 10
 
Theorem5t3e15 11375 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 3) = 15
 
Theorem5t3e15OLD 11376 Obsolete proof of 5t3e15 11375 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 3) = 15
 
Theorem5t4e20 11377 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 4) = 20
 
Theorem5t4e20OLD 11378 Obsolete proof of 5t4e20 11377 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 4) = 20
 
Theorem5t5e25 11379 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 5) = 25
 
Theorem5t5e25OLD 11380 Obsolete proof of 5t5e25 11379 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 5) = 25
 
Theorem6t2e12 11381 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 2) = 12
 
Theorem6t3e18 11382 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18
 
Theorem6t4e24 11383 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24
 
Theorem6t5e30 11384 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 5) = 30
 
Theorem6t5e30OLD 11385 Obsolete proof of 6t5e30 11384 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 · 5) = 30
 
Theorem6t6e36 11386 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 6) = 36
 
Theorem6t6e36OLD 11387 Obsolete proof of 6t6e36 11386 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(6 · 6) = 36
 
Theorem7t2e14 11388 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14
 
Theorem7t3e21 11389 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21
 
Theorem7t4e28 11390 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28
 
Theorem7t5e35 11391 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35
 
Theorem7t6e42 11392 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42
 
Theorem7t7e49 11393 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 7) = 49
 
Theorem8t2e16 11394 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 2) = 16
 
Theorem8t3e24 11395 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24
 
Theorem8t4e32 11396 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32
 
Theorem8t5e40 11397 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 5) = 40
 
Theorem8t5e40OLD 11398 Obsolete proof of 8t5e40 11397 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 · 5) = 40
 
Theorem8t6e48 11399 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 6) = 48
 
Theorem8t6e48OLD 11400 Obsolete proof of 8t6e48 11399 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(8 · 6) = 48
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