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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnumsuc 12101 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 + 1) = 𝐶    &   𝑁 = ((𝑇 · 𝐴) + 𝐵)       (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶)
 
Theoremdeccl 12102 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0
 
Theorem10nn 12103 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ
 
Theorem10pos 12104 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
0 < 10
 
Theorem10nn0 12105 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ0
 
Theorem10re 12106 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
10 ∈ ℝ
 
Theoremdecnncl 12107 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ
 
Theoremdec0u 12108 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0       (10 · 𝐴) = 𝐴0
 
Theoremdec0h 12109 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0       𝐴 = 0𝐴
 
Theoremnumnncl2 12110 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ       ((𝑇 · 𝐴) + 0) ∈ ℕ
 
Theoremdecnncl2 12111 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ       𝐴0 ∈ ℕ
 
Theoremnumlt 12112 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ    &   𝐵 < 𝐶       ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)
 
Theoremnumltc 12113 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 < 𝑇    &   𝐴 < 𝐵       ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)
 
Theoremle9lt10 12114 A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐴 ≤ 9       𝐴 < 10
 
Theoremdeclt 12115 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ    &   𝐵 < 𝐶       𝐴𝐵 < 𝐴𝐶
 
Theoremdecltc 12116 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 < 10    &   𝐴 < 𝐵       𝐴𝐶 < 𝐵𝐷
 
Theoremdeclth 12117 Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 ≤ 9    &   𝐴 < 𝐵       𝐴𝐶 < 𝐵𝐷
 
Theoremdecsuc 12118 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 + 1) = 𝐶    &   𝑁 = 𝐴𝐵       (𝑁 + 1) = 𝐴𝐶
 
Theorem3declth 12119 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐴 < 𝐵    &   𝐶 ≤ 9    &   𝐸 ≤ 9       𝐴𝐶𝐸 < 𝐵𝐷𝐹
 
Theorem3decltc 12120 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐸 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐴 < 𝐵    &   𝐶 < 10    &   𝐸 < 10       𝐴𝐶𝐸 < 𝐵𝐷𝐹
 
Theoremdecle 12121 Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐵𝐶       𝐴𝐵𝐴𝐶
 
Theoremdecleh 12122 Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝐶 ≤ 9    &   𝐴 < 𝐵       𝐴𝐶𝐵𝐷
 
Theoremdeclei 12123 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶𝐴𝐵
 
Theoremnumlti 12124 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 < 𝑇       𝐶 < ((𝑇 · 𝐴) + 𝐵)
 
Theoremdeclti 12125 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 < 10       𝐶 < 𝐴𝐵
 
Theoremdecltdi 12126 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶 < 𝐴𝐵
 
Theoremnumsucc 12127 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑌 ∈ ℕ0    &   𝑇 = (𝑌 + 1)    &   𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   𝑁 = ((𝑇 · 𝐴) + 𝑌)       (𝑁 + 1) = ((𝑇 · 𝐵) + 0)
 
Theoremdecsucc 12128 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   𝑁 = 𝐴9       (𝑁 + 1) = 𝐵0
 
Theorem1e0p1 12129 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
1 = (0 + 1)
 
Theoremdec10p 12130 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 𝐴) = 1𝐴
 
Theoremnumma 12131 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (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)    &   𝑃 ∈ ℕ0    &   ((𝐴 · 𝑃) + 𝐶) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = 𝐹       ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)
 
Theoremnummac 12132 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (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)    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸    &   ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹)       ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)
 
Theoremnumma2c 12133 Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (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)    &   𝑃 ∈ ℕ0    &   𝐹 ∈ ℕ0    &   𝐺 ∈ ℕ0    &   ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸    &   ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹)       ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹)
 
Theoremnumadd 12134 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 12135 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 12136 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 12137 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 12138 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
 
Theoremdecmac 12139 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
 
Theoremdecma2c 12140 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
 
Theoremdecadd 12141 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
 
Theoremdecaddc 12142 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
 
Theoremdecaddc2 12143 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 12144 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 12145 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 12146 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 12147 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 12148 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 12149 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
 
Theoremdecsubi 12150 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
 
Theoremdecmul1 12151 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) Remove hypothesis 𝐷 ∈ ℕ0. (Revised by Steven Nguyen, 7-Dec-2022.)
𝑃 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝑁 = 𝐴𝐵    &   (𝐴 · 𝑃) = 𝐶    &   (𝐵 · 𝑃) = 𝐷       (𝑁 · 𝑃) = 𝐶𝐷
 
Theoremdecmul1c 12152 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
 
Theoremdecmul2c 12153 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
 
Theoremdecmulnc 12154 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 12155 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0       (𝑁 · 11) = 𝑁𝑁
 
Theoremdecmul10add 12156 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)
 
Theorem6p5lem 12157 Lemma for 6p5e11 12160 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 12158 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
 
Theorem6p4e10 12159 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
 
Theorem6p5e11 12160 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 + 5) = 11
 
Theorem6p6e12 12161 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 + 6) = 12
 
Theorem7p3e10 12162 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
 
Theorem7p4e11 12163 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(7 + 4) = 11
 
Theorem7p5e12 12164 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 5) = 12
 
Theorem7p6e13 12165 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 6) = 13
 
Theorem7p7e14 12166 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 7) = 14
 
Theorem8p2e10 12167 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
 
Theorem8p3e11 12168 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 + 3) = 11
 
Theorem8p4e12 12169 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 4) = 12
 
Theorem8p5e13 12170 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 5) = 13
 
Theorem8p6e14 12171 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 6) = 14
 
Theorem8p7e15 12172 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 7) = 15
 
Theorem8p8e16 12173 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 8) = 16
 
Theorem9p2e11 12174 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(9 + 2) = 11
 
Theorem9p3e12 12175 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 3) = 12
 
Theorem9p4e13 12176 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 4) = 13
 
Theorem9p5e14 12177 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 5) = 14
 
Theorem9p6e15 12178 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 6) = 15
 
Theorem9p7e16 12179 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 7) = 16
 
Theorem9p8e17 12180 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 8) = 17
 
Theorem9p9e18 12181 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 9) = 18
 
Theorem10p10e20 12182 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 10) = 20
 
Theorem10m1e9 12183 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
(10 − 1) = 9
 
Theorem4t3lem 12184 Lemma for 4t3e12 12185 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 12185 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 3) = 12
 
Theorem4t4e16 12186 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 4) = 16
 
Theorem5t2e10 12187 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
(5 · 2) = 10
 
Theorem5t3e15 12188 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 3) = 15
 
Theorem5t4e20 12189 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 4) = 20
 
Theorem5t5e25 12190 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 5) = 25
 
Theorem6t2e12 12191 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 2) = 12
 
Theorem6t3e18 12192 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18
 
Theorem6t4e24 12193 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24
 
Theorem6t5e30 12194 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 5) = 30
 
Theorem6t6e36 12195 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 6) = 36
 
Theorem7t2e14 12196 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14
 
Theorem7t3e21 12197 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21
 
Theorem7t4e28 12198 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28
 
Theorem7t5e35 12199 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35
 
Theorem7t6e42 12200 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42
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