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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzaddcld 9201 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
 
Theoremzsubcld 9202 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴𝐵) ∈ ℤ)
 
Theoremzmulcld 9203 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
 
Theoremzadd2cl 9204 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)
 
Theorembtwnapz 9205 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < (𝐴 + 1))       (𝜑𝐵 # 𝐶)
 
4.4.10  Decimal arithmetic
 
Syntaxcdc 9206 Constant used for decimal constructor.
class 𝐴𝐵
 
Definitiondf-dec 9207 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (1000 + 2000) = 3000 1kp2ke3k 13107. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
 
Theorem9p1e10 9208 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
(9 + 1) = 10
 
Theoremdfdec10 9209 Version of the definition of the "decimal constructor" using 10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
𝐴𝐵 = ((10 · 𝐴) + 𝐵)
 
Theoremdeceq1 9210 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 
Theoremdeceq2 9211 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 
Theoremdeceq1i 9212 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐴𝐶 = 𝐵𝐶
 
Theoremdeceq2i 9213 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵       𝐶𝐴 = 𝐶𝐵
 
Theoremdeceq12i 9214 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 = 𝐵𝐷
 
Theoremnumnncl 9215 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       ((𝑇 · 𝐴) + 𝐵) ∈ ℕ
 
Theoremnum0u 9216 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)
 
Theoremnum0h 9217 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0       𝐴 = ((𝑇 · 0) + 𝐴)
 
Theoremnumcl 9218 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑇 ∈ ℕ0    &   𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0
 
Theoremnumsuc 9219 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 9220 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵 ∈ ℕ0
 
Theorem10nn 9221 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ
 
Theorem10pos 9222 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
0 < 10
 
Theorem10nn0 9223 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
10 ∈ ℕ0
 
Theorem10re 9224 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
10 ∈ ℝ
 
Theoremdecnncl 9225 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ       𝐴𝐵 ∈ ℕ
 
Theoremdec0u 9226 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 9227 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 9228 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
𝑇 ∈ ℕ    &   𝐴 ∈ ℕ       ((𝑇 · 𝐴) + 0) ∈ ℕ
 
Theoremdecnncl2 9229 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 9230 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 9231 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 9232 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 9233 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 9234 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 9235 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 9236 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 9237 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 9238 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 9239 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 9240 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 9241 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶𝐴𝐵
 
Theoremnumlti 9242 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 9243 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 9244 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐶 ≤ 9       𝐶 < 𝐴𝐵
 
Theoremnumsucc 9245 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 9246 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 9247 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
1 = (0 + 1)
 
Theoremdec10p 9248 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 𝐴) = 1𝐴
 
Theoremnumma 9249 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 9250 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 9251 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 9252 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 9253 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 9254 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 9255 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 9256 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 9257 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 9258 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 9259 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 9260 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 9261 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 9262 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 9263 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 9264 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 9265 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 9266 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 9267 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 9268 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 9269 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
 
Theoremdecmul1c 9270 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 9271 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 9272 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 9273 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
𝑁 ∈ ℕ0       (𝑁 · 11) = 𝑁𝑁
 
Theoremdecmul10add 9274 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 9275 Lemma for 6p5e11 9278 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 9276 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 9277 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 9278 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 + 5) = 11
 
Theorem6p6e12 9279 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 + 6) = 12
 
Theorem7p3e10 9280 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 9281 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(7 + 4) = 11
 
Theorem7p5e12 9282 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 5) = 12
 
Theorem7p6e13 9283 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 6) = 13
 
Theorem7p7e14 9284 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 + 7) = 14
 
Theorem8p2e10 9285 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 9286 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 + 3) = 11
 
Theorem8p4e12 9287 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 4) = 12
 
Theorem8p5e13 9288 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 5) = 13
 
Theorem8p6e14 9289 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 6) = 14
 
Theorem8p7e15 9290 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 7) = 15
 
Theorem8p8e16 9291 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 8) = 16
 
Theorem9p2e11 9292 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(9 + 2) = 11
 
Theorem9p3e12 9293 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 3) = 12
 
Theorem9p4e13 9294 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 4) = 13
 
Theorem9p5e14 9295 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 5) = 14
 
Theorem9p6e15 9296 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 6) = 15
 
Theorem9p7e16 9297 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 7) = 16
 
Theorem9p8e17 9298 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 8) = 17
 
Theorem9p9e18 9299 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 9) = 18
 
Theorem10p10e20 9300 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(10 + 10) = 20
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