P. 96 of Theorem List - Intuitionistic Logic Explorer

Home	Intuitionistic Logic Explorer Theorem List (p. 96 of 161)	< Previous Next >
		Bad symbols? Try the GIF version.
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List

**Theorem List for Intuitionistic Logic Explorer -** 9501-9600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	nnzd 9501	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ)

Theorem	zred 9502	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ)

Theorem	zcnd 9503	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ)

Theorem	znegcld 9504	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ)

Theorem	peano2zd 9505	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)

Theorem	zaddcld 9506	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)

Theorem	zsubcld 9507	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)

Theorem	zmulcld 9508	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)

Theorem	zadd2cl 9509	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)

Theorem	btwnapz 9510	A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 < (𝐴 + 1)) ⇒ ⊢ (𝜑 → 𝐵 # 𝐶)

4.4.10 Decimal arithmetic

Syntax	cdc 9511	Constant used for decimal constructor.
class ;𝐴𝐵

Definition	df-dec 9512	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 15734. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

Theorem	9p1e10 9513	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

Theorem	dfdec10 9514	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)

Theorem	deceq1 9515	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Theorem	deceq2 9516	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

Theorem	deceq1i 9517	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶

Theorem	deceq2i 9518	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵

Theorem	deceq12i 9519	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷

Theorem	numnncl 9520	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ

Theorem	num0u 9521	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)

Theorem	num0h 9522	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)

Theorem	numcl 9523	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

Theorem	numsuc 9524	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)

Theorem	deccl 9525	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ₀

Theorem	10nn 9526	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ

Theorem	10pos 9527	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10

Theorem	10nn0 9528	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ₀

Theorem	10re 9529	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ ;10 ∈ ℝ

Theorem	decnncl 9530	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ

Theorem	dec0u 9531	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

Theorem	dec0h 9532	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ 𝐴 = ;0𝐴

Theorem	numnncl2 9533	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝑇 ∈ ℕ & ⊢ 𝐴 ∈ ℕ ⇒ ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ

Theorem	decnncl2 9534	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

Theorem	numlt 9535	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)

Theorem	numltc 9536	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)

Theorem	le9lt10 9537	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

Theorem	declt 9538	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

Theorem	decltc 9539	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

Theorem	declth 9540	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

Theorem	decsuc 9541	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

Theorem	3declth 9542	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

Theorem	3decltc 9543	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

Theorem	decle 9544	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 & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶

Theorem	decleh 9545	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 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷

Theorem	declei 9546	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 ≤ ;𝐴𝐵

Theorem	numlti 9547	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)

Theorem	declti 9548	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

Theorem	decltdi 9549	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 < ;𝐴𝐵

Theorem	numsucc 9550	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)

Theorem	decsucc 9551	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

Theorem	1e0p1 9552	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 1 = (0 + 1)

Theorem	dec10p 9553	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴

Theorem	numma 9554	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)

Theorem	nummac 9555	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)

Theorem	numma2c 9556	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)

Theorem	numadd 9557	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)

Theorem	numaddc 9558	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)

Theorem	nummul1c 9559	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)

Theorem	nummul2c 9560	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)

Theorem	decma 9561	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

Theorem	decmac 9562	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

Theorem	decma2c 9563	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

Theorem	decadd 9564	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

Theorem	decaddc 9565	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

Theorem	decaddc2 9566	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

Theorem	decrmanc 9567	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

Theorem	decrmac 9568	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

Theorem	decaddm10 9569	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

Theorem	decaddi 9570	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

Theorem	decaddci 9571	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

Theorem	decaddci2 9572	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

Theorem	decsubi 9573	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 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Theorem	decmul1 9574	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

Theorem	decmul1c 9575	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

Theorem	decmul2c 9576	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

Theorem	decmulnc 9577	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)

Theorem	11multnc 9578	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ₀ ⇒ ⊢ (𝑁 · ;11) = ;𝑁𝑁

Theorem	decmul10add 9579	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)

Theorem	6p5lem 9580	Lemma for 6p5e11 9583 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

Theorem	5p5e10 9581	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

Theorem	6p4e10 9582	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

Theorem	6p5e11 9583	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 + 5) = ;11

Theorem	6p6e12 9584	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (6 + 6) = ;12

Theorem	7p3e10 9585	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

Theorem	7p4e11 9586	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 4) = ;11

Theorem	7p5e12 9587	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 5) = ;12

Theorem	7p6e13 9588	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 6) = ;13

Theorem	7p7e14 9589	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (7 + 7) = ;14

Theorem	8p2e10 9590	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

Theorem	8p3e11 9591	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 3) = ;11

Theorem	8p4e12 9592	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 4) = ;12

Theorem	8p5e13 9593	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 5) = ;13

Theorem	8p6e14 9594	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 6) = ;14

Theorem	8p7e15 9595	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 7) = ;15

Theorem	8p8e16 9596	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 + 8) = ;16

Theorem	9p2e11 9597	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (9 + 2) = ;11

Theorem	9p3e12 9598	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 3) = ;12

Theorem	9p4e13 9599	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 4) = ;13

Theorem	9p5e14 9600	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 + 5) = ;14

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16097

< Previous Next >