P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nneoor 9701	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
Theorem	nneo 9702	A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ))
Theorem	nneoi 9703	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)
Theorem	zeo 9704	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zeo2 9705	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	peano2uz2 9706*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥}) → (𝐵 + 1) ∈ {𝑥 ∈ ℤ ∣ 𝐴 ≤ 𝑥})
Theorem	peano5uzti 9707*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
⊢ (𝑁 ∈ ℤ → ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴))
Theorem	peano5uzi 9708*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ 𝑁 ∈ ℤ    ⇒   ⊢ ((𝑁 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ 𝐴)
Theorem	dfuzi 9709*	An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 9259 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
⊢ 𝑁 ∈ ℤ    ⇒   ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
Theorem	uzind 9710*	Induction on the upper integers that start at 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜏)
Theorem	uzind2 9711*	Induction on the upper integers that start after an integer 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
⊢ (𝑗 = (𝑀 + 1) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏)
Theorem	uzind3 9712*	Induction on the upper integers that start at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑚 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑚 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → 𝜏)
Theorem	nn0ind 9713*	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ0 → 𝜏)
Theorem	fzind 9714*	Induction on the integers from 𝑀 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓)    &   ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃))    ⇒   ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏)
Theorem	fnn0ind 9715*	Induction on the integers from 0 to 𝑁 inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑁 ∈ ℕ0 → 𝜓)    &   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → 𝜏)
Theorem	nn0ind-raph 9716*	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ0 → 𝜏)
Theorem	zindd 9717*	Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏))    &   ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂))    &   ⊢ (𝜁 → 𝜓)    &   ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏)))    &   ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃)))    ⇒   ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂))
Theorem	btwnz 9718*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦))
Theorem	nn0zd 9719	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	nnzd 9720	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	zred 9721	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zcnd 9722	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	znegcld 9723	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℤ)
Theorem	peano2zd 9724	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)
Theorem	zaddcld 9725	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
Theorem	zsubcld 9726	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)
Theorem	zmulcld 9727	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
Theorem	zadd2cl 9728	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 9729	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 9730	Constant used for decimal constructor.
class ;𝐴𝐵
Definition	df-dec 9731	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 16621. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
Theorem	9p1e10 9732	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 9733	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 9734	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 9735	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 9736	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐶
Theorem	deceq2i 9737	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐶𝐴 = ;𝐶𝐵
Theorem	deceq12i 9738	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐷
Theorem	numnncl 9739	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 9740	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 9741	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)
Theorem	numcl 9742	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 9743	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 9744	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 9745	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ
Theorem	10pos 9746	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10
Theorem	10nn0 9747	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ0
Theorem	10re 9748	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ ;10 ∈ ℝ
Theorem	decnncl 9749	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 9750	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 9751	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 9752	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 9753	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 9754	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 9755	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 9756	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 9757	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 9758	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 9759	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 9760	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 9761	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 9762	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 9763	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 9764	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 9765	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 9766	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 9767	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 9768	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 9769	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 9770	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 9771	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 1 = (0 + 1)
Theorem	dec10p 9772	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴
Theorem	numma 9773	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 9774	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 9775	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 9776	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 9777	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 9778	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 9779	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 9780	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 9781	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 9782	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 9783	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 9784	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 9785	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 9786	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 9787	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 9788	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 9789	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 9790	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 9791	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 9792	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 9793	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 9794	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 9795	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 9796	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 9797	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑁 · ;11) = ;𝑁𝑁
Theorem	decmul10add 9798	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 9799	Lemma for 6p5e11 9802 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 9800	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