P. 128 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nneoi 12701	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 12702	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zeo2 12703	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 12704*	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	peano5uzi 12705*	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	peano5uzti 12706*	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	dfuzi 12707*	An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 12279 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 12708*	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 12709*	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 12710*	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 12711*	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	nn0indALT 12712*	Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either nn0ind 12711 or nn0indALT 12712 may be used; see comment for nnind 12284. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑦 ∈ ℕ0 → (𝜒 → 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝐴 ∈ ℕ0 → 𝜏)
Theorem	nn0indd 12713*	Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ (𝑥 = 0 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → 𝜂)
Theorem	fzind 12714*	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 12715*	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 12716*	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 12717*	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	fzindd 12718*	Induction on the integers from M to N inclusive, a deduction version. (Contributed by metakunt, 12-May-2024.)
⊢ (𝑥 = 𝑀 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) ∧ 𝜃) → 𝜏)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → 𝜂)
Theorem	btwnz 12719*	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	zred 12720	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zcnd 12721	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	znegcld 12722	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℤ)
Theorem	peano2zd 12723	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)
Theorem	zaddcld 12724	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
Theorem	zsubcld 12725	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)
Theorem	zmulcld 12726	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
Theorem	znnn0nn 12727	The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ)
Theorem	zadd2cl 12728	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)
Theorem	zriotaneg 12729*	The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.)
⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓))
Theorem	suprfinzcl 12730	The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴)
5.4.10  Decimal arithmetic
Syntax	cdc 12731	Constant used for decimal constructor.
class ;𝐴𝐵
Definition	df-dec 12732	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30382. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
Theorem	9p1e10 12733	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 12734	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	decex 12735	A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;𝐴𝐵 ∈ V
Theorem	deceq1 12736	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 12737	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 12738	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐶
Theorem	deceq2i 12739	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐶𝐴 = ;𝐶𝐵
Theorem	deceq12i 12740	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐷
Theorem	numnncl 12741	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 12742	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 12743	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)
Theorem	numcl 12744	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 12745	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 12746	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 12747	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ
Theorem	10pos 12748	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10
Theorem	10nn0 12749	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ0
Theorem	10re 12750	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
Theorem	decnncl 12751	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 12752	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 12753	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 12754	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 12755	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 12756	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 12757	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 12758	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 12759	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 12760	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 12761	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 12762	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 12763	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 12764	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 12765	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 12766	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 12767	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 12768	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 12769	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 12770	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 12771	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 12772	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 12773	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 1 = (0 + 1)
Theorem	dec10p 12774	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴
Theorem	numma 12775	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 12776	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 12777	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 12778	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 12779	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 12780	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 12781	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 12782	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 12783	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 12784	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 12785	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 12786	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 12787	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 12788	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 12789	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 12790	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 12791	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 12792	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 12793	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 12794	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 12795	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
Theorem	decmul1c 12796	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 12797	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 12798	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 12799	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑁 · ;11) = ;𝑁𝑁
Theorem	decmul10add 12800	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)