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	zeo2 12701	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 12702*	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 12703*	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 12704*	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 12705*	An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 12264 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 12706*	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 12707*	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 12708*	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 12709*	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 12710*	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 12709 or nn0indALT 12710 may be used; see comment for nnind 12269. (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 12711*	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 12712*	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 12713*	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 12714*	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 12715*	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 12716*	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 12717*	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 12718	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zcnd 12719	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	znegcld 12720	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℤ)
Theorem	peano2zd 12721	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)
Theorem	zaddcld 12722	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
Theorem	zsubcld 12723	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)
Theorem	zmulcld 12724	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
Theorem	znnn0nn 12725	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 12726	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 12727*	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 12728	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 12729	Constant used for decimal constructor.
class ;𝐴𝐵
Definition	df-dec 12730	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30827. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
Theorem	9p1e10 12731	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 12732	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 12733	A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;𝐴𝐵 ∈ V
Theorem	deceq1 12734	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 12735	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 12736	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐶
Theorem	deceq2i 12737	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐶𝐴 = ;𝐶𝐵
Theorem	deceq12i 12738	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐷
Theorem	numnncl 12739	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 12740	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 12741	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)
Theorem	numcl 12742	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 12743	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 12744	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    ⇒   ⊢ ;𝐴𝐵 ∈ ℕ0
Theorem	11nn0 12745	11 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;11 ∈ ℕ0
Theorem	12nn0 12746	12 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;12 ∈ ℕ0
Theorem	16nn0 12747	16 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;16 ∈ ℕ0
Theorem	25nn0 12748	25 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;25 ∈ ℕ0
Theorem	10nn 12749	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ
Theorem	10pos 12750	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10
Theorem	10nn0 12751	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ0
Theorem	10re 12752	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 12753	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ ;𝐴𝐵 ∈ ℕ
Theorem	11nn 12754	11 is a positive integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;11 ∈ ℕ
Theorem	dec0u 12755	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 12756	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 12757	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 12758	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 12759	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 12760	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 12761	A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less than 10. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐴 ≤ 9    ⇒   ⊢ 𝐴 < ;10
Theorem	declt 12762	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 12763	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 12764	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 12765	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 12766	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 12767	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 12768	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 12769	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 12770	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 12771	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 12772	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 12773	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 12774	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 12775	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 12776	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 1 = (0 + 1)
Theorem	dec10p 12777	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴
Theorem	numma 12778	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 12779	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 12780	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 12781	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 12782	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 12783	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 12784	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 12785	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 12786	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 12787	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 12788	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 12789	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 12790	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 12791	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 12792	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 12793	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 12794	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 12795	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 12796	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 12797	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 12798	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 12799	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 12800	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