P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sgnmulsgp 32801	If two real numbers are of same signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 < ((sgn‘𝐴) · (sgn‘𝐵))))
21.3.6.2  Integer powers - misc. additions
Theorem	nexple 32802	A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
Theorem	2exple2exp 32803*	If a nonnegative integer 𝑋 is a multiple of a power of two, but less than the next power of two, it is itself a power of two. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (2↑𝐾) ∥ 𝑋)    &   ⊢ (𝜑 → 𝑋 ≤ (2↑(𝐾 + 1)))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 𝑋 = (2↑𝑛))
Theorem	expevenpos 32804	Even powers are positive. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 2 ∥ 𝑁)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴↑𝑁))
Theorem	oexpled 32805	Odd power monomials are monotonic. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝑁)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
21.3.6.3  Indicator Functions
Syntax	cind 32806	Extend class notation with the indicator function generator.
class 𝟭
Definition	df-ind 32807*	Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indv 32808*	Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indval 32809*	Value of the indicator function generator for a set 𝐴 and a domain 𝑂. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
Theorem	indval2 32810	Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))
Theorem	indf 32811	An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1})
Theorem	indfval 32812	Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0))
Theorem	ind1 32813	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)
Theorem	ind0 32814	Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)
Theorem	ind1a 32815	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋 ∈ 𝐴))
Theorem	indpi1 32816	Preimage of the singleton {1} by the indicator function. See i1f1lem 25606. (Contributed by Thierry Arnoux, 21-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
Theorem	indsum 32817*	Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥 ∈ 𝐴 𝐵)
Theorem	indsumin 32818*	Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)
Theorem	prodindf 32819*	The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑂)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (((𝟭‘𝑂)‘𝐵)‘(𝐹‘𝑘)) = if(ran 𝐹 ⊆ 𝐵, 1, 0))
Theorem	indf1o 32820	The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂))
Theorem	indpreima 32821	A function with range {0, 1} as an indicator of the preimage of {1}. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐹:𝑂⟶{0, 1}) → 𝐹 = ((𝟭‘𝑂)‘(◡𝐹 “ {1})))
Theorem	indf1ofs 32822*	The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})
Theorem	indsupp 32823	The support of the indicator function. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (((𝟭‘𝑂)‘𝐴) supp 0) = 𝐴)
21.3.7  Decimal expansion Define a decimal expansion constructor. The decimal expansions built with this constructor are not meant to be used alone outside of this chapter. Rather, they are meant to be used exclusively as part of a decimal number with a decimal fraction, for example (3._1_4_1_59). That decimal point operator is defined in the next section. The bulk of these constructions have originally been proposed by David A. Wheeler on 12-May-2015, and discussed with Mario Carneiro in this thread: https://groups.google.com/g/metamath/c/2AW7T3d2YiQ.
Syntax	cdp2 32824	Constant used for decimal fraction constructor. See df-dp2 32825.
class _𝐴𝐵
Definition	df-dp2 32825	Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12610. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
Theorem	dp2eq1 32826	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)
Theorem	dp2eq2 32827	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)
Theorem	dp2eq1i 32828	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐶
Theorem	dp2eq2i 32829	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐶𝐴 = _𝐶𝐵
Theorem	dp2eq12i 32830	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐷
Theorem	dp20u 32831	Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ _𝐴0 = 𝐴
Theorem	dp20h 32832	Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ _0𝐴 = (𝐴 / ;10)
Theorem	dp2cl 32833	Closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
Theorem	dp2clq 32834	Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℚ    ⇒   ⊢ _𝐴𝐵 ∈ ℚ
Theorem	rpdp2cl 32835	Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ _𝐴𝐵 ∈ ℝ+
Theorem	rpdp2cl2 32836	Closure for a decimal fraction with no decimal expansion in the positive real numbers. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ _𝐴0 ∈ ℝ+
Theorem	dp2lt10 32837	Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐴 < ;10    &   ⊢ 𝐵 < ;10    ⇒   ⊢ _𝐴𝐵 < ;10
Theorem	dp2lt 32838	Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℝ+    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ _𝐴𝐵 < _𝐴𝐶
Theorem	dp2ltsuc 32839	Comparing a decimal fraction with the next integer. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐵 < ;10    &   ⊢ (𝐴 + 1) = 𝐶    ⇒   ⊢ _𝐴𝐵 < 𝐶
Theorem	dp2ltc 32840	Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐵 < ;10    &   ⊢ 𝐴 < 𝐶    ⇒   ⊢ _𝐴𝐵 < _𝐶𝐷
21.3.7.1  Decimal point Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 32842 and df-dp2 32825 for more information; dpval2 32846 and dpfrac1 32845 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12610.
Syntax	cdp 32841	Decimal point operator. See df-dp 32842.
class .
Definition	df-dp 32842*	Define the . (decimal point) operator. For example, (1.5) = (3 / 2), and -(;32._7_18) = -(;;;;32718 / ;;;1000) Unary minus, if applied, should normally be applied in front of the parentheses. Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that Metamath has no built-in way to parse and handle decimal numbers as traditionally written, e.g., "2.54". Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers. The RHS is ℝ, not ℚ; this should simplify some proofs. The LHS is ℕ0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression -(𝐴.𝐵) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)
⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
Theorem	dpval 32843	Define the value of the decimal point operator. See df-dp 32842. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
Theorem	dpcl 32844	Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 32842. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
Theorem	dpfrac1 32845	Prove a simple equivalence involving the decimal point. See df-dp 32842 and dpcl 32844. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))
Theorem	dpval2 32846	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
Theorem	dpval3 32847	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dpmul10 32848	Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
Theorem	decdiv10 32849	Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵)
Theorem	dpmul100 32850	Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
Theorem	dp3mul10 32851	Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶)
Theorem	dpmul1000 32852	Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ    ⇒   ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷
Theorem	dpval3rp 32853	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dp0u 32854	Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (𝐴.0) = 𝐴
Theorem	dp0h 32855	Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ (0.𝐴) = (𝐴 / ;10)
Theorem	rpdpcl 32856	Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) ∈ ℝ+
Theorem	dplt 32857	Comparing two decimal expansions (equal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℝ+    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐴.𝐵) < (𝐴.𝐶)
Theorem	dplti 32858	Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐵 < ;10    &   ⊢ (𝐴 + 1) = 𝐶    ⇒   ⊢ (𝐴.𝐵) < 𝐶
Theorem	dpgti 32859	Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ 𝐴 < (𝐴.𝐵)
Theorem	dpltc 32860	Comparing two decimal integers (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐴 < 𝐶    &   ⊢ 𝐵 < ;10    ⇒   ⊢ (𝐴.𝐵) < (𝐶.𝐷)
Theorem	dpexpp1 32861	Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ (𝑃 + 1) = 𝑄    &   ⊢ 𝑃 ∈ ℤ    &   ⊢ 𝑄 ∈ ℤ    ⇒   ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))
Theorem	0dp2dp 32862	Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵)
Theorem	dpadd2 32863	Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℝ+    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℝ+    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ (𝐺 + 𝐻) = 𝐼    &   ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)    ⇒   ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)
Theorem	dpadd 32864	Addition with one decimal. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹    ⇒   ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
Theorem	dpadd3 32865	Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ 𝐼 ∈ ℕ0    &   ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼    ⇒   ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)
Theorem	dpmul 32866	Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐽 ∈ ℕ0    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ (𝐴 · 𝐶) = 𝐹    &   ⊢ (𝐴 · 𝐷) = 𝑀    &   ⊢ (𝐵 · 𝐶) = 𝐿    &   ⊢ (𝐵 · 𝐷) = ;𝐸𝐾    &   ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽    &   ⊢ (𝐹 + 𝐺) = 𝐼    ⇒   ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)
Theorem	dpmul4 32867	An upper bound to multiplication of decimal numbers with 4 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐽 ∈ ℕ0    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    &   ⊢ 𝐼 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑂 ∈ ℕ0    &   ⊢ 𝑃 ∈ ℕ0    &   ⊢ 𝑄 ∈ ℕ0    &   ⊢ 𝑅 ∈ ℕ0    &   ⊢ 𝑆 ∈ ℕ0    &   ⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝑈 ∈ ℕ0    &   ⊢ 𝑊 ∈ ℕ0    &   ⊢ 𝑋 ∈ ℕ0    &   ⊢ 𝑌 ∈ ℕ0    &   ⊢ 𝑍 ∈ ℕ0    &   ⊢ 𝑈 < ;10    &   ⊢ 𝑃 < ;10    &   ⊢ 𝑄 < ;10    &   ⊢ (;;𝐿𝑀𝑁 + 𝑂) = ;;;𝑅𝑆𝑇𝑈    &   ⊢ ((𝐴.𝐵) · (𝐸.𝐹)) = (𝐼._𝐽𝐾)    &   ⊢ ((𝐶.𝐷) · (𝐺.𝐻)) = (𝑂._𝑃𝑄)    &   ⊢ (;;;𝐼𝐽𝐾1 + ;;𝑅𝑆𝑇) = ;;;𝑊𝑋𝑌𝑍    &   ⊢ (((𝐴.𝐵) + (𝐶.𝐷)) · ((𝐸.𝐹) + (𝐺.𝐻))) = (((𝐼._𝐽𝐾) + (𝐿._𝑀𝑁)) + (𝑂._𝑃𝑄))    ⇒   ⊢ ((𝐴._𝐵_𝐶𝐷) · (𝐸._𝐹_𝐺𝐻)) < (𝑊._𝑋_𝑌𝑍)
Theorem	threehalves 32868	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (3 / 2) = (1.5)
Theorem	1mhdrd 32869	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ ((0._99) + (0._01)) = 1
21.3.7.2  Division in the extended real number system
Syntax	cxdiv 32870	Extend class notation to include division of extended reals.
class /𝑒
Definition	df-xdiv 32871*	Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
Theorem	xdivval 32872*	Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
Theorem	xrecex 32873*	Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)
Theorem	xmulcand 32874	Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xreceu 32875*	Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)
Theorem	xdivcld 32876	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivcl 32877	Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)
Theorem	xdivmul 32878	Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /𝑒 𝐶) = 𝐵 ↔ (𝐶 ·e 𝐵) = 𝐴))
Theorem	rexdiv 32879	The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 / 𝐵))
Theorem	xdivrec 32880	Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 ·e (1 /𝑒 𝐵)))
Theorem	xdivid 32881	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 32882	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 32883	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 32884	Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))
Theorem	elxrge02 32885	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 32886	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 32887	Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)
Theorem	xrpxdivcld 32888	Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞))
21.3.8  Words over a set - misc additions
Theorem	wrdres 32889	Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆)
Theorem	wrdsplex 32890*	Existence of a split of a word at a given index. (Contributed by Thierry Arnoux, 11-Oct-2018.) (Proof shortened by AV, 3-Nov-2022.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))
Theorem	wrdfsupp 32891	A word has finite support. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → 𝑊 finSupp 𝑍)
Theorem	wrdpmcl 32892	Closure of a word with permuted symbols. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝐽 = (0..^(♯‘𝑊))    &   ⊢ (𝜑 → 𝐸:𝐽–1-1-onto→𝐽)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   ⊢ (𝜑 → (𝑊 ∘ 𝐸) ∈ Word 𝑆)
Theorem	pfx1s2 32893	The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)
Theorem	pfxrn2 32894	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14610. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 32895	Express the range of a prefix of a word. Stronger version of pfxrn2 32894. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 32896	Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆)    &   ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆)
Theorem	s1f1 32897	Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
Theorem	s2rnOLD 32898	Obsolete version of s2rn 14888 as of 1-Aug-2025. Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
Theorem	s2f1 32899	Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    ⇒   ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
Theorem	s3rnOLD 32900	Obsolete version of s2rn 14888 as of 1-Aug-2025. Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})