P. 333 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnindf 33201*	Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nn0min 33202*	Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 12709. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ (𝑛 = 0 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑛 = 𝑚 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑛 = (𝑚 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 (¬ 𝜃 ∧ 𝜏))
Theorem	subne0nn 33203	A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
⊢ (𝜑 → 𝑀 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℂ)    &   ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ≠ 𝑁)    ⇒   ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℕ)
Theorem	ltesubnnd 33204	Subtracting an integer number from another number decreases it. See ltsubrpd 13110. (Contributed by Thierry Arnoux, 18-Apr-2017.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)
Theorem	fprodeq02 33205*	If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	fprodex01 33206*	A product of factors equal to zero or one is zero exactly when one of the factors is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝑘 = 𝑙 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {0, 1})    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = if(∀𝑙 ∈ 𝐴 𝐶 = 1, 1, 0))
Theorem	prodpr 33207*	A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)    &   ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹))
Theorem	prodtp 33208*	A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.)
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)    &   ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺))
Theorem	fsumub 33209*	An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (𝑘 = 𝐾 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐷 ≤ 𝐶)
Theorem	fsumiunle 33210*	Upper bound for a sum of nonnegative terms over an indexed union. The inequality may be strict if the indexed union is non-disjoint, since in the right hand side, a summand may be counted several times. (Contributed by Thierry Arnoux, 1-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)
21.3.5.11  Decimal numbers
Theorem	dfdec100 33211	Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶)
21.3.6  Real and complex functions
21.3.6.1  Signum (sgn or sign) function - misc. additions
Theorem	sgnsgn 33212	Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴))
Theorem	sgnmulsgp 33213	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 33214	A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
Theorem	2exple2exp 33215*	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 33216	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 33217	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 (continued)
Theorem	indsumin 33218*	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 33219*	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	indsn 33220*	The indicator function of a singleton. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝑋 ∈ 𝑂) → ((𝟭‘𝑂)‘{𝑋}) = (𝑥 ∈ 𝑂 ↦ if(𝑥 = 𝑋, 1, 0)))
Theorem	indf1o 33221	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 33222	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 33223*	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 33224	The support of the indicator function. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (((𝟭‘𝑂)‘𝐴) supp 0) = 𝐴)
Theorem	indfsd 33225	The indicator function of a finite set has finite support. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ((𝟭‘𝑂)‘𝐴) finSupp 0)
Theorem	indfsid 33226	Conditions for a function to be an indicator function. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑂⟶{0, 1})    ⇒   ⊢ (𝜑 → 𝐹 = ((𝟭‘𝑂)‘(𝐹 supp 0)))
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 33227	Constant used for decimal fraction constructor. See df-dp2 33228.
class _𝐴𝐵
Definition	df-dp2 33228	Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 12730. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
Theorem	dp2eq1 33229	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)
Theorem	dp2eq2 33230	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)
Theorem	dp2eq1i 33231	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐶
Theorem	dp2eq2i 33232	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ _𝐶𝐴 = _𝐶𝐵
Theorem	dp2eq12i 33233	Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ _𝐴𝐶 = _𝐵𝐷
Theorem	dp20u 33234	Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ _𝐴0 = 𝐴
Theorem	dp20h 33235	Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ _0𝐴 = (𝐴 / ;10)
Theorem	dp2cl 33236	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 33237	Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℚ    ⇒   ⊢ _𝐴𝐵 ∈ ℚ
Theorem	rpdp2cl 33238	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 33239	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 33240	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 33241	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 33242	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 33243	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 33245 and df-dp2 33228 for more information; dpval2 33249 and dpfrac1 33248 provide a more convenient way to obtain a value. This is intentionally similar to df-dec 12730.
Syntax	cdp 33244	Decimal point operator. See df-dp 33245.
class .
Definition	df-dp 33245*	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 33246	Define the value of the decimal point operator. See df-dp 33245. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
Theorem	dpcl 33247	Prove that the closure of the decimal point is ℝ as we have defined it. See df-dp 33245. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
Theorem	dpfrac1 33248	Prove a simple equivalence involving the decimal point. See df-dp 33245 and dpcl 33247. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))
Theorem	dpval2 33249	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
Theorem	dpval3 33250	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dpmul10 33251	Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
Theorem	decdiv10 33252	Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵)
Theorem	dpmul100 33253	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 33254	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 33255	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 33256	Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℝ+    ⇒   ⊢ (𝐴.𝐵) = _𝐴𝐵
Theorem	dp0u 33257	Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (𝐴.0) = 𝐴
Theorem	dp0h 33258	Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.)
⊢ 𝐴 ∈ ℝ+    ⇒   ⊢ (0.𝐴) = (𝐴 / ;10)
Theorem	rpdpcl 33259	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 33260	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 33261	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 33262	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 33263	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 33264	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 33265	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 33266	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 33267	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 33268	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 33269	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 33270	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 33271	Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.)
⊢ (3 / 2) = (1.5)
Theorem	1mhdrd 33272	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 33273	Extend class notation to include division of extended reals.
class /𝑒
Definition	df-xdiv 33274*	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 33275*	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 33276*	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 33277	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 33278*	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 33279	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 33280	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 33281	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 33282	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 33283	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 33284	A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)
Theorem	xdiv0 33285	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)
Theorem	xdiv0rp 33286	Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)
Theorem	eliccioo 33287	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 33288	Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞))
Theorem	xdivpnfrp 33289	Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)
Theorem	rpxdivcld 33290	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 33291	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 33292	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 33293*	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 33294	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 33295	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 33296	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 33297	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14747. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 33298	Express the range of a prefix of a word. Stronger version of pfxrn2 33297. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 33299	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	s2f1 33300	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)