P. 487 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48601-48700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2ffzoeq 48601*	Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
Theorem	elfzo2nn 48602	A member of a half-open range of integers starting at 2 is a positive integer. (Contributed by AV, 5-Apr-2026.)
⊢ (𝐾 ∈ (2..^𝑁) → 𝐾 ∈ ℕ)
Theorem	nnmul2 48603	If one factor of a product of integers is at least 2 and less then the product, so is the second factor. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝐴 ∈ (2..^𝑁) ∧ 𝐵 ∈ ℕ ∧ (𝐴 · 𝐵) = 𝑁) → 𝐵 ∈ (2..^𝑁))
Theorem	nnmul2b 48604	A factor of a product of integers is at least 2 and less then the product iff the second factor is at least 2 and less then the product. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 · 𝐵) = 𝑁) → (𝐴 ∈ (2..^𝑁) ↔ 𝐵 ∈ (2..^𝑁)))
21.52.6.18  The floor and ceiling functions - extension
Theorem	2ltceilhalf 48605	The ceiling of half of an integer greater than 2 is greater than or equal to 2. (Contributed by AV, 4-Sep-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ≤ (⌈‘(𝑁 / 2)))
Theorem	ceilhalfgt1 48606	The ceiling of half of an integer greater than two is greater than one. (Contributed by AV, 2-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 1 < (⌈‘(𝑁 / 2)))
Theorem	ceilhalfelfzo1 48607	A positive integer less than (the ceiling of) half of another integer is in the half-open range of positive integers up to the other integer. (Contributed by AV, 7-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^𝑁)))
Theorem	gpgedgvtx1lem 48608	Lemma for gpgedgvtx1 49363. (Contributed by AV, 1-Sep-2025.) (Proof shortened by AV, 8-Sep-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ 𝐽) → 𝑋 ∈ 𝐼)
Theorem	2tceilhalfelfzo1 48609	Two times a positive integer less than (the ceiling of) half of another integer is less than the other integer. This theorem would hold even for integers less than 3, but then a corresponding 𝐾 would not exist. (Contributed by AV, 9-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (2 · 𝐾) < 𝑁)
Theorem	ceilbi 48610	A condition equivalent to ceiling. Analogous to flbi 13980. (Contributed by AV, 2-Nov-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌈‘𝐴) = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 < (𝐴 + 1))))
Theorem	ceilhalf1 48611	The ceiling of one half is one. (Contributed by AV, 2-Nov-2025.)
⊢ (⌈‘(1 / 2)) = 1
Theorem	rehalfge1 48612	Half of a real number greater than or equal to two is greater than or equal to one. (Contributed by AV, 2-Nov-2025.)
⊢ (𝑋 ∈ (2[,)+∞) → 1 ≤ (𝑋 / 2))
Theorem	ceilhalfnn 48613	The ceiling of half of a positive integer is a positive integer. (Contributed by AV, 2-Nov-2025.)
⊢ (𝑁 ∈ ℕ → (⌈‘(𝑁 / 2)) ∈ ℕ)
Theorem	1elfzo1ceilhalf1 48614	1 is in the half-open integer range from 1 to the ceiling of half of an integer greater than two is greater than one. (Contributed by AV, 2-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(𝑁 / 2))))
Theorem	nnge2recfl0 48615	The floor of the reciprocal of an integer greater than 1 is 0. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(1 / 𝑁)) = 0)
Theorem	flmrecm1 48616	The floor of an integer minus the reciprocal of a positive integer is the integer minus 1. (Contributed by AV, 10-Apr-2026.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (⌊‘(𝑀 − (1 / 𝑁))) = (𝑀 − 1))
21.52.6.19  The modulo (remainder) operation - extension
Theorem	fldivmod 48617	Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵))
Theorem	ceildivmod 48618	Expressing the ceiling of a division by the modulo operator. (Contributed by AV, 7-Sep-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌈‘(𝐴 / 𝐵)) = ((𝐴 + ((𝐵 − 𝐴) mod 𝐵)) / 𝐵))
Theorem	ceil5half3 48619	The ceiling of half of 5 is 3. (Contributed by AV, 7-Sep-2025.)
⊢ (⌈‘(5 / 2)) = 3
Theorem	submodaddmod 48620	Subtraction and addition modulo a positive integer. (Contributed by AV, 7-Sep-2025.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 + 𝐵) mod 𝑁) = ((𝐴 − 𝐶) mod 𝑁) ↔ ((𝐴 + (𝐵 + 𝐶)) mod 𝑁) = (𝐴 mod 𝑁)))
Theorem	difltmodne 48621	Two nonnegative integers are not equal modulo a positive modulus if their difference is greater than 0 and less than the modulus. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (1 ≤ (𝐴 − 𝐵) ∧ (𝐴 − 𝐵) < 𝑁)) → (𝐴 mod 𝑁) ≠ (𝐵 mod 𝑁))
Theorem	zplusmodne 48622	A nonnegative integer is not itself plus a positive integer modulo an integer greater than 1 and the positive integer. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐴 + 𝐾) mod 𝑁) ≠ (𝐴 mod 𝑁))
Theorem	addmodne 48623	The sum of a nonnegative integer and a positive integer modulo a number greater than both integers is not equal to the nonnegative integer. (Contributed by AV, 27-Aug-2025.) (Proof shortened by AV, 6-Sep-2025.)
⊢ ((𝑀 ∈ ℕ ∧ (𝐴 ∈ ℕ0 ∧ 𝐴 < 𝑀) ∧ (𝐵 ∈ ℕ ∧ 𝐵 < 𝑀)) → ((𝐴 + 𝐵) mod 𝑀) ≠ 𝐴)
Theorem	plusmod5ne 48624	A nonnegative integer is not itself plus a positive integer less than 5 modulo 5. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝐴 ∈ (0..^5) ∧ 𝐾 ∈ (1..^5)) → ((𝐴 + 𝐾) mod 5) ≠ 𝐴)
Theorem	zp1modne 48625	An integer is not itself plus 1 modulo an integer greater than 1. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ) → ((𝐴 + 1) mod 𝑁) ≠ (𝐴 mod 𝑁))
Theorem	p1modne 48626	A nonnegative integer is not itself plus 1 modulo an integer greater than 1 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 + 1) mod 𝑁) ≠ 𝐴)
Theorem	m1modne 48627	A nonnegative integer is not itself minus 1 modulo an integer greater than 1 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 − 1) mod 𝑁) ≠ 𝐴)
Theorem	minusmod5ne 48628	A nonnegative integer is not itself minus a positive integer less than 5 modulo 5. (Contributed by AV, 7-Sep-2025.)
⊢ ((𝐴 ∈ (0..^5) ∧ 𝐾 ∈ (1..^5)) → ((𝐴 − 𝐾) mod 5) ≠ 𝐴)
Theorem	submodlt 48629	The difference of an element of a half-open range of nonnegative integers and the upper bound of this range modulo an integer greater than the upper bound. (Contributed by AV, 1-Sep-2025.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (0..^𝐵) ∧ 𝐵 < 𝑁) → ((𝐴 − 𝐵) mod 𝑁) = ((𝑁 + 𝐴) − 𝐵))
Theorem	submodneaddmod 48630	An integer minus 𝐵 is not itself plus 𝐶 modulo an integer greater than the sum of 𝐵 and 𝐶. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (1 ≤ (𝐵 + 𝐶) ∧ (𝐵 + 𝐶) < 𝑁)) → ((𝐴 + 𝐵) mod 𝑁) ≠ ((𝐴 − 𝐶) mod 𝑁))
Theorem	m1modnep2mod 48631	A nonnegative integer minus 1 is not itself plus 2 modulo an integer greater than 3 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐴 ∈ ℤ) → ((𝐴 − 1) mod 𝑁) ≠ ((𝐴 + 2) mod 𝑁))
Theorem	minusmodnep2tmod 48632	A nonnegative integer minus a positive integer 1 or 2 is not itself plus 2 times the positive integer modulo 5. (Contributed by AV, 8-Sep-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (1..^3)) → ((𝐴 − 𝐵) mod 5) ≠ ((𝐴 + (2 · 𝐵)) mod 5))
Theorem	m1mod0mod1 48633	An integer decreased by 1 is 0 modulo a positive integer iff the integer is 1 modulo the same modulus. (Contributed by AV, 6-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 < 𝑁) → (((𝐴 − 1) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = 1))
Theorem	elmod2 48634	An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ {0, 1})
Theorem	mod0mul 48635*	If an integer is 0 modulo a positive integer, this integer must be a multiple of the modulus. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁)))
Theorem	modn0mul 48636*	If an integer is not 0 modulo a positive integer, this integer must be the sum of a multiple of the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) ≠ 0 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦)))
Theorem	m1modmmod 48637	An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) mod 𝑁) − (𝐴 mod 𝑁)) = if((𝐴 mod 𝑁) = 0, (𝑁 − 1), -1))
Theorem	difmodm1lt 48638	The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd 𝐴 and 𝑁 = 2, since ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) would be (1 − 0) = 1 which is not less than (𝑁 − 1) = 1. (Contributed by AV, 6-Jun-2012.) (Proof shortened by SN, 27-Nov-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) < (𝑁 − 1))
Theorem	8mod5e3 48639	8 modulo 5 is 3. (Contributed by AV, 20-Nov-2025.)
⊢ (8 mod 5) = 3
Theorem	modmkpkne 48640	If an integer minus a constant equals another integer plus the constant modulo 𝑁, then the first integer plus the constant equals the second integer minus the constant modulo 𝑁 iff the fourfold of the constant is a multiple of 𝑁. (Contributed by AV, 15-Nov-2025.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))
Theorem	modmknepk 48641	A nonnegative integer less than the modulus plus/minus a positive integer less than (the ceiling of) half of the modulus are not equal modulo the modulus. For this theorem, it is essential that 𝐾 < (𝑁 / 2)! (Contributed by AV, 3-Sep-2025.) (Revised by AV, 15-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑌 ∈ 𝐼 ∧ 𝐾 ∈ 𝐽) → ((𝑌 − 𝐾) mod 𝑁) ≠ ((𝑌 + 𝐾) mod 𝑁))
Theorem	modlt0b 48642	An integer with an absolute value less than a positive integer is 0 modulo the positive integer iff it is 0. (Contributed by AV, 21-Nov-2025.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℤ ∧ (abs‘𝑋) < 𝑁) → ((𝑋 mod 𝑁) = 0 ↔ 𝑋 = 0))
Theorem	mod2addne 48643	The sums of a nonnegative integer less than the modulus and two integers whose difference is less than the modulus are not equal modulo the modulus. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ 𝐼 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (abs‘(𝐴 − 𝐵)) ∈ (1..^𝑁)) → ((𝑋 + 𝐴) mod 𝑁) ≠ ((𝑋 + 𝐵) mod 𝑁))
Theorem	modm1nep1 48644	A nonnegative integer less than a modulus greater than 2 plus/minus one are not equal modulo the modulus. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
Theorem	modm2nep1 48645	A nonnegative integer less than a modulus greater than 4 plus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 2) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
Theorem	modp2nep1 48646	A nonnegative integer less than a modulus greater than 4 plus one/plus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 + 2) mod 𝑁) ≠ ((𝑌 + 1) mod 𝑁))
Theorem	modm1nep2 48647	A nonnegative integer less than a modulus greater than 4 plus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 + 2) mod 𝑁))
Theorem	modm1nem2 48648	A nonnegative integer less than a modulus greater than 4 minus one/minus two are not equal modulo the modulus. (Contributed by AV, 22-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑌 ∈ 𝐼) → ((𝑌 − 1) mod 𝑁) ≠ ((𝑌 − 2) mod 𝑁))
Theorem	modm1p1ne 48649	If an integer minus one equals another integer plus one modulo an integer greater than 4, then the first integer plus one is not equal to the second integer minus one modulo the same modulus. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))
21.52.6.20  The infinite sequence builder "seq"
Theorem	smonoord 48650*	Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 14199 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 46516? (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))
21.52.6.21  Integer powers - extension
Theorem	2timesltsq 48651	Two times an integer greater than 2 is less than the square of the integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝐴 ∈ (ℤ≥‘3) → (2 · 𝐴) < (𝐴↑2))
Theorem	2timesltsqm1 48652	Two times an integer greater than 2 is less than the square of the integer minus 1. (Contributed by AV, 7-Apr-2026.)
⊢ (𝐴 ∈ (ℤ≥‘3) → (2 · 𝐴) < ((𝐴↑2) − 1))
21.52.6.22  Finite and infinite sums - extension
Theorem	fsummsndifre 48653*	A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ)
Theorem	fsumsplitsndif 48654*	Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + ⦋𝑋 / 𝑘⦌𝐵))
Theorem	fsummmodsndifre 48655*	A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ)
Theorem	fsummmodsnunz 48656*	A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ)
21.52.6.23  The divides relation - extension
Theorem	nndivides2 48657*	Definition of the divides relation for divisors greater than 1. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝑀 ∈ (2..^𝑁) ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ (2..^𝑁)(𝑛 · 𝑀) = 𝑁))
Theorem	facnn0dvdsfac 48658	The factorial of a nonnegative integer divides the factorial of an integer which is greater than or equal to the first integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝑀 ∈ (0...𝑁) → (!‘𝑀) ∥ (!‘𝑁))
Theorem	muldvdsfacgt 48659	The product of two different positive integers divides the factorial of the bigger integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝐴 ∈ (1..^𝐵) → (𝐴 · 𝐵) ∥ (!‘𝐵))
Theorem	muldvdsfacm1 48660	The product of two different positive integers less than a third integer divides the factorial of the third integer decreased by 1. By assumption, the third integer must be greater than 3. (Contributed by AV, 6-Apr-2026.)
⊢ ((𝐴 ∈ (1..^𝐵) ∧ 𝐵 ∈ (1..^𝑁)) → (𝐴 · 𝐵) ∥ (!‘(𝑁 − 1)))
21.52.6.24  Extensible structures - extension
Theorem	setsidel 48661	The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	setsnidel 48662	The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)
Theorem	setsv 48663	The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
21.52.7  Preimages of function values According to Wikipedia ("Image (mathematics)", 17-Mar-2024, https://en.wikipedia.org/wiki/ImageSupport_(mathematics)): "... evaluating a given function 𝑓 at each element of a given subset 𝐴 of its domain produces a set, called the "image of 𝐴 under (or through) 𝑓". Similarly, the inverse image (or preimage) of a given subset 𝐵 of the codomain of 𝑓 is the set of all elements of the domain that map to the members of 𝐵." The preimage of a set 𝐵 under a function 𝑓 is often denoted as "f^-1 (B)", but in set.mm, the idiom (◡𝑓 “ 𝐵) is used. As a special case, the idiom for the preimage of a function value at 𝑋 under a function 𝐹 is (◡𝐹 “ {(𝐹‘𝑋)}) (according to Wikipedia, the preimage of a singleton is also called a "fiber"). We use the label fragment "preima" (as in mptpreima 6251) for theorems about preimages (sometimes, also "imacnv" is used as in fvimacnvi 7071), and "preimafv" (as in preimafvn0 48665) for theorems about preimages of a function value. In this section, 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} will be the set of all preimages of function values of a function 𝐹, that means 𝑆 ∈ 𝑃 is a preimage of a function value (see, for example, elsetpreimafv 48670): 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}). With the help of such a set, it is shown that every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function (see fundcmpsurinj 48694) by constructing a surjective function 𝑔:𝐴⟶onto𝑃 and an injective function ℎ:𝑃⟶1-1𝐵 so that 𝐹 = (ℎ ∘ 𝑔) ( see fundcmpsurinjpreimafv 48693). See also Wikipedia ("Surjective function", 17-Mar-2024, https://en.wikipedia.org/wiki/Surjective_function 48693 (section "Composition and decomposition"). This is different from the decomposition of 𝐹 into the surjective function 𝑔:𝐴⟶onto(𝐹 “ 𝐴) (with (𝑔‘𝑥) = (𝐹‘𝑥) for 𝑥 ∈ 𝐴) and the injective function ℎ = ( I ↾ (𝐹 “ 𝐴)), ( see fundcmpsurinjimaid 48696), see also Wikipedia ("Bijection, injection and surjection", 17-Mar-2024, https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection 48696 (section "Properties"). Finally, it is shown that every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective, a bijective and an injective function (see fundcmpsurbijinj 48695), by showing that there is a bijection between the set of all preimages of values of a function and the range of the function (see imasetpreimafvbij 48691). From this, both variants of decompositions of a function into a surjective and an injective function can be derived: Let 𝐹 = ((𝐼 ∘ 𝐵) ∘ 𝑆) be a decomposition of a function into a surjective, a bijective and an injective function, then 𝐹 = (𝐽 ∘ 𝑆) with 𝐽 = (𝐼 ∘ 𝐵) (an injective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinj 48694, and 𝐹 = (𝐼 ∘ 𝑂) with 𝑂 = (𝐵 ∘ 𝑆) (a surjective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinjimaid 48696.
Theorem	preimafvsnel 48664	The preimage of a function value at 𝑋 contains 𝑋. (Contributed by AV, 7-Mar-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
Theorem	preimafvn0 48665	The preimage of a function value is not empty. (Contributed by AV, 7-Mar-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ≠ ∅)
Theorem	uniimafveqt 48666*	The union of the image of a subset 𝑆 of the domain of a function with elements having the same function value is the function value at one of the elements of 𝑆. (Contributed by AV, 5-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))
Theorem	uniimaprimaeqfv 48667	The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))
Theorem	setpreimafvex 48668*	The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)
Theorem	elsetpreimafvb 48669*	The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
Theorem	elsetpreimafv 48670*	An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
Theorem	elsetpreimafvssdm 48671*	An element of the class 𝑃 of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
Theorem	fvelsetpreimafv 48672*	There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
Theorem	preimafvelsetpreimafv 48673*	The preimage of a function value is an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)
Theorem	preimafvsspwdm 48674*	The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)
Theorem	0nelsetpreimafv 48675*	The empty set is not an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 6-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
Theorem	elsetpreimafvbi 48676*	An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
Theorem	elsetpreimafveqfv 48677*	The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐹‘𝑋) = (𝐹‘𝑌))
Theorem	eqfvelsetpreimafv 48678*	If an element of the domain of the function has the same function value as an element of the preimage of a function value, then it is an element of the same preimage. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)) → 𝑌 ∈ 𝑆))
Theorem	elsetpreimafvrab 48679*	An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})
Theorem	imaelsetpreimafv 48680*	The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
Theorem	uniimaelsetpreimafv 48681*	The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
Theorem	elsetpreimafveq 48682*	If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑆 = 𝑅))
Theorem	fundcmpsurinjlem1 48683*	Lemma 1 for fundcmpsurinj 48694. (Contributed by AV, 4-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ⊢ ran 𝐺 = 𝑃
Theorem	fundcmpsurinjlem2 48684*	Lemma 2 for fundcmpsurinj 48694. (Contributed by AV, 4-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴⟶onto𝑃)
Theorem	fundcmpsurinjlem3 48685*	Lemma 3 for fundcmpsurinj 48694. (Contributed by AV, 3-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))
Theorem	imasetpreimafvbijlemf 48686*	Lemma for imasetpreimafvbij 48691: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
Theorem	imasetpreimafvbijlemfv 48687*	Lemma for imasetpreimafvbij 48691: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))
Theorem	imasetpreimafvbijlemfv1 48688*	Lemma for imasetpreimafvbij 48691: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))
Theorem	imasetpreimafvbijlemf1 48689*	Lemma for imasetpreimafvbij 48691: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶1-1(𝐹 “ 𝐴))
Theorem	imasetpreimafvbijlemfo 48690*	Lemma for imasetpreimafvbij 48691: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃⟶onto(𝐹 “ 𝐴))
Theorem	imasetpreimafvbij 48691*	The mapping 𝐻 is a bijective function between the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃⟶1-1onto(𝐹 “ 𝐴))
Theorem	fundcmpsurbijinjpreimafv 48692*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴⟶onto𝑃 ∧ ℎ:𝑃⟶1-1onto(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)⟶1-1𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjpreimafv 48693*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴⟶onto𝑃 ∧ ℎ:𝑃⟶1-1𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurinj 48694*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴⟶onto𝑝 ∧ ℎ:𝑝⟶1-1𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurbijinj 48695*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴⟶onto𝑝 ∧ ℎ:𝑝⟶1-1onto𝑞 ∧ 𝑖:𝑞⟶1-1𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjimaid 48696*	Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto the image (𝐹 “ 𝐴) of the domain of 𝐹 and an injective function from the image (𝐹 “ 𝐴). (Contributed by AV, 17-Mar-2024.)
⊢ 𝐼 = (𝐹 “ 𝐴)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))    &   ⊢ 𝐻 = ( I ↾ 𝐼)    ⇒   ⊢ (𝐹:𝐴⟶𝐵 → (𝐺:𝐴⟶onto𝐼 ∧ 𝐻:𝐼⟶1-1𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))
Theorem	fundcmpsurinjALT 48697*	Alternate proof of fundcmpsurinj 48694, based on fundcmpsurinjimaid 48696: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴⟶onto𝑝 ∧ ℎ:𝑝⟶1-1𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
21.52.8  Partitions of real intervals Based on the theorems of the fourierdlem* series of GS's mathbox.
Syntax	ciccp 48698	Extend class notation with the partitions of a closed interval of extended reals.
class RePart
Definition	df-iccp 48699*	Define partitions of a closed interval of extended reals. Such partitions are finite increasing sequences of extended reals. (Contributed by AV, 8-Jul-2020.)
⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
Theorem	iccpval 48700*	Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)
⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})