P. 486 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48501-48600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ceilbi 48501	A condition equivalent to ceiling. Analogous to flbi 13960. (Contributed by AV, 2-Nov-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌈‘𝐴) = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 < (𝐴 + 1))))
Theorem	ceilhalf1 48502	The ceiling of one half is one. (Contributed by AV, 2-Nov-2025.)
⊢ (⌈‘(1 / 2)) = 1
Theorem	rehalfge1 48503	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 48504	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 48505	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 48506	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 48507	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 48508	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 48509	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 48510	The ceiling of half of 5 is 3. (Contributed by AV, 7-Sep-2025.)
⊢ (⌈‘(5 / 2)) = 3
Theorem	submodaddmod 48511	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 48512	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 48513	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 48514	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 48515	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 48516	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 48517	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 48518	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 48519	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 48520	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 48521	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 48522	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 48523	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 48524	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 48525	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 48526*	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 48527*	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 48528	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 48529	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 48530	8 modulo 5 is 3. (Contributed by AV, 20-Nov-2025.)
⊢ (8 mod 5) = 3
Theorem	modmkpkne 48531	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 48532	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 48533	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 48534	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 48535	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 48536	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 48537	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 48538	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 48539	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 48540	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 48541*	Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 14179 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 46407? (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 48542	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 48543	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 48544*	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 48545*	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 48546*	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 48547*	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 48548*	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 48549	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 48550	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 48551	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 48552	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 48553	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 48554	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 6239) for theorems about preimages (sometimes, also "imacnv" is used as in fvimacnvi 7052), and "preimafv" (as in preimafvn0 48556) 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 48561): 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 48585) by constructing a surjective function 𝑔:𝐴–onto→𝑃 and an injective function ℎ:𝑃–1-1→𝐵 so that 𝐹 = (ℎ ∘ 𝑔) ( see fundcmpsurinjpreimafv 48584). See also Wikipedia ("Surjective function", 17-Mar-2024, https://en.wikipedia.org/wiki/Surjective_function 48584 (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 48587), see also Wikipedia ("Bijection, injection and surjection", 17-Mar-2024, https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection 48587 (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 48586), 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 48582). 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 48585, and 𝐹 = (𝐼 ∘ 𝑂) with 𝑂 = (𝐵 ∘ 𝑆) (a surjective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinjimaid 48587.
Theorem	preimafvsnel 48555	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 48556	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 48557*	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 48558	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 48559*	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 48560*	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 48561*	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 48562*	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 48563*	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 48564*	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 48565*	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 48566*	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 48567*	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 48568*	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 48569*	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 48570*	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 48571*	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 48572*	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 48573*	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 48574*	Lemma 1 for fundcmpsurinj 48585. (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 48575*	Lemma 2 for fundcmpsurinj 48585. (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–onto→𝑃)
Theorem	fundcmpsurinjlem3 48576*	Lemma 3 for fundcmpsurinj 48585. (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 48577*	Lemma for imasetpreimafvbij 48582: 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 48578*	Lemma for imasetpreimafvbij 48582: 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 48579*	Lemma for imasetpreimafvbij 48582: 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 48580*	Lemma for imasetpreimafvbij 48582: 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 48581*	Lemma for imasetpreimafvbij 48582: 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 48582*	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–1-1-onto→(𝐹 “ 𝐴))
Theorem	fundcmpsurbijinjpreimafv 48583*	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–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjpreimafv 48584*	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–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurinj 48585*	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–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
Theorem	fundcmpsurbijinj 48586*	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–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
Theorem	fundcmpsurinjimaid 48587*	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–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))
Theorem	fundcmpsurinjALT 48588*	Alternate proof of fundcmpsurinj 48585, based on fundcmpsurinjimaid 48587: 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–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
21.52.8  Partitions of real intervals Based on the theorems of the fourierdlem* series of GS's mathbox.
Syntax	ciccp 48589	Extend class notation with the partitions of a closed interval of extended reals.
class RePart
Definition	df-iccp 48590*	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 48591*	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))})
Theorem	iccpart 48592*	A special partition. Corresponds to fourierdlem2 47213 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
Theorem	iccpartimp 48593	Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
Theorem	iccpartres 48594	The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
Theorem	iccpartxr 48595	If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)
Theorem	iccpartgtprec 48596	If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼))
Theorem	iccpartipre 48597	If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ)
Theorem	iccpartiltu 48598*	If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
Theorem	iccpartigtl 48599*	If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
Theorem	iccpartlt 48600	If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 47222 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))    ⇒   ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))