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Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.41.6.19  The modulo (remainder) operation - extension

Theoremm1mod0mod1 43901 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))

Theoremelmod2 43902 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})

20.41.6.20  The infinite sequence builder "seq"

Theoremsmonoord 43903* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 13398 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 41944? (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) < (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) < (𝐹𝑁))

20.41.6.21  Finite and infinite sums - extension

Theoremfsummsndifre 43904* 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 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ)

Theoremfsumsplitsndif 43905* 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 ∧ 𝑋𝐴 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + 𝑋 / 𝑘𝐵))

Theoremfsummmodsndifre 43906* 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 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ)

Theoremfsummmodsnunz 43907* 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)

20.41.6.22  Extensible structures - extension

Theoremsetsidel 43908 The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
(𝜑𝑆𝑉)    &   (𝜑𝐵𝑊)    &   𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Theoremsetsnidel 43909 The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
(𝜑𝑆𝑉)    &   (𝜑𝐵𝑊)    &   𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)    &   (𝜑𝐴𝐶)       (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Theoremsetsv 43910 The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)

20.41.7  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 (𝑓𝐵) is used. As a special case, the idiom for the preimage of a function value at 𝑋 under a function 𝐹 is (𝐹 “ {(𝐹𝑋)}) (according to Wikipedia, the preimage of a singleton is also called a "fiber").

We use the label fragment "preima" (as in mptpreima 6059) for theorems about preimages (sometimes, also "imacnv" is used as in fvimacnvi 6799), and "preimafv" (as in preimafvn0 43912) for theorems about preimages of a function value.

In this section, 𝑃 = {𝑧 ∣ ∃𝑥𝐴𝑧 = (𝐹 “ {(𝐹𝑥)})} will be the set of all preimages of function values of a function 𝐹, that means 𝑆𝑃 is a preimage of a function value (see, for example, elsetpreimafv 43917): 𝑆 = (𝐹 “ {(𝐹𝑥)}).

With the help of such a set, it is shown that every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function (see fundcmpsurinj 43941) by constructing a surjective function 𝑔:𝐴onto𝑃 and an injective function :𝑃1-1𝐵 so that 𝐹 = (𝑔) ( see fundcmpsurinjpreimafv 43940). See also Wikipedia ("Surjective function", 17-Mar-2024, https://en.wikipedia.org/wiki/Surjective_function 43940 (section "Composition and decomposition"). This is different from the decomposition of 𝐹 into the surjective function 𝑔:𝐴onto→(𝐹𝐴) (with (𝑔𝑥) = (𝐹𝑥) for 𝑥𝐴) and the injective function = ( I ↾ (𝐹𝐴)), ( see fundcmpsurinjimaid 43943), see also Wikipedia ("Bijection, injection and surjection", 17-Mar-2024, https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection 43943 (section "Properties").

Finally, it is shown that every function 𝐹:𝐴𝐵 can be decomposed into a surjective, a bijective and an injective function (see fundcmpsurbijinj 43942), 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 43938). From this, both variants of decompositions of a function into a surjective and an injective function can be derived:

Let 𝐹 = ((𝐼𝐵) ∘ 𝑆) be a decomposition of a function into a surjective, a bijective and an injective function, then 𝐹 = (𝐽𝑆) with 𝐽 = (𝐼𝐵) (an injective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinj 43941, and 𝐹 = (𝐼𝑂) with 𝑂 = (𝐵𝑆) (a surjective function) is a decomposition into a surjective and an injective function corresponding to fundcmpsurinjimaid 43943.

Theorempreimafvsnel 43911 The preimage of a function value at 𝑋 contains 𝑋. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → 𝑋 ∈ (𝐹 “ {(𝐹𝑋)}))

Theorempreimafvn0 43912 The preimage of a function value is not empty. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ≠ ∅)

Theoremuniimafveqt 43913* 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:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))

Theoremuniimaprimaeqfv 43914 The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑋)})) = (𝐹𝑋))

Theoremsetpreimafvex 43915* The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐴𝑉𝑃 ∈ V)

Theoremelsetpreimafvb 43916* The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝑆𝑉 → (𝑆𝑃 ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))

Theoremelsetpreimafv 43917* An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))

Theoremelsetpreimafvssdm 43918* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)

Theoremfvelsetpreimafv 43919* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑥𝑆 𝑆 = (𝐹 “ {(𝐹𝑥)}))

Theorempreimafvelsetpreimafv 43920* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ 𝑃)

Theorempreimafvsspwdm 43921* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐹 Fn 𝐴𝑃 ⊆ 𝒫 𝐴)

Theorem0nelsetpreimafv 43922* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)

Theoremelsetpreimafvbi 43923* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))

Theoremelsetpreimafveqfv 43924* The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))

Theoremeqfvelsetpreimafv 43925* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → 𝑌𝑆))

Theoremelsetpreimafvrab 43926* An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})

Theoremimaelsetpreimafv 43927* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝐹𝑆) = {(𝐹𝑋)})

Theoremuniimaelsetpreimafv 43928* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)

Theoremelsetpreimafveq 43929* 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑆 = 𝑅))

Theoremfundcmpsurinjlem1 43930* Lemma 1 for fundcmpsurinj 43941. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))       ran 𝐺 = 𝑃

Theoremfundcmpsurinjlem2 43931* Lemma 2 for fundcmpsurinj 43941. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)

Theoremfundcmpsurinjlem3 43932* Lemma 3 for fundcmpsurinj 43941. (Contributed by AV, 3-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((Fun 𝐹𝑋𝑃) → (𝐻𝑋) = (𝐹𝑋))

Theoremimasetpreimafvbijlemf 43933* Lemma for imasetpreimafvbij 43938: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))

Theoremimasetpreimafvbijlemfv 43934* Lemma for imasetpreimafvbij 43938: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))

Theoremimasetpreimafvbijlemfv1 43935* Lemma for imasetpreimafvbij 43938: 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝑋𝑃) → ∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦))

Theoremimasetpreimafvbijlemf1 43936* Lemma for imasetpreimafvbij 43938: 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))

Theoremimasetpreimafvbijlemfo 43937* Lemma for imasetpreimafvbij 43938: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))

Theoremimasetpreimafvbij 43938* The mapping 𝐻 is a bijective function betwen the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1-onto→(𝐹𝐴))

Theoremfundcmpsurbijinjpreimafv 43939* Every function 𝐹:𝐴𝐵 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖((𝑔:𝐴onto𝑃:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))

Theoremfundcmpsurinjpreimafv 43940* Every function 𝐹:𝐴𝐵 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)))

Theoremfundcmpsurinj 43941* Every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))

Theoremfundcmpsurbijinj 43942* Every function 𝐹:𝐴𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))

Theoremfundcmpsurinjimaid 43943* Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto the image (𝐹𝐴) of the domain of 𝐹 and an injective function from the image (𝐹𝐴). (Contributed by AV, 17-Mar-2024.)
𝐼 = (𝐹𝐴)    &   𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))    &   𝐻 = ( I ↾ 𝐼)       (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))

TheoremfundcmpsurinjALT 43944* Alternate proof of fundcmpsurinj 43941, based on fundcmpsurinjimaid 43943: Every function 𝐹:𝐴𝐵 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:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))

20.41.8  Partitions of real intervals

Based on the theorems of the fourierdlem* series of GS's mathbox.

Syntaxciccp 43945 Extend class notation with the partitions of a closed interval of extended reals.
class RePart

Definitiondf-iccp 43946* 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)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})

Theoremiccpval 43947* 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)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})

Theoremiccpart 43948* A special partition. Corresponds to fourierdlem2 42766 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
(𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))

Theoremiccpartimp 43949 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)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))

Theoremiccpartres 43950 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))

Theoremiccpartxr 43951 If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (0...𝑀))       (𝜑 → (𝑃𝐼) ∈ ℝ*)

Theoremiccpartgtprec 43952 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.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (1...𝑀))       (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃𝐼))

Theoremiccpartipre 43953 If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (1..^𝑀))       (𝜑 → (𝑃𝐼) ∈ ℝ)

Theoremiccpartiltu 43954* If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃𝑖) < (𝑃𝑀))

Theoremiccpartigtl 43955* If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑖))

Theoremiccpartlt 43956 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 42775 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → (𝑃‘0) < (𝑃𝑀))

Theoremiccpartltu 43957* If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃𝑀))

Theoremiccpartgtl 43958* If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃𝑖))

Theoremiccpartgt 43959* If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))

Theoremiccpartleu 43960* If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃𝑖) ≤ (𝑃𝑀))

Theoremiccpartgel 43961* If there is a partition, then all intermediate points and the upper and the lower bound are greater than or equal to the lower bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑖))

Theoremiccpartrn 43962 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccpartf 43963 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 42779 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃𝑀)))

Theoremiccpartel 43964 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       ((𝜑𝐼 ∈ (0...𝑀)) → (𝑃𝐼) ∈ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccelpart 43965* An element of any partitioned half-open interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
(𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))

Theoremiccpartiun 43966* A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ((𝑃‘0)[,)(𝑃𝑀)) = 𝑖 ∈ (0..^𝑀)((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremicceuelpartlem 43967 Lemma for icceuelpart 43968. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 < 𝐽 → (𝑃‘(𝐼 + 1)) ≤ (𝑃𝐽))))

Theoremicceuelpart 43968* An element of a partitioned half-open interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremiccpartdisj 43969* The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑Disj 𝑖 ∈ (0..^𝑀)((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremiccpartnel 43970 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 42776 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝑋 ∈ ran 𝑃)       ((𝜑𝐼 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑃𝐼)(,)(𝑃‘(𝐼 + 1))))

20.41.9  Shifting functions with an integer range domain

Theoremfargshiftfv 43971* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺𝑋) = (𝐹‘(𝑋 + 1))))

Theoremfargshiftf 43972* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)

Theoremfargshiftf1 43973* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸)

Theoremfargshiftfo 43974* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸)

Theoremfargshiftfva 43975* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))

20.41.10  Words over a set (extension)

20.41.10.1  Last symbol of a word - extension

Theoremlswn0 43976 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( is the last symbol) and invalid cases ( means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (♯‘𝑊) ≠ 0) → (lastS‘𝑊) ≠ ∅)

20.41.11  Unordered pairs

20.41.11.1  Interchangeable setvar variables

Syntaxwich 43977 Extend wff notation to include the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. Read this notation as "𝑥 and 𝑦 are interchangeable in wff 𝜑".
wff [𝑥𝑦]𝜑

Definitiondf-ich 43978* Define the property of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. For an alternate definition using implicit substitution and a temporary setvar variable see ichcircshi 43986. Another, equivalent definition using two temporary setvar variables is provided in dfich2 43990. (Contributed by AV, 29-Jul-2023.)
([𝑥𝑦]𝜑 ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑))

Theoremnfich1 43979 The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
𝑥[𝑥𝑦]𝜑

Theoremnfich2 43980 The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)
𝑦[𝑥𝑦]𝜑

Theoremichv 43981* Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]𝜑

Theoremichf 43982 Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
𝑥𝜑    &   𝑦𝜑       [𝑥𝑦]𝜑

Theoremichid 43983 A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
[𝑥𝑥]𝜑

Theoremicht 43984 A theorem is interchangeable. (Contributed by SN, 4-May-2024.)
𝜑       [𝑥𝑦]𝜑

Theoremichbidv 43985* Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑥𝑦]𝜓 ↔ [𝑥𝑦]𝜒))

Theoremichcircshi 43986* The setvar variables are interchangeable if they can be circularily shifted using a third setvar variable, using implicit substitution. (Contributed by AV, 29-Jul-2023.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑥 → (𝜓𝜒))    &   (𝑧 = 𝑦 → (𝜒𝜑))       [𝑥𝑦]𝜑

Theoremichan 43987 If two setvar variables are interchangeable in two wffs, then they are interchangeable in the conjunction of these two wffs. Notice that the reverse implication is not necessarily true. Corresponding theorems will hold for other commutative operations, too. (Contributed by AV, 31-Jul-2023.) Use df-ich 43978 instead of dfich2 43990 to reduce axioms. (Revised by SN, 4-May-2024.)
(([𝑎𝑏]𝜑 ∧ [𝑎𝑏]𝜓) → [𝑎𝑏](𝜑𝜓))

Theoremichn 43988 Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.)
([𝑎𝑏]𝜑 ↔ [𝑎𝑏] ¬ 𝜑)

Theoremichim 43989 Formula building rule for implication in interchangeability. (Contributed by SN, 4-May-2024.)
(([𝑎𝑏]𝜑 ∧ [𝑎𝑏]𝜓) → [𝑎𝑏](𝜑𝜓))

Theoremdfich2 43990* Alternate definition of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. (Contributed by AV and WL, 6-Aug-2023.)
([𝑥𝑦]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

Theoremichcom 43991* The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)
([𝑥𝑦]𝜓 ↔ [𝑦𝑥]𝜓)

Theoremichbi12i 43992* Equivalence for interchangeable setvar variables. (Contributed by AV, 29-Jul-2023.)
((𝑥 = 𝑎𝑦 = 𝑏) → (𝜓𝜒))       ([𝑥𝑦]𝜓 ↔ [𝑎𝑏]𝜒)

Theoremicheqid 43993 In an equality for the same setvar variable, the setvar variable is interchangeable by itself. Special case of ichid 43983 and icheq 43994 without distinct variables restriction. (Contributed by AV, 29-Jul-2023.)
[𝑥𝑥]𝑥 = 𝑥

Theoremicheq 43994* In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.)
[𝑥𝑦]𝑥 = 𝑦

Theoremichnfimlem 43995* Lemma for ichnfim 43996: A substitution of a non-free variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2379. (Revised by Gino Giotto, 1-May-2024.)
(∀𝑦𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Theoremichnfim 43996* If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)

Theoremichnfb 43997* If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑))

Theoremichal 43998* Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)
(∀𝑥[𝑎𝑏]𝜑 → [𝑎𝑏]∀𝑥𝜑)

Theoremich2al 43999 Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]∀𝑥𝑦𝜑

Theoremich2ex 44000 Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]∃𝑥𝑦𝜑

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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