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Theorem List for Metamath Proof Explorer - 41601-41700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfaovcl 41601 Closure law for an operation, analogous to fovcl 6807. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Theoremaovmpt4g 41602* Value of a function given by the "maps to" notation, analogous to ovmpt4g 6825. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Theoremaoprssdm 41603* Domain of closure of an operation. In contrast to oprssdm 6857, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)       (𝑆 × 𝑆) ⊆ dom 𝐹

Theoremndmaovcl 41604 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6861 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)    &   ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)    &    ((𝐴𝐹𝐵)) ∈ V        ((𝐴𝐹𝐵)) ∈ 𝑆

Theoremndmaovrcl 41605 Reverse closure law, in contrast to ndmovrcl 6862 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))

Theoremndmaovcom 41606 Any operation is commutative outside its domain, analogous to ndmovcom 6863. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Theoremndmaovass 41607 Any operation is associative outside its domain. In contrast to ndmovass 6864 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )

Theoremndmaovdistr 41608 Any operation is distributive outside its domain. In contrast to ndmovdistr 6865 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)    &   dom 𝐺 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )

20.35.3  General auxiliary theorems

20.35.3.1  Logical disjunction and conjunction - extension

Theoreman4com24 41609 Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜃) ∧ (𝜒𝜓)))

20.35.3.2  Abbreviated conjunction and disjunction of three wff's - extension

Theorem3an4ancom24 41610 Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜃𝜒) ∧ 𝜓))

Theorem4an21 41611 Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
(((𝜑𝜓) ∧ 𝜒𝜃) ↔ (𝜓 ∧ (𝜑𝜒𝜃)))

20.35.3.3  Negated membership (alternative)

Syntaxcnelbr 41612 Extend wff notation to include the 'not elemet of' relation.
class _∉

Definitiondf-nelbr 41613* Define negated membership as binary relation. Analogous to df-eprel 5058 (the epsilon relation). (Contributed by AV, 26-Dec-2021.)
_∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}

Theoremdfnelbr2 41614 Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
_∉ = ((V × V) ∖ E )

Theoremnelbr 41615 The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))

Theoremnelbrim 41616 If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.)
(𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)

Theoremnelbrnel 41617 A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))

Theoremnelbrnelim 41618 If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
(𝐴 _∉ 𝐵𝐴𝐵)

20.35.3.4  The empty set - extension

Theoremralralimp 41619* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))

20.35.3.5  Unordered and ordered pairs - extension

Theoremelprneb 41620 An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵𝐶) → (𝐴 = 𝐵𝐴𝐶))

20.35.3.6  Indexed union and intersection - extension

TheoremotiunsndisjX 41621* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
(𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

20.35.3.7  Functions - extension

Theoremfvifeq 41622 Equality of function values with conditional arguments, see also fvif 6242. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))

Theoremrnfdmpr 41623 The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))

Theoremimarnf1pr 41624 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function of a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Theoremfunop1 41625* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
(∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Theoremfun2dmnopgexmpl 41626 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Theoremopabresex0d 41627* A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfex0d 41628* A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Theoremopabresexd 41629* A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfexd 41630* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

20.35.3.8  "Maps to" notation - extension

Theoremfvmptrab 41631* Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 6348, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))    &   (𝑥 = 𝑋𝑀 = 𝑁)    &   (𝑋𝑉𝑁 ∈ V)    &   (𝑋𝑉𝑁 = ∅)       (𝐹𝑋) = {𝑦𝑁𝜓}

Theoremfvmptrabdm 41632* Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6348. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)
𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))    &   (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)       (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}

20.35.3.9  Ordering on reals - extension

Theoremleltletr 41633 Transitive law, weaker form of lelttr 10166. (Contributed by AV, 14-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴𝐶))

20.35.3.10  Subtraction - extension

Theoremcnambpcma 41634 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) − 𝐴) = (𝐶𝐵))

Theoremcnapbmcpd 41635 ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) − 𝐶) + 𝐷) = (((𝐴 + 𝐷) + 𝐵) − 𝐶))

20.35.3.11  Ordering on reals (cont.) - extension

Theoremleaddsuble 41636 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))

Theorem2leaddle2 41637 If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶)))

Theoremltnltne 41638 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴)))

Theoremp1lep2 41639 A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ ℝ → (𝑁 + 1) ≤ (𝑁 + 2))

Theoremltsubsubaddltsub 41640 If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝐽 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → (𝐽 < ((𝐿𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿𝑁)))

Theoremzm1nn 41641 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝑁 ∈ ℕ0𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽𝐽 < ((𝐿𝑁) − 1)) → (𝐿 − 1) ∈ ℕ))

20.35.3.12  Nonnegative integers (as a subset of complex numbers) - extension

Theoremnn0resubcl 41642 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ ℝ)

20.35.3.13  Integers (as a subset of complex numbers) - extension

Theoremzgeltp1eq 41643 If an integer is between another integer and its successor, the integer is equal to the other integer. (Contributed by AV, 30-May-2020.)
((𝐼 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴𝐼𝐼 < (𝐴 + 1)) → 𝐼 = 𝐴))

20.35.3.14  Decimal arithmetic - extension

Theorem1t10e1p1e11 41644 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
11 = ((1 · (10↑1)) + 1)

Theorem1t10e1p1e11OLD 41645 Obsolete version of 1t10e1p1e11 41644 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
11 = ((1 · (10↑1)) + 1)

Theoremdeccarry 41646 Add 1 to a 2 digit number with carry. This is a special case of decsucc 11588, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get (999 + 1) = 1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.)
(𝐴 ∈ ℕ → (𝐴9 + 1) = (𝐴 + 1)0)

20.35.3.15  Upper sets of integers - extension

Theoremeluzge0nn0 41647 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
(𝑁 ∈ (ℤ𝑀) → (0 ≤ 𝑀𝑁 ∈ ℕ0))

20.35.3.16  Infinity and the extended real number system (cont.) - extension

Theoremnltle2tri 41648 Negated extended trichotomy law for 'less than' and 'less than or equal to'. (Contributed by AV, 18-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵𝐶𝐶𝐴))

20.35.3.17  Finite intervals of integers - extension

Theoremssfz12 41649 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀𝐾𝐿𝑁)))

Theoremelfz2z 41650 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ (0 ≤ 𝐾𝐾𝑁)))

Theorem2elfz3nn0 41651 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0))

Theoremfz0addcom 41652 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theorem2elfz2melfz 41653 If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁𝐴)) ∈ (0...𝐴)))

Theoremfz0addge0 41654 The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ (0...𝑀) ∧ 𝐵 ∈ (0...𝑁)) → 0 ≤ (𝐴 + 𝐵))

Theoremelfzlble 41655 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ((𝑁𝑀)...𝑁))

Theoremelfzelfzlble 41656 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑁 < (𝑀 + 𝐾)) → 𝐾 ∈ ((𝑁𝑀)...𝑁))

20.35.3.18  Half-open integer ranges - extension

Theoremfzopred 41657 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 12597. (Contributed by AV, 14-Jul-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → (𝑀..^𝑁) = ({𝑀} ∪ ((𝑀 + 1)..^𝑁)))

Theoremfzopredsuc 41658 Join a predecessor and a successor to the beginning and the end of an open integer interval. This theorem holds even if 𝑁 = 𝑀 (then (𝑀...𝑁) = {𝑀} = ({𝑀} ∪ ∅) ∪ {𝑀}). (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = (({𝑀} ∪ ((𝑀 + 1)..^𝑁)) ∪ {𝑁}))

Theorem1fzopredsuc 41659 Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁}))

Theoremel1fzopredsuc 41660 An element of an open integer interval starting at 1 joined by 0 and a successor at the beginning and the end is either 0 or an element of the open integer interval or the successor. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (𝐼 ∈ (0...𝑁) ↔ (𝐼 = 0 ∨ 𝐼 ∈ (1..^𝑁) ∨ 𝐼 = 𝑁)))

Theoremsubsubelfzo0 41661 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐼 ∈ (0..^𝑁) ∧ ¬ 𝐼 < (𝑁𝐴)) → (𝐼 − (𝑁𝐴)) ∈ (0..^𝐴))

Theoremfzoopth 41662 A half-open integer range can represent an ordered pair, analogous to fzopth 12416. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

Theorem2ffzoeq 41663* 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 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))

20.35.3.19  The modulo (remainder) operation - extension

Theoremm1mod0mod1 41664 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 41665 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})

20.35.3.20  The infinite sequence builder "seq"

Theoremsmonoord 41666* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 12871 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 39825? (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))

20.35.3.21  Finite and infinite sums - extension

Theoremfsummsndifre 41667* 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 41668* 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 41669* 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 41670* 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)

20.35.3.22  Extensible structures - extension

Theoremsetsidel 41671 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 41672 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 41673 The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)

20.35.4  Partitions of real intervals

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

Syntaxciccp 41674 Extend class notation with the partitions of a closed interval of extended reals.
class RePart

Definitiondf-iccp 41675* 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 41676* 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 41677* A special partition. Corresponds to fourierdlem2 40644 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 41678 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 41679 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 41680 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 41681 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 41682 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 41683* 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 41684* 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 41685 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 40653 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → (𝑃‘0) < (𝑃𝑀))

Theoremiccpartltu 41686* 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 41687* 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 41688* 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 41689* 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 41690* 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 41691 If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccpartf 41692 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 40657 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 41693 If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       ((𝜑𝐼 ∈ (0...𝑀)) → (𝑃𝐼) ∈ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccelpart 41694* An element of any partitioned half opened 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 41695* A half opened 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 41696 Lemma for icceuelpart 41697. (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 41697* An element of a partitioned half opened 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 41698* The segments of a partitioned half opened 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 41699 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 40654 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))))

20.35.5  Shifting functions with an integer range domain

Theoremfargshiftfv 41700* 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))))

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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