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Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremafv2ndeffv0 42301 If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
 
Theoremdfatafv2eqfv 42302 If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv2rnfveq 42303 If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv20fv0 42304 If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
 
Theoremafv2fvn0fveq 42305 If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv2fv0 42306 If the function's value at an argument is the empty set, then the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
 
Theoremafv2fv0b 42307 The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
 
Theoremafv2fv0xorb 42308 If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.)
(∅ ∈ ran 𝐹 → ((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
 
20.36.6  General auxiliary theorems (2)
 
20.36.6.1  Logical conjunction - extension
 
Theoreman4com24 42309 Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜃) ∧ (𝜒𝜓)))
 
20.36.6.2  Abbreviated conjunction and disjunction of three wff's - extension
 
Theorem3an4ancom24 42310 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 42311 Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
(((𝜑𝜓) ∧ 𝜒𝜃) ↔ (𝜓 ∧ (𝜑𝜒𝜃)))
 
20.36.6.3  Negated membership (alternative)
 
Syntaxcnelbr 42312 Extend wff notation to include the 'not elemet of' relation.
class _∉
 
Definitiondf-nelbr 42313* Define negated membership as binary relation. Analogous to df-eprel 5266 (the epsilon relation). (Contributed by AV, 26-Dec-2021.)
_∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
 
Theoremdfnelbr2 42314 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 42315 The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
 
Theoremnelbrim 42316 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 42317 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 42318 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)
 
20.36.6.4  The empty set - extension
 
Theoremralralimp 42319* 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))
 
20.36.6.5  Indexed union and intersection - extension
 
TheoremotiunsndisjX 42320* 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⟩})
 
20.36.6.6  Functions - extension
 
Theoremfvifeq 42321 Equality of function values with conditional arguments, see also fvif 6462. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
 
Theoremrnfdmpr 42322 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 42323 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from 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 42324* 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 42325 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Avoid depending on this detail.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
 
Theoremopabresex0d 42326* 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 42327* 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 42328* 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 42329* 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)
 
Theoremf1oresf1orab 42330* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)
𝐹 = (𝑥𝐴𝐶)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝑥𝐷))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1oresf1o 42331* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1oresf1o2 42332* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
20.36.6.7  Maps-to notation - extension
 
Theoremfvmptrab 42333* 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 6571, 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 42334* Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6571. (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}
 
20.36.6.8  Ordering on reals - extension
 
Theoremleltletr 42335 Transitive law, weaker form of lelttr 10467. (Contributed by AV, 14-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴𝐶))
 
20.36.6.9  Subtraction - extension
 
Theoremcnambpcma 42336 ((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 42337 ((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))
 
Theoremaddsubeq0 42338 The sum of two complex numbers is equal to the difference of these two complex numbers iff the subtrahend is 0. (Contributed by AV, 8-May-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴𝐵) ↔ 𝐵 = 0))
 
20.36.6.10  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 42339 Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))
 
Theorem2leaddle2 42340 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 42341 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 42342 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 42343 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 42344 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))
 
20.36.6.11  Imaginary and complex number properties - extension
 
Theoremreaddcnnred 42345 The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))       (𝜑 → (𝐴 + 𝐵) ∉ ℝ)
 
Theoremresubcnnred 42346 The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))       (𝜑 → (𝐴𝐵) ∉ ℝ)
 
Theoremrecnmulnred 42347 The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 · 𝐵) ∉ ℝ)
 
Theoremcndivrenred 42348 The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐵 / 𝐴) ∉ ℝ)
 
Theoremsqrtnegnre 42349 The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023.)
((𝑋 ∈ ℝ ∧ 𝑋 < 0) → (√‘𝑋) ∉ ℝ)
 
20.36.6.12  Nonnegative integers (as a subset of complex numbers) - extension
 
Theoremnn0resubcl 42350 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ ℝ)
 
20.36.6.13  Integers (as a subset of complex numbers) - extension
 
Theoremzgeltp1eq 42351 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))
 
20.36.6.14  Decimal arithmetic - extension
 
Theorem1t10e1p1e11 42352 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)
 
Theoremdeccarry 42353 Add 1 to a 2 digit number with carry. This is a special case of decsucc 11887, 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)
 
20.36.6.15  Upper sets of integers - extension
 
Theoremeluzge0nn0 42354 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))
 
20.36.6.16  Infinity and the extended real number system (cont.) - extension
 
Theoremnltle2tri 42355 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))
 
20.36.6.17  Finite intervals of integers - extension
 
Theoremssfz12 42356 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 42357 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 42358 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 42359 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 42360 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 42361 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 42362 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 42363 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))
 
20.36.6.18  Half-open integer ranges - extension
 
Theoremfzopred 42364 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 12876. (Contributed by AV, 14-Jul-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → (𝑀..^𝑁) = ({𝑀} ∪ ((𝑀 + 1)..^𝑁)))
 
Theoremfzopredsuc 42365 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 42366 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 42367 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 42368 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 42369 A half-open integer range can represent an ordered pair, analogous to fzopth 12695. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
 
Theorem2ffzoeq 42370* 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))))
 
20.36.6.19  The modulo (remainder) operation - extension
 
Theoremm1mod0mod1 42371 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 42372 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})
 
20.36.6.20  The infinite sequence builder "seq"
 
Theoremsmonoord 42373* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 13149 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 40420? (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))
 
20.36.6.21  Finite and infinite sums - extension
 
Theoremfsummsndifre 42374* 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 42375* 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 42376* 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 42377* 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)
 
20.36.6.22  Extensible structures - extension
 
Theoremsetsidel 42378 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 42379 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 42380 The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
 
20.36.7  Partitions of real intervals

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

 
Syntaxciccp 42381 Extend class notation with the partitions of a closed interval of extended reals.
class RePart
 
Definitiondf-iccp 42382* 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 42383* 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 42384* A special partition. Corresponds to fourierdlem2 41253 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 42385 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 42386 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 42387 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 42388 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 42389 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 42390* 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 42391* 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 42392 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 41262 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → (𝑃‘0) < (𝑃𝑀))
 
Theoremiccpartltu 42393* 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 42394* 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 42395* 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 42396* 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 42397* 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 42398 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 42399 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 41266 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 42400 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)))
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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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