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Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

3.6.10  The ` # ` (set size) function

Syntaxchash 10101 Extend the definition of a class to include the set size function.
class

Definitiondf-ihash 10102* Define the set size function , which gives the cardinality of a finite set as a member of 0, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3.

Note that we use the sharp sign () for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8003). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of and ) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))

Theoremhashinfuni 10103* The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
(ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)

Theoremhashinfom 10104 The value of the function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
(ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Theoremhashennnuni 10105* The ordinal size of a set equinumerous to an element of ω is that element of ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = 𝑁)

Theoremhashennn 10106* The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Theoremhashcl 10107 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
(𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)

Theoremhashfiv01gt1 10108 The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)
(𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))

Theoremhashfz1 10109 The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
(𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)

Theoremhashen 10110 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴𝐵))

Theoremhasheqf1o 10111* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))

Theoremfiinfnf1o 10112* There is no bijection between a finite set and an infinite set. By infnfi 6565 the theorem would also hold if "infinite" were expressed as ω ≼ 𝐵. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)

Theoremfocdmex 10113 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)

Theoremfihasheqf1oi 10114 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → (♯‘𝐴) = (♯‘𝐵))

Theoremfihashf1rn 10115 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Theoremfihasheqf1od 10116 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴1-1-onto𝐵)       (𝜑 → (♯‘𝐴) = (♯‘𝐵))

Theoremfz1eqb 10117 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))

Theoremfiltinf 10118 The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < (♯‘𝐵))

Theoremisfinite4im 10119 A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
(𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)

Theoremfihasheq0 10120 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
(𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

Theoremfihashneq0 10121 Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6555. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
(𝐴 ∈ Fin → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅))

Theoremhashnncl 10122 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))

Theoremhash0 10123 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
(♯‘∅) = 0

Theoremhashsng 10124 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
(𝐴𝑉 → (♯‘{𝐴}) = 1)

Theoremfihashen1 10125 A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
(𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1𝑜))

Theoremfihashfn 10126 A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → (♯‘𝐹) = (♯‘𝐴))

Theoremfseq1hash 10127 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)

Theoremomgadd 10128 Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))

Theoremfihashdom 10129 Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴𝐵))

Theoremhashunlem 10130 Lemma for hashun 10131. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑁 ∈ ω)    &   (𝜑𝑀 ∈ ω)    &   (𝜑𝐴𝑁)    &   (𝜑𝐵𝑀)       (𝜑 → (𝐴𝐵) ≈ (𝑁 +𝑜 𝑀))

Theoremhashun 10131 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (♯‘(𝐴𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))

Theorem1elfz0hash 10132 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(♯‘𝐴)))

Theoremhashunsng 10133 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
(𝐵𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1)))

Theoremhashprg 10134 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))

Theoremprhash2ex 10135 There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 10141, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
(♯‘{0, 1}) = 2

Theoremhashp1i 10136 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝐴 ∈ ω    &   𝐵 = suc 𝐴    &   (♯‘𝐴) = 𝑀    &   (𝑀 + 1) = 𝑁       (♯‘𝐵) = 𝑁

Theoremhash1 10137 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘1𝑜) = 1

Theoremhash2 10138 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘2𝑜) = 2

Theoremhash3 10139 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘3𝑜) = 3

Theoremhash4 10140 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘4𝑜) = 4

Theorempr0hash2ex 10141 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
(♯‘{∅, {∅}}) = 2

Theoremfihashss 10142 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))

Theoremfiprsshashgt1 10143 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
(((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐶 ∈ Fin) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶)))

Theoremfihashssdif 10144 The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → (♯‘(𝐴𝐵)) = ((♯‘𝐴) − (♯‘𝐵)))

Theoremhashdifsn 10145 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))

Theoremhashdifpr 10146 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
((𝐴 ∈ Fin ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))

Theoremhashfz 10147 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
(𝐵 ∈ (ℤ𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵𝐴) + 1))

Theoremhashfzo 10148 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵𝐴))

Theoremhashfzo0 10149 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = 𝐵)

Theoremhashfzp1 10150 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))

Theoremhashfz0 10151 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1))

Theoremhashxp 10152 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Theoremfimaxq 10153* A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)
((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)

Theoremresunimafz0 10154 The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
(𝜑 → Fun 𝐼)    &   (𝜑𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)    &   (𝜑𝑁 ∈ (0..^(♯‘𝐹)))       (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))

Theoremfnfz0hash 10155 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0...𝑁)) → (♯‘𝐹) = (𝑁 + 1))

Theoremffz0hash 10156 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶𝐵) → (♯‘𝐹) = (𝑁 + 1))

Theoremffzo0hash 10157 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0..^𝑁)) → (♯‘𝐹) = 𝑁)

Theoremfnfzo0hash 10158 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0..^𝑁)⟶𝐵) → (♯‘𝐹) = 𝑁)

Theoremhashfacen 10159* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴𝐵𝐶𝐷) → {𝑓𝑓:𝐴1-1-onto𝐶} ≈ {𝑓𝑓:𝐵1-1-onto𝐷})

Theoremleisorel 10160 Version of isorel 5550 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))

Theoremzfz1isolemsplit 10161 Lemma for zfz1iso 10164. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑀𝑋)       (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))

Theoremzfz1isolemiso 10162* Lemma for zfz1iso 10164. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ⊆ ℤ)    &   (𝜑𝑀𝑋)    &   (𝜑 → ∀𝑧𝑋 𝑧𝑀)    &   (𝜑𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))    &   (𝜑𝐴 ∈ (1...(♯‘𝑋)))    &   (𝜑𝐵 ∈ (1...(♯‘𝑋)))       (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Theoremzfz1isolem1 10163* Lemma for zfz1iso 10164. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
(𝜑𝐾 ∈ ω)    &   (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))    &   (𝜑𝑋 ⊆ ℤ)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≈ suc 𝐾)    &   (𝜑𝑀𝑋)    &   (𝜑 → ∀𝑧𝑋 𝑧𝑀)       (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))

Theoremzfz1iso 10164* A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Theoremiseqcoll 10165* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))    &   (𝜑𝑁 ∈ (1...(♯‘𝐴)))    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ‘1)) → (𝐻𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝐺𝑁)) = (seq1( + , 𝐻, 𝑆)‘𝑁))

3.7  Elementary real and complex functions

3.7.1  The "shift" operation

Syntaxcshi 10166 Extend class notation with function shifter.
class shift

Definitiondf-shft 10167* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 10177 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})

Theoremshftlem 10168* Two ways to write a shifted set (𝐵 + 𝐴). (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ⊆ ℂ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = (𝑦 + 𝐴)})

Theoremshftuz 10169* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ (ℤ𝐵)} = (ℤ‘(𝐵 + 𝐴)))

Theoremshftfvalg 10170* The value of the sequence shifter operation is a function on . 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐹𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})

Theoremovshftex 10171 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V)

Theoremshftfibg 10172 Value of a fiber of the relation 𝐹. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵𝐴)}))

Theoremshftfval 10173* The value of the sequence shifter operation is a function on . 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V       (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})

Theoremshftdm 10174* Domain of a relation shifted by 𝐴. The set on the right is more commonly notated as (dom 𝐹 + 𝐴) (meaning add 𝐴 to every element of dom 𝐹). (Contributed by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V       (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})

Theoremshftfib 10175 Value of a fiber of the relation 𝐹. (Contributed by Mario Carneiro, 4-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵𝐴)}))

Theoremshftfn 10176* Functionality and domain of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V       ((𝐹 Fn 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})

Theoremshftval 10177 Value of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵𝐴)))

Theoremshftval2 10178 Value of a sequence shifted by 𝐴𝐵. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶)))

Theoremshftval3 10179 Value of a sequence shifted by 𝐴𝐵. (Contributed by NM, 20-Jul-2005.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴𝐵))‘𝐴) = (𝐹𝐵))

Theoremshftval4 10180 Value of a sequence shifted by -𝐴. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))

Theoremshftval5 10181 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘(𝐵 + 𝐴)) = (𝐹𝐵))

Theoremshftf 10182* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐹:𝐵𝐶𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵}⟶𝐶)

Theorem2shfti 10183 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) shift 𝐵) = (𝐹 shift (𝐴 + 𝐵)))

Theoremshftidt2 10184 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐹 shift 0) = (𝐹 ↾ ℂ)

Theoremshftidt 10185 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐴 ∈ ℂ → ((𝐹 shift 0)‘𝐴) = (𝐹𝐴))

Theoremshftcan1 10186 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift 𝐴) shift -𝐴)‘𝐵) = (𝐹𝐵))

Theoremshftcan2 10187 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift -𝐴) shift 𝐴)‘𝐵) = (𝐹𝐵))

Theoremshftvalg 10188 Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵𝐴)))

Theoremshftval4g 10189 Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))

3.7.2  Real and imaginary parts; conjugate

Syntaxccj 10190 Extend class notation to include complex conjugate function.
class

Syntaxcre 10191 Extend class notation to include real part of a complex number.
class

Syntaxcim 10192 Extend class notation to include imaginary part of a complex number.
class

Definitiondf-cj 10193* Define the complex conjugate function. See cjcli 10264 for its closure and cjval 10196 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))

Definitiondf-re 10194 Define a function whose value is the real part of a complex number. See reval 10200 for its value, recli 10262 for its closure, and replim 10210 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))

Definitiondf-im 10195 Define a function whose value is the imaginary part of a complex number. See imval 10201 for its value, imcli 10263 for its closure, and replim 10210 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))

Theoremcjval 10196* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))

Theoremcjth 10197 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))

Theoremcjf 10198 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ

Theoremcjcl 10199 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)

Theoremreval 10200 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2))

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