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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnambpcma 42901 | ((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) → (((𝐴 − 𝐵) + 𝐶) − 𝐴) = (𝐶 − 𝐵)) | ||
Theorem | cnapbmcpd 42902 | ((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)) | ||
Theorem | addsubeq0 42903 | 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) = (𝐴 − 𝐵) ↔ 𝐵 = 0)) | ||
Theorem | leaddsuble 42904 | 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) → (𝐵 ≤ 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴)) | ||
Theorem | 2leaddle2 42905 | 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 < 𝐶 ∧ 𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶))) | ||
Theorem | ltnltne 42906 | 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))) | ||
Theorem | p1lep2 42907 | 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)) | ||
Theorem | ltsubsubaddltsub 42908 | 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 < ((𝐿 − 𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿 − 𝑁))) | ||
Theorem | zm1nn 42909 | An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽 ∧ 𝐽 < ((𝐿 − 𝑁) − 1)) → (𝐿 − 1) ∈ ℕ)) | ||
Theorem | readdcnnred 42910 | The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∉ ℝ) | ||
Theorem | resubcnnred 42911 | The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∉ ℝ) | ||
Theorem | recnmulnred 42912 | The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∉ ℝ) | ||
Theorem | cndivrenred 42913 | The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ℝ)) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐵 / 𝐴) ∉ ℝ) | ||
Theorem | sqrtnegnre 42914 | The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023.) |
⊢ ((𝑋 ∈ ℝ ∧ 𝑋 < 0) → (√‘𝑋) ∉ ℝ) | ||
Theorem | nn0resubcl 42915 | Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 − 𝐵) ∈ ℝ) | ||
Theorem | zgeltp1eq 42916 | 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) → ((𝐴 ≤ 𝐼 ∧ 𝐼 < (𝐴 + 1)) → 𝐼 = 𝐴)) | ||
Theorem | 1t10e1p1e11 42917 | 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) | ||
Theorem | deccarry 42918 | Add 1 to a 2 digit number with carry. This is a special case of decsucc 11956, 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) | ||
Theorem | eluzge0nn0 42919 | 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 ∈ (ℤ≥‘𝑀) → (0 ≤ 𝑀 → 𝑁 ∈ ℕ0)) | ||
Theorem | nltle2tri 42920 | 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 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) | ||
Theorem | ssfz12 42921 | Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 ≤ 𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁))) | ||
Theorem | elfz2z 42922 | 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 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | 2elfz3nn0 42923 | 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) | ||
Theorem | fz0addcom 42924 | 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)) | ||
Theorem | 2elfz2melfz 42925 | 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 − (𝑁 − 𝐴)) ∈ (0...𝐴))) | ||
Theorem | fz0addge0 42926 | 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)) | ||
Theorem | elfzlble 42927 | 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 ∈ ((𝑁 − 𝑀)...𝑁)) | ||
Theorem | elfzelfzlble 42928 | 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 ∈ ((𝑁 − 𝑀)...𝑁)) | ||
Theorem | fzopred 42929 | Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 12944. (Contributed by AV, 14-Jul-2020.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → (𝑀..^𝑁) = ({𝑀} ∪ ((𝑀 + 1)..^𝑁))) | ||
Theorem | fzopredsuc 42930 | 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 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = (({𝑀} ∪ ((𝑀 + 1)..^𝑁)) ∪ {𝑁})) | ||
Theorem | 1fzopredsuc 42931 | 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})) | ||
Theorem | el1fzopredsuc 42932 | 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))) | ||
Theorem | subsubelfzo0 42933 | 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 < (𝑁 − 𝐴)) → (𝐼 − (𝑁 − 𝐴)) ∈ (0..^𝐴)) | ||
Theorem | fzoopth 42934 | A half-open integer range can represent an ordered pair, analogous to fzopth 12763. (Contributed by Alexander van der Vekens, 1-Jul-2018.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) | ||
Theorem | 2ffzoeq 42935* | 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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))) | ||
Theorem | m1mod0mod1 42936 | An integer decreased by 1 is 0 modulo a positive integer iff the integer is 1 modulo the same modulus. (Contributed by AV, 6-Jun-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 < 𝑁) → (((𝐴 − 1) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = 1)) | ||
Theorem | elmod2 42937 | An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ {0, 1}) | ||
Theorem | smonoord 42938* | Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 13218 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 40994? (Contributed by AV, 12-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁)) | ||
Theorem | fsummsndifre 42939* | A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ) | ||
Theorem | fsumsplitsndif 42940* | Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + ⦋𝑋 / 𝑘⦌𝐵)) | ||
Theorem | fsummmodsndifre 42941* | A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ) | ||
Theorem | fsummmodsnunz 42942* | 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) | ||
Theorem | setsidel 42943 | The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑅) | ||
Theorem | setsnidel 42944 | The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝑅 = (𝑆 sSet 〈𝐴, 𝐵〉) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ 𝑅) | ||
Theorem | setsv 42945 | The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | ||
Based on the theorems of the fourierdlem* series of GS's mathbox. | ||
Syntax | ciccp 42946 | Extend class notation with the partitions of a closed interval of extended reals. |
class RePart | ||
Definition | df-iccp 42947* | Define partitions of a closed interval of extended reals. Such partitions are finite increasing sequences of extended reals. (Contributed by AV, 8-Jul-2020.) |
⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | ||
Theorem | iccpval 42948* | Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.) |
⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | ||
Theorem | iccpart 42949* | A special partition. Corresponds to fourierdlem2 41826 in GS's mathbox. (Contributed by AV, 9-Jul-2020.) |
⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) | ||
Theorem | iccpartimp 42950 | Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) | ||
Theorem | iccpartres 42951 | The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)) | ||
Theorem | iccpartxr 42952 | If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) & ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*) | ||
Theorem | iccpartgtprec 42953 | If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) | ||
Theorem | iccpartipre 42954 | If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) & ⊢ (𝜑 → 𝐼 ∈ (1..^𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ) | ||
Theorem | iccpartiltu 42955* | If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) | ||
Theorem | iccpartigtl 42956* | If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)) | ||
Theorem | iccpartlt 42957 | If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 41835 in GS's mathbox. (Contributed by AV, 12-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) | ||
Theorem | iccpartltu 42958* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) | ||
Theorem | iccpartgtl 42959* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃‘𝑖)) | ||
Theorem | iccpartgt 42960* | If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))) | ||
Theorem | iccpartleu 42961* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) | ||
Theorem | iccpartgel 42962* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑖)) | ||
Theorem | iccpartrn 42963 | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀))) | ||
Theorem | iccpartf 42964 | The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 41839 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → 𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀))) | ||
Theorem | iccpartel 42965 | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐼 ∈ (0...𝑀)) → (𝑃‘𝐼) ∈ ((𝑃‘0)[,](𝑃‘𝑀))) | ||
Theorem | iccelpart 42966* | 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)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) | ||
Theorem | iccpartiun 42967* | A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ((𝑃‘0)[,)(𝑃‘𝑀)) = ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))) | ||
Theorem | icceuelpartlem 42968 | Lemma for icceuelpart 42969. (Contributed by AV, 19-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 < 𝐽 → (𝑃‘(𝐼 + 1)) ≤ (𝑃‘𝐽)))) | ||
Theorem | icceuelpart 42969* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))) | ||
Theorem | iccpartdisj 42970* | The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → Disj 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))) | ||
Theorem | iccpartnel 42971 | A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 41836 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑃) ⇒ ⊢ ((𝜑 ∧ 𝐼 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑃‘𝐼)(,)(𝑃‘(𝐼 + 1)))) | ||
Theorem | fargshiftfv 42972* | 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺‘𝑋) = (𝐹‘(𝑋 + 1)))) | ||
Theorem | fargshiftf 42973* | 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) | ||
Theorem | fargshiftf1 42974* | 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸) | ||
Theorem | fargshiftfo 42975* | 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸) | ||
Theorem | fargshiftfva 42976* | 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 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃)) | ||
Theorem | lswn0 42977 | 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) ≠ ∅) | ||
Syntax | wich 42978 | 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 [𝑥⇄𝑦]𝜑 | ||
Definition | df-ich 42979* | Define the propery 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 42983. Another, equivalent definition using two temporary setvar variables is provided in dfich2 42984. (Contributed by AV, 29-Jul-2023.) |
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | nfich1 42980 | The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.) |
⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑 | ||
Theorem | nfich2 42981 | The second interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.) |
⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑 | ||
Theorem | ichid 42982 | A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.) |
⊢ [𝑥⇄𝑥]𝜑 | ||
Theorem | ichcircshi 42983* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑)) ⇒ ⊢ [𝑥⇄𝑦]𝜑 | ||
Theorem | dfich2 42984* | 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.) |
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑)) | ||
Theorem | dfich2ai 42985* | Obsolete version of dfich2 42984 as of 18-Sep-2023. The definition df-ich 42979 of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable implies the alternate definition dfich2 42984. (Contributed by AV, 6-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑)) | ||
Theorem | dfich2bi 42986* | Obsolete version of dfich2 42984 as of 18-Sep-2023. The alternate definition dfich2 42984 of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable implies the definition df-ich 42979. (Contributed by AV, 6-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | dfich2OLD 42987* | Obsolete version of dfich2 42984 as of 18-Sep-2023. Alternate definition of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. (Contributed by AV, 6-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑)) | ||
Theorem | ichcom 42988* | The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.) |
⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓) | ||
Theorem | ichbi12i 42989* | Equivalence for interchangeable setvar variables. (Contributed by AV, 29-Jul-2023.) |
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜓 ↔ 𝜒)) ⇒ ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑎⇄𝑏]𝜒) | ||
Theorem | icheqid 42990 | In an equality for the same setvar variable, the setvar variable is interchangeable by itself. Special case of ichid 42982 and icheq 42991 without distinct variables restriction. (Contributed by AV, 29-Jul-2023.) |
⊢ [𝑥⇄𝑥]𝑥 = 𝑥 | ||
Theorem | icheq 42991* | In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.) |
⊢ [𝑥⇄𝑦]𝑥 = 𝑦 | ||
Theorem | ichnfimlem1 42992* | Lemma for ichnfimlem3 42994: A substitution of a non-free variable has no effect. Give the distinctor in a form that can be easily eliminiated. (Contributed by Wolf Lammen, 6-Aug-2023.) |
⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | ||
Theorem | ichnfimlem2 42993* | Lemma for ichnfimlem3 42994: When substituting successively for two always equal variables, the second substitution has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | ||
Theorem | ichnfimlem3 42994* | Lemma for ichnfim 42995: A substitution of a non-free variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) |
⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | ||
Theorem | ichnfim 42995* | 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.) |
⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑) | ||
Theorem | ichnfb 42996* | 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.) |
⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑)) | ||
Theorem | ichn 42997 | Negation does not affect interchangability. (Contributed by SN, 30-Aug-2023.) |
⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑) | ||
Theorem | ichal 42998* | Move a universal quantifier inside interchangability. (Contributed by SN, 30-Aug-2023.) |
⊢ (∀𝑥[𝑎⇄𝑏]𝜑 → [𝑎⇄𝑏]∀𝑥𝜑) | ||
Theorem | ichan 42999* | 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.) |
⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓)) | ||
Theorem | ichexmpl1 43000* | Example for interchangeable setvar variables in a statement of predicate calculus with equality. (Contributed by AV, 31-Jul-2023.) |
⊢ [𝑎⇄𝑏]∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) |
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