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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | afv2ndeffv0 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 → (𝐹‘𝐴) = ∅) | ||
Theorem | dfatafv2eqfv 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) = (𝐹‘𝐴)) | ||
Theorem | afv2rnfveq 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) = (𝐹‘𝐴)) | ||
Theorem | afv20fv0 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) = ∅ → (𝐹‘𝐴) = ∅) | ||
Theorem | afv2fvn0fveq 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.) |
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afv2fv0 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.) |
⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | ||
Theorem | afv2fv0b 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.) |
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | ||
Theorem | afv2fv0xorb 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 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹))) | ||
Theorem | an4com24 42309 | Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓))) | ||
Theorem | 3an4ancom24 42310 | Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓)) | ||
Theorem | 4an21 42311 | Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃))) | ||
Syntax | cnelbr 42312 | Extend wff notation to include the 'not elemet of' relation. |
class _∉ | ||
Definition | df-nelbr 42313* | Define negated membership as binary relation. Analogous to df-eprel 5266 (the epsilon relation). (Contributed by AV, 26-Dec-2021.) |
⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | ||
Theorem | dfnelbr2 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 ) | ||
Theorem | nelbr 42315 | The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)) | ||
Theorem | nelbrim 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 ¬ 𝐴 ∈ 𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.) |
⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) | ||
Theorem | nelbrnel 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵)) | ||
Theorem | nelbrnelim 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 _∉ 𝐵 → 𝐴 ∉ 𝐵) | ||
Theorem | ralralimp 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.) |
⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) | ||
Theorem | otiunsndisjX 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.) |
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {〈𝑎, 𝐵, 𝑐〉}) | ||
Theorem | fvifeq 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) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))) | ||
Theorem | rnfdmpr 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})) | ||
Theorem | imarnf1pr 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})) | ||
Theorem | funop1 42324* | A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) |
⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) | ||
Theorem | fun2dmnopgexmpl 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)) | ||
Theorem | opabresex0d 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.) |
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | ||
Theorem | opabbrfex0d 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.) |
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) | ||
Theorem | opabresexd 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.) |
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | ||
Theorem | opabbrfexd 42329* | A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.) |
⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) | ||
Theorem | f1oresf1orab 42330* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝑥 ∈ 𝐷)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | f1oresf1o 42331* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | f1oresf1o2 42332* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑥 ∈ 𝐷 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | fvmptrab 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 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) & ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁) & ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) & ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) ⇒ ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓} | ||
Theorem | fvmptrabdm 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 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑}) & ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) & ⊢ (𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹) ⇒ ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} | ||
Theorem | leltletr 42335 | Transitive law, weaker form of lelttr 10467. (Contributed by AV, 14-Oct-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | cnambpcma 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) → (((𝐴 − 𝐵) + 𝐶) − 𝐴) = (𝐶 − 𝐵)) | ||
Theorem | cnapbmcpd 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)) | ||
Theorem | addsubeq0 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) = (𝐴 − 𝐵) ↔ 𝐵 = 0)) | ||
Theorem | leaddsuble 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) → (𝐵 ≤ 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴)) | ||
Theorem | 2leaddle2 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 < 𝐶 ∧ 𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶))) | ||
Theorem | ltnltne 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))) | ||
Theorem | p1lep2 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)) | ||
Theorem | ltsubsubaddltsub 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 < ((𝐿 − 𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿 − 𝑁))) | ||
Theorem | zm1nn 42344 | 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 42345 | 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 42346 | 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 42347 | 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 42348 | 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 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) | ||
Theorem | nn0resubcl 42350 | 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 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) → ((𝐴 ≤ 𝐼 ∧ 𝐼 < (𝐴 + 1)) → 𝐼 = 𝐴)) | ||
Theorem | 1t10e1p1e11 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) | ||
Theorem | deccarry 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) | ||
Theorem | eluzge0nn0 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 ∈ (ℤ≥‘𝑀) → (0 ≤ 𝑀 → 𝑁 ∈ ℕ0)) | ||
Theorem | nltle2tri 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 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) | ||
Theorem | ssfz12 42356 | 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 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 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | 2elfz3nn0 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) | ||
Theorem | fz0addcom 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)) | ||
Theorem | 2elfz2melfz 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 − (𝑁 − 𝐴)) ∈ (0...𝐴))) | ||
Theorem | fz0addge0 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)) | ||
Theorem | elfzlble 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 ∈ ((𝑁 − 𝑀)...𝑁)) | ||
Theorem | elfzelfzlble 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 ∈ ((𝑁 − 𝑀)...𝑁)) | ||
Theorem | fzopred 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))) | ||
Theorem | fzopredsuc 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 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = (({𝑀} ∪ ((𝑀 + 1)..^𝑁)) ∪ {𝑁})) | ||
Theorem | 1fzopredsuc 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})) | ||
Theorem | el1fzopredsuc 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))) | ||
Theorem | subsubelfzo0 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 < (𝑁 − 𝐴)) → (𝐼 − (𝑁 − 𝐴)) ∈ (0..^𝐴)) | ||
Theorem | fzoopth 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 = 𝐽 ∧ 𝑁 = 𝐾))) | ||
Theorem | 2ffzoeq 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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))) | ||
Theorem | m1mod0mod1 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)) | ||
Theorem | elmod2 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}) | ||
Theorem | smonoord 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.) |
⊢ (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 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 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ) | ||
Theorem | fsumsplitsndif 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 ∧ 𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + ⦋𝑋 / 𝑘⦌𝐵)) | ||
Theorem | fsummmodsndifre 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 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ) | ||
Theorem | fsummmodsnunz 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) | ||
Theorem | setsidel 42378 | 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 42379 | 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 42380 | 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 42381 | Extend class notation with the partitions of a closed interval of extended reals. |
class RePart | ||
Definition | df-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)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | ||
Theorem | iccpval 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)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | ||
Theorem | iccpart 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)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))) | ||
Theorem | iccpartimp 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)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))) | ||
Theorem | iccpartres 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)) | ||
Theorem | iccpartxr 42387 | 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 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃‘𝐼)) | ||
Theorem | iccpartipre 42389 | 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 42390* | 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 42391* | 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 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) | ||
Theorem | iccpartltu 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) | ||
Theorem | iccpartgtl 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃‘𝑖)) | ||
Theorem | iccpartgt 42395* | 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 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘𝑖) ≤ (𝑃‘𝑀)) | ||
Theorem | iccpartgel 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑖)) | ||
Theorem | iccpartrn 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀))) | ||
Theorem | iccpartf 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ (𝜑 → 𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃‘𝑀))) | ||
Theorem | iccpartel 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐼 ∈ (0...𝑀)) → (𝑃‘𝐼) ∈ ((𝑃‘0)[,](𝑃‘𝑀))) |
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