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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | fsupprnfi 32701 | Finite support implies finite range. (Contributed by Thierry Arnoux, 24-Jun-2024.) |
| ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin) | ||
| Theorem | mptiffisupp 32702* | Conditions for a mapping function defined with a conditional to have finite support. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍) | ||
| Theorem | cosnopne 32703 | Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐷〉}) = ∅) | ||
| Theorem | cosnop 32704 | Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉}) = {〈𝐶, 𝐵〉}) | ||
| Theorem | cnvprop 32705 | Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {〈𝐵, 𝐴〉, 〈𝐷, 𝐶〉}) | ||
| Theorem | brprop 32706 | Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))) | ||
| Theorem | mptprop 32707* | Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷))) | ||
| Theorem | coprprop 32708 | Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐸 ≠ 𝐹) ⇒ ⊢ (𝜑 → ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∘ {〈𝐸, 𝐴〉, 〈𝐹, 𝐶〉}) = {〈𝐸, 𝐵〉, 〈𝐹, 𝐷〉}) | ||
| Theorem | fmptunsnop 32709* | Two ways to express a function with a value replaced. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | ||
| Theorem | gtiso 32710 | Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
| ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → (𝐹 Isom < , ◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ◡ ≤ (𝐴, 𝐵))) | ||
| Theorem | isoun 32711* | Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.) |
| ⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤) & ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) ⇒ ⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷))) | ||
| Theorem | disjdsct 32712* | A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 6635) (Contributed by Thierry Arnoux, 28-Feb-2017.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅})) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) | ||
| Theorem | df1stres 32713* | Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) | ||
| Theorem | df2ndres 32714* | Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) | ||
| Theorem | 1stpreimas 32715 | The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
| ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋}))) | ||
| Theorem | 1stpreima 32716 | The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
| ⊢ (𝐴 ⊆ 𝐵 → (◡(1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶)) | ||
| Theorem | 2ndpreima 32717 | The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
| ⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴)) | ||
| Theorem | curry2ima 32718* | The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
| ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) ⇒ ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)}) | ||
| Theorem | preiman0 32719 | The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅) | ||
| Theorem | intimafv 32720* | The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) | ||
| Theorem | supssd 32721* | Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
| ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅)) | ||
| Theorem | infssd 32722* | Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.) |
| ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)) | ||
| Theorem | imafi2 32723 | The image by a finite set is finite. See also imafi 9353. (Contributed by Thierry Arnoux, 25-Apr-2020.) |
| ⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin) | ||
| Theorem | unifi3 32724 | If a union is finite, then all its elements are finite. See unifi 9384. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
| ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin) | ||
| Theorem | snct 32725 | A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
| ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) | ||
| Theorem | prct 32726 | An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω) | ||
| Theorem | mpocti 32727* | An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
| ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 ⇒ ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ≼ ω) | ||
| Theorem | abrexct 32728* | An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
| ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) | ||
| Theorem | mptctf 32729 | A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | ||
| Theorem | abrexctf 32730* | An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) | ||
| Theorem | padct 32731* | Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.) |
| ⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) | ||
| Theorem | f1od2 32732* | Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌)) & ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) ⇒ ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) | ||
| Theorem | fcobij 32733* | Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.) |
| ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑m 𝑅)–1-1-onto→(𝑇 ↑m 𝑅)) | ||
| Theorem | fcobijfs 32734* | Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9448. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) & ⊢ (𝜑 → 𝑂 ∈ 𝑆) & ⊢ 𝑄 = (𝐺‘𝑂) & ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} & ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄} ⇒ ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) | ||
| Theorem | suppss3 32735* | Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍) ⇒ ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | ||
| Theorem | fsuppcurry1 32736* | Finite support of a curried function with a constant first argument. (Contributed by Thierry Arnoux, 7-Jul-2023.) |
| ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) ⇒ ⊢ (𝜑 → 𝐺 finSupp 𝑍) | ||
| Theorem | fsuppcurry2 32737* | Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.) |
| ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 finSupp 𝑍) ⇒ ⊢ (𝜑 → 𝐺 finSupp 𝑍) | ||
| Theorem | offinsupp1 32738* | Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑇) & ⊢ (𝜑 → 𝐹 finSupp 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) | ||
| Theorem | ffs2 32739 | Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8200. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ 𝐶 = (𝐵 ∖ {𝑍}) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶)) | ||
| Theorem | ffsrn 32740 | The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
| ⊢ (𝜑 → 𝑍 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ Fin) | ||
| Theorem | resf1o 32741* | Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
| ⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶} & ⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶)) ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶)) | ||
| Theorem | maprnin 32742* | Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} | ||
| Theorem | fpwrelmapffslem 32743* | Lemma for fpwrelmapffs 32745. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵) & ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}) ⇒ ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin))) | ||
| Theorem | fpwrelmap 32744* | Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 10488 and marypha2lem1 9475. (Contributed by Thierry Arnoux, 28-Aug-2017.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ⇒ ⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵) | ||
| Theorem | fpwrelmapffs 32745* | Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) & ⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} ⇒ ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) | ||
| Theorem | creq0 32746 | The real representation of complex numbers is zero iff both its terms are zero. Cf. crne0 12259. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ (𝐴 + (i · 𝐵)) = 0)) | ||
| Theorem | 1nei 32747 | The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ 1 ≠ i | ||
| Theorem | 1neg1t1neg1 32748 | An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
| ⊢ (𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1) | ||
| Theorem | nnmulge 32749 | Multiplying by a positive integer 𝑀 yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁)) | ||
| Theorem | submuladdd 32750 | The product of a difference and a sum. Cf. addmulsub 11725. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐴 · 𝐷)) − ((𝐵 · 𝐶) + (𝐵 · 𝐷)))) | ||
| Theorem | muldivdid 32751 | Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (((𝐴 · 𝐵) + 𝐶) / 𝐵) = (𝐴 + (𝐶 / 𝐵))) | ||
| Theorem | cjsubd 32752 | Complex conjugate distributes over subtraction. (Contributed by Thierry Arnoux, 1-Jul-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) | ||
| Theorem | re0cj 32753 | The conjugate of a pure imaginary number is its negative. (Contributed by Thierry Arnoux, 25-Jun-2025.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (ℜ‘𝐴) = 0) ⇒ ⊢ (𝜑 → (∗‘𝐴) = -𝐴) | ||
| Theorem | quad3d 32754 | Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) Deduction version. (Revised by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0) ⇒ ⊢ (𝜑 → (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))) | ||
| Theorem | lt2addrd 32755* | If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶)) | ||
| Theorem | xrlelttric 32756 | Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
| Theorem | xaddeq0 32757 | Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵)) | ||
| Theorem | xrinfm 32758 | The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
| ⊢ inf(ℝ*, ℝ*, < ) = -∞ | ||
| Theorem | le2halvesd 32759 | A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ (𝐶 / 2)) & ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 2)) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≤ 𝐶) | ||
| Theorem | xraddge02 32760 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +𝑒 𝐵))) | ||
| Theorem | xrge0addge 32761 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵)) | ||
| Theorem | xlt2addrd 32762* | If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ≠ -∞) & ⊢ (𝜑 → 𝐶 ≠ -∞) & ⊢ (𝜑 → 𝐴 < (𝐵 +𝑒 𝐶)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ* ∃𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶)) | ||
| Theorem | xrsupssd 32763 | Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
| ⊢ (𝜑 → 𝐵 ⊆ 𝐶) & ⊢ (𝜑 → 𝐶 ⊆ ℝ*) ⇒ ⊢ (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < )) | ||
| Theorem | xrge0infss 32764* | Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
| ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | ||
| Theorem | xrge0infssd 32765 | Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
| ⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ (0[,]+∞)) ⇒ ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < )) | ||
| Theorem | xrge0addcld 32766 | Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞)) | ||
| Theorem | xrge0subcld 32767 | Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) | ||
| Theorem | infxrge0lb 32768 | A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) | ||
| Theorem | infxrge0glb 32769* | The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) | ||
| Theorem | infxrge0gelb 32770* | The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
| ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) | ||
| Theorem | xrofsup 32771 | The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.) |
| ⊢ (𝜑 → 𝑋 ⊆ ℝ*) & ⊢ (𝜑 → 𝑌 ⊆ ℝ*) & ⊢ (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞) & ⊢ (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞) & ⊢ (𝜑 → 𝑍 = ( +𝑒 “ (𝑋 × 𝑌))) ⇒ ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < ))) | ||
| Theorem | supxrnemnf 32772 | The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
| ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞) | ||
| Theorem | xnn0gt0 32773 | Nonzero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝑁 ∈ ℕ0* ∧ 𝑁 ≠ 0) → 0 < 𝑁) | ||
| Theorem | xnn01gt 32774 | An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.) |
| ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁)) | ||
| Theorem | nn0xmulclb 32775 | Finite multiplication in the extended nonnegative integers. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 ·e 𝐵) ∈ ℕ0 ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0))) | ||
| Theorem | joiniooico 32776 | Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))) | ||
| Theorem | ubico 32777 | A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴[,)𝐵)) | ||
| Theorem | xeqlelt 32778 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
| Theorem | eliccelico 32779 | Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵))) | ||
| Theorem | elicoelioo 32780 | Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)))) | ||
| Theorem | iocinioc2 32781 | Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶)) | ||
| Theorem | xrdifh 32782 | Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.) |
| ⊢ 𝐴 ∈ ℝ* ⇒ ⊢ (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴) | ||
| Theorem | iocinif 32783 | Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶))) | ||
| Theorem | difioo 32784 | The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶)) | ||
| Theorem | difico 32785 | The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵)) | ||
| Theorem | uzssico 32786 | Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) | ||
| Theorem | fz2ssnn0 32787 | A finite set of sequential integers that is a subset of ℕ0. (Contributed by Thierry Arnoux, 8-Dec-2021.) |
| ⊢ (𝑀 ∈ ℕ0 → (𝑀...𝑁) ⊆ ℕ0) | ||
| Theorem | nndiffz1 32788 | Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.) |
| ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1))) | ||
| Theorem | ssnnssfz 32789* | For any finite subset of ℕ, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.) |
| ⊢ (𝐴 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛)) | ||
| Theorem | fzm1ne1 32790 | Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
| ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1))) | ||
| Theorem | fzspl 32791 | Split the last element of a finite set of sequential integers. More generic than fzsuc 13611. (Contributed by Thierry Arnoux, 7-Nov-2016.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) | ||
| Theorem | fzdif2 32792 | Split the last element of a finite set of sequential integers. More generic than fzsuc 13611. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1))) | ||
| Theorem | fzodif2 32793 | Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀..^(𝑁 + 1)) ∖ {𝑁}) = (𝑀..^𝑁)) | ||
| Theorem | fzodif1 32794 | Set difference of two half-open range of sequential integers sharing the same starting value. (Contributed by Thierry Arnoux, 2-Oct-2023.) |
| ⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀..^𝑁) ∖ (𝑀..^𝐾)) = (𝐾..^𝑁)) | ||
| Theorem | fzsplit3 32795 | Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.) |
| ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁))) | ||
| Theorem | elfzodif0 32796 | If an integer 𝑀 is in an open interval starting at 0, except 0, then (𝑀 − 1) is also in that interval. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ (𝜑 → 𝑀 ∈ ((0..^𝑁) ∖ {0})) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑀 − 1) ∈ (0..^𝑁)) | ||
| Theorem | bcm1n 32797 | The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.) |
| ⊢ ((𝐾 ∈ (0...(𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (((𝑁 − 1)C𝐾) / (𝑁C𝐾)) = ((𝑁 − 𝐾) / 𝑁)) | ||
| Theorem | iundisjfi 32798* | Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 25583. (Contributed by Thierry Arnoux, 15-Feb-2017.) |
| ⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
| Theorem | iundisj2fi 32799* | A disjoint union is disjoint, finite version. Cf. iundisj2 25584. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
| ⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) ⇒ ⊢ Disj 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) | ||
| Theorem | iundisjcnt 32800* | Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.) |
| ⊢ Ⅎ𝑛𝐵 & ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀))) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) | ||
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