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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | czn 14101 | The ring of integers modulo 𝑛. |
class ℤ/nℤ | ||
Definition | df-zrh 14102 | Define the unique homomorphism from the integers into a ring. This encodes the usual notation of 𝑛 = 1r + 1r + ... + 1r for integers (see also df-mulg 13190). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | ||
Definition | df-zlm 14103 | Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉)) | ||
Definition | df-zn 14104* | Define the ring of integers mod 𝑛. This is literally the quotient ring of ℤ by the ideal 𝑛ℤ, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤ/nℤ = (𝑛 ∈ ℕ0 ↦ ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet 〈(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)〉)) | ||
Theorem | zrhval 14105 | Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) | ||
Theorem | zrhvalg 14106 | Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐿 = ∪ (ℤring RingHom 𝑅)) | ||
Theorem | zrhval2 14107* | Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) | ||
Theorem | zrhmulg 14108 | Value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁 · 1 )) | ||
Theorem | zrhex 14109 | Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐿 ∈ V) | ||
Theorem | zrhrhmb 14110 | The ℤRHom homomorphism is the unique ring homomorphism from ℤ. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐹 ∈ (ℤring RingHom 𝑅) ↔ 𝐹 = 𝐿)) | ||
Theorem | zrhrhm 14111 | The ℤRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) | ||
Theorem | zrh1 14112 | Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿‘1) = 1 ) | ||
Theorem | zrh0 14113 | Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿‘0) = 0 ) | ||
Theorem | zrhpropd 14114* | The ℤ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿)) | ||
Theorem | zlmval 14115 | Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) | ||
Theorem | zlmlemg 14116 | Lemma for zlmbasg 14117 and zlmplusgg 14118. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ∈ ℕ & ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) & ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑊)) | ||
Theorem | zlmbasg 14117 | Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐵 = (Base‘𝑊)) | ||
Theorem | zlmplusgg 14118 | Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → + = (+g‘𝑊)) | ||
Theorem | zlmmulrg 14119 | Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.r‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → · = (.r‘𝑊)) | ||
Theorem | zlmsca 14120 | Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.) |
⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) | ||
Theorem | zlmvscag 14121 | Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → · = ( ·𝑠 ‘𝑊)) | ||
Theorem | znlidl 14122 | The set 𝑛ℤ is an ideal in ℤ. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) ⇒ ⊢ (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) | ||
Theorem | zncrng2 14123 | Making a commutative ring as a quotient of ℤ and 𝑛ℤ. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ⇒ ⊢ (𝑁 ∈ ℤ → 𝑈 ∈ CRing) | ||
Theorem | znval 14124 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znle 14125 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znval2 14126 | Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znbaslemnn 14127 | Lemma for znbas 14132. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ∈ ℕ & ⊢ (𝐸‘ndx) ≠ (le‘ndx) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) | ||
Theorem | znbas2 14128 | The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) | ||
Theorem | znadd 14129 | The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) | ||
Theorem | znmul 14130 | The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) | ||
Theorem | znzrh 14131 | The ℤ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) | ||
Theorem | znbas 14132 | The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑅 = (ℤring ~QG (𝑆‘{𝑁})) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤ / 𝑅) = (Base‘𝑌)) | ||
Theorem | zncrng 14133 | ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) | ||
Theorem | znzrh2 14134* | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) | ||
Theorem | znzrhval 14135 | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴] ∼ ) | ||
Theorem | znzrhfo 14136 | The ℤ ring homomorphism is a surjection onto ℤ/nℤ. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) | ||
Theorem | zndvds 14137 | Express equality of equivalence classes in ℤ / 𝑛ℤ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵))) | ||
Theorem | zndvds0 14138 | Special case of zndvds 14137 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 0 = (0g‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) | ||
Theorem | znf1o 14139 | The function 𝐹 enumerates all equivalence classes in ℤ/nℤ for each 𝑛. When 𝑛 = 0, ℤ / 0ℤ = ℤ / {0} ≈ ℤ so we let 𝑊 = ℤ; otherwise 𝑊 = {0, ..., 𝑛 − 1} enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝑊–1-1-onto→𝐵) | ||
Theorem | znle2 14140 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znleval 14141 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) | ||
Theorem | znleval2 14142 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) | ||
Theorem | znfi 14143 | The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin) | ||
Theorem | znhash 14144 | The ℤ/nℤ structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁) | ||
Theorem | znidom 14145 | The ℤ/nℤ structure is an integral domain when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ IDomn) | ||
Theorem | znidomb 14146 | The ℤ/nℤ structure is a domain precisely when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑌 ∈ IDomn ↔ 𝑁 ∈ ℙ)) | ||
Theorem | znunit 14147 | The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) ∈ 𝑈 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | znrrg 14148 | The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for 𝑁 = 0, where all nonzero integers are regular, but only ±1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐸 = 𝑈) | ||
According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 13495), but the existence of a unity element is always assumed (our rings are unital, see df-ring 13494). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space. | ||
Syntax | cmps 14149 | Multivariate power series. |
class mPwSer | ||
Definition | df-psr 14150* | Define the algebra of power series over the index set 𝑖 and with coefficients from the ring 𝑟. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))〉} ∪ {〈(Scalar‘ndx), 𝑟〉, 〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))〉, 〈(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))〉})) | ||
Theorem | reldmpsr 14151 | The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ Rel dom mPwSer | ||
Theorem | psrval 14152* | Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (TopOpen‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷)) & ⊢ ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵)) & ⊢ × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) & ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) & ⊢ (𝜑 → 𝐽 = (∏t‘(𝐷 × {𝑂}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑆 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), ✚ 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑅〉, 〈( ·𝑠 ‘ndx), ∙ 〉, 〈(TopSet‘ndx), 𝐽〉})) | ||
Theorem | fnpsr 14153 | The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.) |
⊢ mPwSer Fn (V × V) | ||
Theorem | psrvalstrd 14154 | The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → + ∈ 𝑌) & ⊢ (𝜑 → × ∈ 𝑍) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → · ∈ 𝑃) & ⊢ (𝜑 → 𝐽 ∈ 𝑄) ⇒ ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑅〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(TopSet‘ndx), 𝐽〉}) Struct 〈1, 9〉) | ||
Theorem | psrbag 14155* | Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) | ||
Theorem | psrbagf 14156* | A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.) |
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ⇒ ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) | ||
Theorem | fczpsrbag 14157* | The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.) |
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ 0) ∈ 𝐷) | ||
Theorem | psrbaglesuppg 14158* | The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ)) | ||
Theorem | psrbasg 14159* | The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷)) | ||
Theorem | psrelbas 14160* | An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋:𝐷⟶𝐾) | ||
Theorem | psrelbasfun 14161 | An element of the set of power series is a function. (Contributed by AV, 17-Jul-2019.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝑋 ∈ 𝐵 → Fun 𝑋) | ||
Theorem | psrplusgg 14162 | The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑆) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵))) | ||
Theorem | psradd 14163 | The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ✚ 𝑌) = (𝑋 ∘𝑓 + 𝑌)) | ||
Theorem | psraddcl 14164 | Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Mgm) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) | ||
A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set. | ||
Syntax | ctop 14165 | Syntax for the class of topologies. |
class Top | ||
Definition | df-top 14166* |
Define the class of topologies. It is a proper class. See istopg 14167 and
istopfin 14168 for the corresponding characterizations,
using respectively
binary intersections like in this definition and nonempty finite
intersections.
The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.) |
⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} | ||
Theorem | istopg 14167* |
Express the predicate "𝐽 is a topology". See istopfin 14168 for
another characterization using nonempty finite intersections instead of
binary intersections.
Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | ||
Theorem | istopfin 14168* | Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 14167. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.) |
⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽))) | ||
Theorem | uniopn 14169 | The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) | ||
Theorem | iunopn 14170* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | inopn 14171 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
Theorem | fiinopn 14172 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽)) | ||
Theorem | unopn 14173 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) | ||
Theorem | 0opn 14174 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | ||
Theorem | 0ntop 14175 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
⊢ ¬ ∅ ∈ Top | ||
Theorem | topopn 14176 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | ||
Theorem | eltopss 14177 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Syntax | ctopon 14178 | Syntax for the function of topologies on sets. |
class TopOn | ||
Definition | df-topon 14179* | Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | ||
Theorem | funtopon 14180 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
⊢ Fun TopOn | ||
Theorem | istopon 14181 | Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | ||
Theorem | topontop 14182 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | ||
Theorem | toponuni 14183 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | ||
Theorem | topontopi 14184 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | toponunii 14185 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐵 = ∪ 𝐽 | ||
Theorem | toptopon 14186 | Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | toptopon2 14187 | A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | topontopon 14188 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | toponrestid 14189 | Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
⊢ 𝐴 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐴 = (𝐴 ↾t 𝐵) | ||
Theorem | toponsspwpwg 14190 | The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.) |
⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) | ||
Theorem | dmtopon 14191 | The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.) |
⊢ dom TopOn = V | ||
Theorem | fntopon 14192 | The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.) |
⊢ TopOn Fn V | ||
Theorem | toponmax 14193 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) | ||
Theorem | toponss 14194 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | toponcom 14195 | If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾)) | ||
Theorem | toponcomb 14196 | Biconditional form of toponcom 14195. (Contributed by BJ, 5-Dec-2021.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
Theorem | topgele 14197 | The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | ||
Syntax | ctps 14198 | Syntax for the class of topological spaces. |
class TopSp | ||
Definition | df-topsp 14199 | Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.) |
⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | ||
Theorem | istps 14200 | Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
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