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Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxczn 14101 The ring of integers modulo 𝑛.
class ℤ/n
 
Definitiondf-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))
 
Definitiondf-zlm 14103 Augment an abelian group with vector space operations to turn it into a -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), (.g𝑔)⟩))
 
Definitiondf-zn 14104* Define the ring of integers mod 𝑛. This is literally the quotient ring of by the ideal 𝑛, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
ℤ/nℤ = (𝑛 ∈ ℕ0ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩))
 
Theoremzrhval 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)
 
Theoremzrhvalg 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)       (𝑅𝑉𝐿 = (ℤring RingHom 𝑅))
 
Theoremzrhval2 14107* Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )))
 
Theoremzrhmulg 14108 Value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿𝑁) = (𝑁 · 1 ))
 
Theoremzrhex 14109 Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
𝐿 = (ℤRHom‘𝑅)       (𝑅𝑉𝐿 ∈ V)
 
Theoremzrhrhmb 14110 The ℤRHom homomorphism is the unique ring homomorphism from . (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (ℤRHom‘𝑅)       (𝑅 ∈ Ring → (𝐹 ∈ (ℤring RingHom 𝑅) ↔ 𝐹 = 𝐿))
 
Theoremzrhrhm 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))
 
Theoremzrh1 14112 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐿‘1) = 1 )
 
Theoremzrh0 14113 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿‘0) = 0 )
 
Theoremzrhpropd 14114* The ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿))
 
Theoremzlmval 14115 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)       (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))
 
Theoremzlmlemg 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)       (𝐺𝑉 → (𝐸𝐺) = (𝐸𝑊))
 
Theoremzlmbasg 14117 Base set of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)       (𝐺𝑉𝐵 = (Base‘𝑊))
 
Theoremzlmplusgg 14118 Group operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g𝐺)       (𝐺𝑉+ = (+g𝑊))
 
Theoremzlmmulrg 14119 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r𝐺)       (𝐺𝑉· = (.r𝑊))
 
Theoremzlmsca 14120 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
𝑊 = (ℤMod‘𝐺)       (𝐺𝑉 → ℤring = (Scalar‘𝑊))
 
Theoremzlmvscag 14121 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)       (𝐺𝑉· = ( ·𝑠𝑊))
 
Theoremznlidl 14122 The set 𝑛 is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
𝑆 = (RSpan‘ℤring)       (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring))
 
Theoremzncrng2 14123 Making a commutative ring as a quotient of and 𝑛. (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)
 
Theoremznval 14124 The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added. (In other words, ℤ/n 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 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
 
Theoremznle 14125 The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added. (In other words, ℤ/n 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))
 
Theoremznval2 14126 Self-referential expression for the ℤ/n 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 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
 
Theoremznbaslemnn 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 → (𝐸𝑈) = (𝐸𝑌))
 
Theoremznbas2 14128 The base set of ℤ/n 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))
 
Theoremznadd 14129 The additive structure of ℤ/n 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 → (+g𝑈) = (+g𝑌))
 
Theoremznmul 14130 The multiplicative structure of ℤ/n 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 → (.r𝑈) = (.r𝑌))
 
Theoremznzrh 14131 The ring homomorphism of ℤ/n 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))
 
Theoremznbas 14132 The base set of ℤ/n 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))
 
Theoremzncrng 14133 ℤ/n is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)       (𝑁 ∈ ℕ0𝑌 ∈ CRing)
 
Theoremznzrh2 14134* The 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 ∈ ℕ0𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ))
 
Theoremznzrhval 14135 The 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 ∈ ℕ0𝐴 ∈ ℤ) → (𝐿𝐴) = [𝐴] )
 
Theoremznzrhfo 14136 The ring homomorphism is a surjection onto ℤ/n. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝐿 = (ℤRHom‘𝑌)       (𝑁 ∈ ℕ0𝐿:ℤ–onto𝐵)
 
Theoremzndvds 14137 Express equality of equivalence classes in ℤ / 𝑛 in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐿 = (ℤRHom‘𝑌)       ((𝑁 ∈ ℕ0𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿𝐴) = (𝐿𝐵) ↔ 𝑁 ∥ (𝐴𝐵)))
 
Theoremzndvds0 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 = (0g𝑌)       ((𝑁 ∈ ℕ0𝐴 ∈ ℤ) → ((𝐿𝐴) = 0𝑁𝐴))
 
Theoremznf1o 14139 The function 𝐹 enumerates all equivalence classes in ℤ/n 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 ∈ ℕ0𝐹:𝑊1-1-onto𝐵)
 
Theoremznle2 14140 The ordering of the ℤ/n 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))
 
Theoremznleval 14141 The ordering of the ℤ/n 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 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐹𝐴) ≤ (𝐹𝐵))))
 
Theoremznleval2 14142 The ordering of the ℤ/n 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 ∈ ℕ0𝐴𝑋𝐵𝑋) → (𝐴 𝐵 ↔ (𝐹𝐴) ≤ (𝐹𝐵)))
 
Theoremznfi 14143 The ℤ/n structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)       (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
 
Theoremznhash 14144 The ℤ/n structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)       (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁)
 
Theoremznidom 14145 The ℤ/n 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)
 
Theoremznidomb 14146 The ℤ/n 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))
 
Theoremznunit 14147 The units of ℤ/n 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 ∈ ℕ0𝐴 ∈ ℤ) → ((𝐿𝐴) ∈ 𝑈 ↔ (𝐴 gcd 𝑁) = 1))
 
Theoremznrrg 14148 The regular elements of ℤ/n 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)
 
PART 8  BASIC LINEAR ALGEBRA

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.

 
8.1  Abstract multivariate polynomials
 
8.1.1  Definition and basic properties
 
Syntaxcmps 14149 Multivariate power series.
class mPwSer
 
Definitiondf-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 ↦ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑𝑚 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
 
Theoremreldmpsr 14151 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPwSer
 
Theorempsrval 14152* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝑂 = (TopOpen‘𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐵 = (𝐾𝑚 𝐷))    &    = ( ∘𝑓 + ↾ (𝐵 × 𝐵))    &    × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦𝑟𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘𝑓𝑥)))))))    &    = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))    &   (𝜑𝐽 = (∏t‘(𝐷 × {𝑂})))    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑋)       (𝜑𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
 
Theoremfnpsr 14153 The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
mPwSer Fn (V × V)
 
Theorempsrvalstrd 14154 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵𝑋)    &   (𝜑+𝑌)    &   (𝜑×𝑍)    &   (𝜑𝑅𝑊)    &   (𝜑·𝑃)    &   (𝜑𝐽𝑄)       (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) Struct ⟨1, 9⟩)
 
Theorempsrbag 14155* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝐹𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (𝐹 “ ℕ) ∈ Fin)))
 
Theorempsrbagf 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐹𝐷𝐹:𝐼⟶ℕ0)
 
Theoremfczpsrbag 14157* The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.)
𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝑥𝐼 ↦ 0) ∈ 𝐷)
 
Theorempsrbaglesuppg 14158* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉 ∧ (𝐹𝐷𝐺:𝐼⟶ℕ0𝐺𝑟𝐹)) → (𝐺 “ ℕ) ⊆ (𝐹 “ ℕ))
 
Theorempsrbasg 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝐵 = (𝐾𝑚 𝐷))
 
Theorempsrelbas 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 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)       (𝜑𝑋:𝐷𝐾)
 
Theorempsrelbasfun 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)       (𝑋𝐵 → Fun 𝑋)
 
Theorempsrplusgg 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)    &    + = (+g𝑅)    &    = (+g𝑆)       ((𝐼𝑉𝑅𝑊) → = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
 
Theorempsradd 14163 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))
 
Theorempsraddcl 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)    &    + = (+g𝑆)    &   (𝜑𝑅 ∈ Mgm)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
PART 9  BASIC TOPOLOGY
 
9.1  Topology
 
9.1.1  Topological spaces

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.

 
9.1.1.1  Topologies
 
Syntaxctop 14165 Syntax for the class of topologies.
class Top
 
Definitiondf-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 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
 
Theoremistopg 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.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
 
Theoremistopfin 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(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽)))
 
Theoremuniopn 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 ∧ 𝐴𝐽) → 𝐴𝐽)
 
Theoremiunopn 14170* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
 
Theoreminopn 14171 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremfiinopn 14172 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))
 
Theoremunopn 14173 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theorem0opn 14174 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)
 
Theorem0ntop 14175 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top
 
Theoremtopopn 14176 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)
 
Theoremeltopss 14177 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
 
9.1.1.2  Topologies on sets
 
Syntaxctopon 14178 Syntax for the function of topologies on sets.
class TopOn
 
Definitiondf-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})
 
Theoremfuntopon 14180 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn
 
Theoremistopon 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))
 
Theoremtopontop 14182 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
 
Theoremtoponuni 14183 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
 
Theoremtopontopi 14184 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top
 
Theoremtoponunii 14185 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽
 
Theoremtoptopon 14186 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
 
Theoremtoptopon2 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))
 
Theoremtopontopon 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))
 
Theoremtoponrestid 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 = (𝐴t 𝐵)
 
Theoremtoponsspwpwg 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.)
(𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
 
Theoremdmtopon 14191 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V
 
Theoremfntopon 14192 The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V
 
Theoremtoponmax 14193 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)
 
Theoremtoponss 14194 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremtoponcom 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))
 
Theoremtoponcomb 14196 Biconditional form of toponcom 14195. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))
 
Theoremtopgele 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} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
 
9.1.1.3  Topological spaces
 
Syntaxctps 14198 Syntax for the class of topological spaces.
class TopSp
 
Definitiondf-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))}
 
Theoremistps 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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