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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremresvval 29801 Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))

Theoremresvid2 29802 General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremresvval2 29803 Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))

Theoremresvsca 29804 Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝐴𝑉 → (𝐹s 𝐴) = (Scalar‘𝑅))

Theoremresvlem 29805 Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 5       (𝐴𝑉𝐶 = (𝐸𝑅))

Theoremresvbas 29806 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))

Theoremresvplusg 29807 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))

Theoremresvvsca 29808 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremresvmulr 29809 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))

Theoremresv0g 29810 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))

Theoremresv1r 29811 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))

Theoremresvcmn 29812 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))

20.3.9.11  The commutative ring of gaussian integers

Theoremgzcrng 29813 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing

20.3.9.12  The archimedean ordered field of real numbers

Theoremreofld 29814 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField

Theoremnn0omnd 29815 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd

Theoremrearchi 29816 The field of the real numbers is Archimedean. See also arch 11274. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi

Theoremnn0archi 29817 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi

Theoremxrge0slmod 29818 The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   𝑊 = (𝐺v (0[,)+∞))       𝑊 ∈ SLMod

20.3.10  Matrices

20.3.10.1  The symmetric group

Theoremsymgfcoeu 29819* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))

20.3.10.2  Permutation Signs

Theorempsgndmfi 29820 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgn‘𝐷)    &   𝐺 = (Base‘(SymGrp‘𝐷))       (𝐷 ∈ Fin → 𝑆 Fn 𝐺)

Theorempsgnid 29821 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)

Theorempmtrprfv2 29822 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)

Theorempmtrto1cl 29823 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑇 = (pmTrsp‘𝐷)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)

Theorempsgnfzto1stlem 29824* Lemma for psgnfzto1st 29829. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖𝐾, (𝑖 − 1), 𝑖)))))

Theoremfzto1stfv1 29825* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼𝐷 → (𝑃‘1) = 𝐼)

Theoremfzto1st1 29826* Special case where the permutation defined in psgnfzto1st 29829 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷))

Theoremfzto1st 29827* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷𝑃𝐵)

Theoremfzto1stinvn 29828* Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷 → (𝑃𝐼) = 1)

Theorempsgnfzto1st 29829* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))

20.3.10.3  Transpositions

Theorempmtridf1o 29830 Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑𝑇:𝐴1-1-onto𝐴)

Theorempmtridfv1 29831 Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑋) = 𝑌)

Theorempmtridfv2 29832 Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑌) = 𝑋)

20.3.10.4  Submatrices

Syntaxcsmat 29833 Syntax for a function generating submatrixes.
class subMat1

Definitiondf-smat 29834* Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 20364. (Contributed by Thierry Arnoux, 18-Aug-2020.)
subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))

Theoremsmatfval 29835* Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))

Theoremsmatrcl 29836 Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))       (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Theoremsmatlem 29837 Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)       (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Theoremsmattl 29838 Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))

Theoremsmattr 29839 Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))

Theoremsmatbl 29840 Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))

Theoremsmatbr 29841 Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))

Theoremsmatcl 29842 Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indexes in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑𝑆𝐶)

Theoremmatmpt2 29843* Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀𝑗)))

Theorem1smat1 29844 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20370. (Contributed by Thierry Arnoux, 19-Aug-2020.)
1 = (1r‘((1...𝑁) Mat 𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))       (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Theoremsubmat1n 29845 One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))

Theoremsubmatres 29846 Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))

Theoremsubmateqlem1 29847 Lemma for submateq 29849. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝐾𝑀)       (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateqlem2 29848 Lemma for submateq 29849. (Contributed by Thierry Arnoux, 26-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 < 𝐾)       (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateq 29849* Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))       (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Theoremsubmatminr1 29850 If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝐵)    &   𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)       (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))

20.3.10.5  Matrix literals

Syntaxclmat 29851 Extend class notation with the literal matrix conversion function.
class litMat

Definitiondf-lmat 29852* Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))

Theoremlmatval 29853* Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
(𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))

Theoremlmatfval 29854* Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))       (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))

Theoremlmatfvlem 29855* Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝐾 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   𝐼𝑁    &   𝐽𝑁    &   (𝐾 + 1) = 𝐼    &   (𝐿 + 1) = 𝐽    &   (𝑊𝐾) = 𝑋    &   (𝜑 → (𝑋𝐿) = 𝑌)       (𝜑 → (𝐼𝑀𝐽) = 𝑌)

Theoremlmatcl 29856* Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝑉 = (Base‘𝑅)    &   𝑂 = ((1...𝑁) Mat 𝑅)    &   𝑃 = (Base‘𝑂)    &   (𝜑𝑅𝑋)       (𝜑𝑀𝑃)

Theoremlmat22lem 29857* Lemma for lmat22e11 29858 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       ((𝜑𝑖 ∈ (0..^2)) → (#‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)

Theoremlmat22e11 29858 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀1) = 𝐴)

Theoremlmat22e12 29859 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀2) = 𝐵)

Theoremlmat22e21 29860 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀1) = 𝐶)

Theoremlmat22e22 29861 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀2) = 𝐷)

Theoremlmat22det 29862 The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑉 = (Base‘𝑅)    &   𝐽 = ((1...2) maDet 𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝐽𝑀) = ((𝐴 · 𝐷) (𝐶 · 𝐵)))

20.3.10.6  Laplace expansion of determinants

Theoremmdetpmtr1 29863* The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐺 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑍 = (ℤRHom‘𝑅)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑃𝑖)𝑀𝑗))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))

Theoremmdetpmtr2 29864* The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐺 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑍 = (ℤRHom‘𝑅)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀(𝑃𝑗)))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))

Theoremmdetpmtr12 29865* The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐺 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑍 = (ℤRHom‘𝑅)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑃𝑖)𝑀(𝑄𝑗)))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑀𝐵)    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)       (𝜑 → (𝐷𝑀) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝐸)))

Theoremmdetlap1 29866* A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐾 = (𝑁 maAdju 𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝐼𝑁) → (𝐷𝑀) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾𝑀)𝐼)))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   𝑆 = (pmSgn‘(1...𝑁))    &   𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗)))    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)    &   (𝜑 → (𝑃𝑁) = 𝐼)    &   (𝜑 → (𝑄𝑁) = 𝐽)    &   (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))       ((𝜑𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃𝑆)‘𝑋))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))    &   (𝜑𝑈𝐵)       (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmadjusmdet 29871 Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrixes. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmdetlap 29872* Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐷𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))

20.3.11  Topology

20.3.11.1  Open maps

Theoremfvproj 29873* Value of a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)

Theoremfimaproj 29874* Image of a cartesian product for a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))

Theoremtxomap 29875* Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.)
(𝜑𝐹:𝑋𝑍)    &   (𝜑𝐺:𝑌𝑇)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑇))    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐿)    &   ((𝜑𝑦𝐾) → (𝐺𝑦) ∈ 𝑀)    &   (𝜑𝐴 ∈ (𝐽 ×t 𝐾))    &   𝐻 = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)       (𝜑 → (𝐻𝐴) ∈ (𝐿 ×t 𝑀))

20.3.11.2  Topology of the unit circle

Theoremqtopt1 29876* If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Fre)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))       (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

Theoremqtophaus 29877* If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
𝑋 = 𝐽    &    = (𝐹𝐹)    &   𝐻 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)    &   (𝜑𝐽 ∈ Haus)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))    &   (𝜑 ∈ (Clsd‘(𝐽 ×t 𝐽)))       (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)

Theoremcirctopn 29878* The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.)
𝐼 = (0[,](2 · π))    &   𝐽 = (topGen‘ran (,))    &   𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   𝐶 = (abs “ {1})       (𝐽 qTop 𝐹) = (TopOpen‘(𝐹sfld))

Theoremcirccn 29879* The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝐼 = (0[,](2 · π))    &   𝐽 = (topGen‘ran (,))    &   𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   𝐶 = (abs “ {1})       𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))

20.3.11.3  Refinements

Theoremreff 29880* For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a defintion of refinement. Note that this definition uses the axiom of choice through ac6sg 9295. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))

Theoremlocfinreflem 29881* A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))

Theoremlocfinref 29882* A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

20.3.11.4  Open cover refinement property

Syntaxccref 29883 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴

Definitiondf-cref 29884* Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}

Theoremiscref 29885* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽       (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))

Theoremcrefeq 29886 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)

Theoremcreftop 29887 A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)

Theoremcrefi 29888* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽       ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)

Theoremcrefdf 29889* A formulation of crefi 29888 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))

Theoremcrefss 29890 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Theoremcmpcref 29891 Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Comp = CovHasRefFin

Theoremcmpfiref 29892* Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))

20.3.11.5  Lindelöf spaces

Syntaxcldlf 29893 Extend class notation with the class of all Lindelöf spaces.
class Ldlf

Definitiondf-ldlf 29894 Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
Ldlf = CovHasRef{𝑥𝑥 ≼ ω}

Theoremldlfcntref 29895* Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

20.3.11.6  Paracompact spaces

Syntaxcpcmp 29896 Extend class notation with the class of all paracompact topologies.
class Paracomp

Definitiondf-pcmp 29897 Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}

Theoremispcmp 29898 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Theoremcmppcmp 29899 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Theoremdispcmp 29900 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)

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