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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prdsdsval3 17301* | Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.) |
β’ π = (πXs(π₯ β πΌ β¦ π )) & β’ π΅ = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β βπ₯ β πΌ π β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ πΎ = (Baseβπ ) & β’ πΈ = ((distβπ ) βΎ (πΎ Γ πΎ)) & β’ π· = (distβπ) β β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) βͺ {0}), β*, < )) | ||
Theorem | pwsval 17302 | Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ πΉ = (Scalarβπ ) β β’ ((π β π β§ πΌ β π) β π = (πΉXs(πΌ Γ {π }))) | ||
Theorem | pwsbas 17303 | Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) β β’ ((π β π β§ πΌ β π) β (π΅ βm πΌ) = (Baseβπ)) | ||
Theorem | pwselbasb 17304 | Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ π = (Baseβπ) β β’ ((π β π β§ πΌ β π) β (π β π β π:πΌβΆπ΅)) | ||
Theorem | pwselbas 17305 | An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π β π) β β’ (π β π:πΌβΆπ΅) | ||
Theorem | pwsplusgval 17306 | Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπ) β β’ (π β (πΉ β πΊ) = (πΉ βf + πΊ)) | ||
Theorem | pwsmulrval 17307 | Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ (π β (πΉ β πΊ) = (πΉ βf Β· πΊ)) | ||
Theorem | pwsle 17308 | Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ π = (leβπ ) & β’ β€ = (leβπ) β β’ ((π β π β§ πΌ β π) β β€ = ( βr π β© (π΅ Γ π΅))) | ||
Theorem | pwsleval 17309* | Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ π = (leβπ ) & β’ β€ = (leβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) β β’ (π β (πΉ β€ πΊ β βπ₯ β πΌ (πΉβπ₯)π(πΊβπ₯))) | ||
Theorem | pwsvscafval 17310 | Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) & β’ πΉ = (Scalarβπ ) & β’ πΎ = (BaseβπΉ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) β β’ (π β (π΄ β π) = ((πΌ Γ {π΄}) βf Β· π)) | ||
Theorem | pwsvscaval 17311 | Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) & β’ πΉ = (Scalarβπ ) & β’ πΎ = (BaseβπΉ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) β β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) | ||
Theorem | pwssca 17312 | The ring of scalars of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π = (Scalarβπ ) β β’ ((π β π β§ πΌ β π) β π = (Scalarβπ)) | ||
Theorem | pwsdiagel 17313 | Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ πΆ = (Baseβπ) β β’ (((π β π β§ πΌ β π) β§ π΄ β π΅) β (πΌ Γ {π΄}) β πΆ) | ||
Theorem | pwssnf1o 17314* | Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
β’ π = (π βs {πΌ}) & β’ π΅ = (Baseβπ ) & β’ πΉ = (π₯ β π΅ β¦ ({πΌ} Γ {π₯})) & β’ πΆ = (Baseβπ) β β’ ((π β π β§ πΌ β π) β πΉ:π΅β1-1-ontoβπΆ) | ||
Syntax | cordt 17315 | Extend class notation with the order topology. |
class ordTop | ||
Syntax | cxrs 17316 | Extend class notation with the extended real number structure. |
class β*π | ||
Definition | df-ordt 17317* | Define the order topology, given an order β€, written as π below. A closed subbasis for the order topology is given by the closed rays [π¦, +β) = {π§ β π β£ π¦ β€ π§} and (-β, π¦] = {π§ β π β£ π§ β€ π¦}, along with (-β, +β) = π itself. (Contributed by Mario Carneiro, 3-Sep-2015.) |
β’ ordTop = (π β V β¦ (topGenβ(fiβ({dom π} βͺ ran ((π₯ β dom π β¦ {π¦ β dom π β£ Β¬ π¦ππ₯}) βͺ (π₯ β dom π β¦ {π¦ β dom π β£ Β¬ π₯ππ¦})))))) | ||
Definition | df-xrs 17318* | The extended real number structure. Unlike df-cnfld 20720, the extended real numbers do not have good algebraic properties, so this is not actually a group or anything higher, even though it has just as many operations as df-cnfld 20720. The main interest in this structure is in its ordering, which is complete and compact. The metric described here is an extension of the absolute value metric, but it is not itself a metric because +β is infinitely far from all other points. The topology is based on the order and not the extended metric (which would make +β an isolated point since there is nothing else in the 1 -ball around it). All components of this structure agree with βfld when restricted to β. (Contributed by Mario Carneiro, 20-Aug-2015.) |
β’ β*π = ({β¨(Baseβndx), β*β©, β¨(+gβndx), +π β©, β¨(.rβndx), Β·e β©} βͺ {β¨(TopSetβndx), (ordTopβ β€ )β©, β¨(leβndx), β€ β©, β¨(distβndx), (π₯ β β*, π¦ β β* β¦ if(π₯ β€ π¦, (π¦ +π -ππ₯), (π₯ +π -ππ¦)))β©}) | ||
Syntax | cqtop 17319 | Extend class notation with the quotient topology function. |
class qTop | ||
Syntax | cimas 17320 | Image structure function. |
class βs | ||
Syntax | cqus 17321 | Quotient structure function. |
class /s | ||
Syntax | cxps 17322 | Binary product structure function. |
class Γs | ||
Definition | df-qtop 17323* | Define the quotient topology given a function π and topology π on the domain of π. (Contributed by Mario Carneiro, 23-Mar-2015.) |
β’ qTop = (π β V, π β V β¦ {π β π« (π β βͺ π) β£ ((β‘π β π ) β© βͺ π) β π}) | ||
Definition | df-imas 17324* |
Define an image structure, which takes a structure and a function on the
base set, and maps all the operations via the function. For this to
work properly π must either be injective or satisfy
the
well-definedness condition π(π) = π(π) β§ π(π) = π(π) β
π(π + π) = π(π + π) for each relevant operation.
Note that although we call this an "image" by association to df-ima 5643, in order to keep the definition simple we consider only the case when the domain of πΉ is equal to the base set of π . Other cases can be achieved by restricting πΉ (with df-res 5642) and/or π ( with df-ress 17047) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by AV, 6-Oct-2020.) |
β’ βs = (π β V, π β V β¦ β¦(Baseβπ) / π£β¦(({β¨(Baseβndx), ran πβ©, β¨(+gβndx), βͺ π β π£ βͺ π β π£ {β¨β¨(πβπ), (πβπ)β©, (πβ(π(+gβπ)π))β©}β©, β¨(.rβndx), βͺ π β π£ βͺ π β π£ {β¨β¨(πβπ), (πβπ)β©, (πβ(π(.rβπ)π))β©}β©} βͺ {β¨(Scalarβndx), (Scalarβπ)β©, β¨( Β·π βndx), βͺ π β π£ (π β (Baseβ(Scalarβπ)), π₯ β {(πβπ)} β¦ (πβ(π( Β·π βπ)π)))β©, β¨(Β·πβndx), βͺ π β π£ βͺ π β π£ {β¨β¨(πβπ), (πβπ)β©, (π(Β·πβπ)π)β©}β©}) βͺ {β¨(TopSetβndx), ((TopOpenβπ) qTop π)β©, β¨(leβndx), ((π β (leβπ)) β β‘π)β©, β¨(distβndx), (π₯ β ran π, π¦ β ran π β¦ inf(βͺ π β β ran (π β {β β ((π£ Γ π£) βm (1...π)) β£ ((πβ(1st β(ββ1))) = π₯ β§ (πβ(2nd β(ββπ))) = π¦ β§ βπ β (1...(π β 1))(πβ(2nd β(ββπ))) = (πβ(1st β(ββ(π + 1)))))} β¦ (β*π Ξ£g ((distβπ) β π))), β*, < ))β©})) | ||
Definition | df-qus 17325* | Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 17324 where the image function is π₯ β¦ [π₯]π. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ /s = (π β V, π β V β¦ ((π₯ β (Baseβπ) β¦ [π₯]π) βs π)) | ||
Definition | df-xps 17326* | Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
β’ Γs = (π β V, π β V β¦ (β‘(π₯ β (Baseβπ), π¦ β (Baseβπ ) β¦ {β¨β , π₯β©, β¨1o, π¦β©}) βs ((Scalarβπ)Xs{β¨β , πβ©, β¨1o, π β©}))) | ||
Theorem | imasval 17327* | Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ + = (+gβπ ) & β’ Γ = (.rβπ ) & β’ πΊ = (Scalarβπ ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ , = (Β·πβπ ) & β’ π½ = (TopOpenβπ ) & β’ πΈ = (distβπ ) & β’ π = (leβπ ) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π + π))β©}) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Γ π))β©}) & β’ (π β β = βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))) & β’ (π β πΌ = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}) & β’ (π β π = (π½ qTop πΉ)) & β’ (π β π· = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π₯ β§ (πΉβ(2nd β(ββπ))) = π¦ β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < ))) & β’ (π β β€ = ((πΉ β π) β β‘πΉ)) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) β β’ (π β π = (({β¨(Baseβndx), π΅β©, β¨(+gβndx), β β©, β¨(.rβndx), β β©} βͺ {β¨(Scalarβndx), πΊβ©, β¨( Β·π βndx), β β©, β¨(Β·πβndx), πΌβ©}) βͺ {β¨(TopSetβndx), πβ©, β¨(leβndx), β€ β©, β¨(distβndx), π·β©})) | ||
Theorem | imasbas 17328 | The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) β β’ (π β π΅ = (Baseβπ)) | ||
Theorem | imasds 17329* | The distance function of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΈ = (distβπ ) & β’ π· = (distβπ) β β’ (π β π· = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π₯ β§ (πΉβ(2nd β(ββπ))) = π¦ β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < ))) | ||
Theorem | imasdsfn 17330 | The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΈ = (distβπ ) & β’ π· = (distβπ) β β’ (π β π· Fn (π΅ Γ π΅)) | ||
Theorem | imasdsval 17331* | The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΈ = (distβπ ) & β’ π· = (distβπ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π β§ (πΉβ(2nd β(ββπ))) = π β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} β β’ (π β (ππ·π) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (πΈ β π))), β*, < )) | ||
Theorem | imasdsval2 17332* | The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΈ = (distβπ ) & β’ π· = (distβπ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π β§ (πΉβ(2nd β(ββπ))) = π β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} & β’ π = (πΈ βΎ (π Γ π)) β β’ (π β (ππ·π) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (π β π))), β*, < )) | ||
Theorem | imasplusg 17333* | The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ + = (+gβπ ) & β’ β = (+gβπ) β β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π + π))β©}) | ||
Theorem | imasmulr 17334* | The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) | ||
Theorem | imassca 17335 | The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΊ = (Scalarβπ ) β β’ (π β πΊ = (Scalarβπ)) | ||
Theorem | imasvsca 17336* | The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΊ = (Scalarβπ ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) β β’ (π β β = βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))) | ||
Theorem | imasip 17337* | The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ , = (Β·πβπ ) & β’ πΌ = (Β·πβπ) β β’ (π β πΌ = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}) | ||
Theorem | imastset 17338 | The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ π½ = (TopOpenβπ ) & β’ π = (TopSetβπ) β β’ (π β π = (π½ qTop πΉ)) | ||
Theorem | imasle 17339 | The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ π = (leβπ ) & β’ β€ = (leβπ) β β’ (π β β€ = ((πΉ β π) β β‘πΉ)) | ||
Theorem | f1ocpbllem 17340 | Lemma for f1ocpbl 17341. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β πΉ:πβ1-1-ontoβπ) β β’ ((π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (((πΉβπ΄) = (πΉβπΆ) β§ (πΉβπ΅) = (πΉβπ·)) β (π΄ = πΆ β§ π΅ = π·))) | ||
Theorem | f1ocpbl 17341 | An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β πΉ:πβ1-1-ontoβπ) β β’ ((π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (((πΉβπ΄) = (πΉβπΆ) β§ (πΉβπ΅) = (πΉβπ·)) β (πΉβ(π΄ + π΅)) = (πΉβ(πΆ + π·)))) | ||
Theorem | f1ovscpbl 17342 | An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ (π β πΉ:πβ1-1-ontoβπ) β β’ ((π β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((πΉβπ΅) = (πΉβπΆ) β (πΉβ(π΄ + π΅)) = (πΉβ(π΄ + πΆ)))) | ||
Theorem | f1olecpbl 17343 | An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β πΉ:πβ1-1-ontoβπ) β β’ ((π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (((πΉβπ΄) = (πΉβπΆ) β§ (πΉβπ΅) = (πΉβπ·)) β (π΄ππ΅ β πΆππ·))) | ||
Theorem | imasaddfnlem 17344* | The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) β β’ (π β β Fn (π΅ Γ π΅)) | ||
Theorem | imasaddvallem 17345* | The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) β β’ ((π β§ π β π β§ π β π) β ((πΉβπ) β (πΉβπ)) = (πΉβ(π Β· π))) | ||
Theorem | imasaddflem 17346* | The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) β β’ (π β β :(π΅ Γ π΅)βΆπ΅) | ||
Theorem | imasaddfn 17347* | The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (+gβπ ) & β’ β = (+gβπ) β β’ (π β β Fn (π΅ Γ π΅)) | ||
Theorem | imasaddval 17348* | The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (+gβπ ) & β’ β = (+gβπ) β β’ ((π β§ π β π β§ π β π) β ((πΉβπ) β (πΉβπ)) = (πΉβ(π Β· π))) | ||
Theorem | imasaddf 17349* | The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (+gβπ ) & β’ β = (+gβπ) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) β β’ (π β β :(π΅ Γ π΅)βΆπ΅) | ||
Theorem | imasmulfn 17350* | The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ (π β β Fn (π΅ Γ π΅)) | ||
Theorem | imasmulval 17351* | The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ ((π β§ π β π β§ π β π) β ((πΉβπ) β (πΉβπ)) = (πΉβ(π Β· π))) | ||
Theorem | imasmulf 17352* | The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β πΉ:πβontoβπ΅) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β π β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) β β’ (π β β :(π΅ Γ π΅)βΆπ΅) | ||
Theorem | imasvscafn 17353* | The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΊ = (Scalarβπ ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) & β’ ((π β§ (π β πΎ β§ π β π β§ π β π)) β ((πΉβπ) = (πΉβπ) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) β β’ (π β β Fn (πΎ Γ π΅)) | ||
Theorem | imasvscaval 17354* | The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΊ = (Scalarβπ ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) & β’ ((π β§ (π β πΎ β§ π β π β§ π β π)) β ((πΉβπ) = (πΉβπ) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) β β’ ((π β§ π β πΎ β§ π β π) β (π β (πΉβπ)) = (πΉβ(π Β· π))) | ||
Theorem | imasvscaf 17355* | The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ πΊ = (Scalarβπ ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ β = ( Β·π βπ) & β’ ((π β§ (π β πΎ β§ π β π β§ π β π)) β ((πΉβπ) = (πΉβπ) β (πΉβ(π Β· π)) = (πΉβ(π Β· π)))) & β’ ((π β§ (π β πΎ β§ π β π)) β (π Β· π) β π) β β’ (π β β :(πΎ Γ π΅)βΆπ΅) | ||
Theorem | imasless 17356 | The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ β€ = (leβπ) β β’ (π β β€ β (π΅ Γ π΅)) | ||
Theorem | imasleval 17357* | The value of the image structure's ordering when the order is compatible with the mapping function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (πΉ βs π )) & β’ (π β π = (Baseβπ )) & β’ (π β πΉ:πβontoβπ΅) & β’ (π β π β π) & β’ β€ = (leβπ) & β’ π = (leβπ ) & β’ ((π β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β (((πΉβπ) = (πΉβπ) β§ (πΉβπ) = (πΉβπ)) β (πππ β πππ))) β β’ ((π β§ π β π β§ π β π) β ((πΉβπ) β€ (πΉβπ) β πππ)) | ||
Theorem | qusval 17358* | Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β βΌ β π) & β’ (π β π β π) β β’ (π β π = (πΉ βs π )) | ||
Theorem | quslem 17359* | The function in qusval 17358 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β βΌ β π) & β’ (π β π β π) β β’ (π β πΉ:πβontoβ(π / βΌ )) | ||
Theorem | qusin 17360 | Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ β π) & β’ (π β π β π) & β’ (π β ( βΌ β π) β π) β β’ (π β π = (π /s ( βΌ β© (π Γ π)))) | ||
Theorem | qusbas 17361 | Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ β π) & β’ (π β π β π) β β’ (π β (π / βΌ ) = (Baseβπ)) | ||
Theorem | quss 17362 | The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ β π) & β’ (π β π β π) & β’ πΎ = (Scalarβπ ) β β’ (π β πΎ = (Scalarβπ)) | ||
Theorem | divsfval 17363* | Value of the function in qusval 17358. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
β’ (π β βΌ Er π) & β’ (π β π β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) β β’ (π β (πΉβπ΄) = [π΄] βΌ ) | ||
Theorem | ercpbllem 17364* | Lemma for ercpbl 17365. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.) |
β’ (π β βΌ Er π) & β’ (π β π β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β π΄ β π) β β’ (π β ((πΉβπ΄) = (πΉβπ΅) β π΄ βΌ π΅)) | ||
Theorem | ercpbl 17365* | Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
β’ (π β βΌ Er π) & β’ (π β π β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ ((π β§ (π β π β§ π β π)) β (π + π) β π) & β’ (π β ((π΄ βΌ πΆ β§ π΅ βΌ π·) β (π΄ + π΅) βΌ (πΆ + π·))) β β’ ((π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (((πΉβπ΄) = (πΉβπΆ) β§ (πΉβπ΅) = (πΉβπ·)) β (πΉβ(π΄ + π΅)) = (πΉβ(πΆ + π·)))) | ||
Theorem | erlecpbl 17366* | Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
β’ (π β βΌ Er π) & β’ (π β π β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β ((π΄ βΌ πΆ β§ π΅ βΌ π·) β (π΄ππ΅ β πΆππ·))) β β’ ((π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (((πΉβπ΄) = (πΉβπΆ) β§ (πΉβπ΅) = (πΉβπ·)) β (π΄ππ΅ β πΆππ·))) | ||
Theorem | qusaddvallem 17367* | Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) β β’ ((π β§ π β π β§ π β π) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) | ||
Theorem | qusaddflem 17368* | The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) & β’ (π β β = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π Β· π))β©}) β β’ (π β β :((π / βΌ ) Γ (π / βΌ ))βΆ(π / βΌ )) | ||
Theorem | qusaddval 17369* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ Β· = (+gβπ ) & β’ β = (+gβπ) β β’ ((π β§ π β π β§ π β π) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) | ||
Theorem | qusaddf 17370* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ Β· = (+gβπ ) & β’ β = (+gβπ) β β’ (π β β :((π / βΌ ) Γ (π / βΌ ))βΆ(π / βΌ )) | ||
Theorem | qusmulval 17371* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ ((π β§ π β π β§ π β π) β ([π] βΌ β [π] βΌ ) = [(π Β· π)] βΌ ) | ||
Theorem | qusmulf 17372* | The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
β’ (π β π = (π /s βΌ )) & β’ (π β π = (Baseβπ )) & β’ (π β βΌ Er π) & β’ (π β π β π) & β’ (π β ((π βΌ π β§ π βΌ π) β (π Β· π) βΌ (π Β· π))) & β’ ((π β§ (π β π β§ π β π)) β (π Β· π) β π) & β’ Β· = (.rβπ ) & β’ β = (.rβπ) β β’ (π β β :((π / βΌ ) Γ (π / βΌ ))βΆ(π / βΌ )) | ||
Theorem | fnpr2o 17373 | Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.) |
β’ ((π΄ β π β§ π΅ β π) β {β¨β , π΄β©, β¨1o, π΅β©} Fn 2o) | ||
Theorem | fnpr2ob 17374 | Biconditional version of fnpr2o 17373. (Contributed by Jim Kingdon, 27-Sep-2023.) |
β’ ((π΄ β V β§ π΅ β V) β {β¨β , π΄β©, β¨1o, π΅β©} Fn 2o) | ||
Theorem | fvpr0o 17375 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
β’ (π΄ β π β ({β¨β , π΄β©, β¨1o, π΅β©}ββ ) = π΄) | ||
Theorem | fvpr1o 17376 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
β’ (π΅ β π β ({β¨β , π΄β©, β¨1o, π΅β©}β1o) = π΅) | ||
Theorem | fvprif 17377 | The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ ((π΄ β π β§ π΅ β π β§ πΆ β 2o) β ({β¨β , π΄β©, β¨1o, π΅β©}βπΆ) = if(πΆ = β , π΄, π΅)) | ||
Theorem | xpsfrnel 17378* | Elementhood in the target space of the function πΉ appearing in xpsval 17386. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (πΊ β Xπ β 2o if(π = β , π΄, π΅) β (πΊ Fn 2o β§ (πΊββ ) β π΄ β§ (πΊβ1o) β π΅)) | ||
Theorem | xpsfeq 17379 | A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.) |
β’ (πΊ Fn 2o β {β¨β , (πΊββ )β©, β¨1o, (πΊβ1o)β©} = πΊ) | ||
Theorem | xpsfrnel2 17380* | Elementhood in the target space of the function πΉ appearing in xpsval 17386. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ ({β¨β , πβ©, β¨1o, πβ©} β Xπ β 2o if(π = β , π΄, π΅) β (π β π΄ β§ π β π΅)) | ||
Theorem | xpscf 17381 | Equivalent condition for the pair function to be a proper function on π΄. (Contributed by Mario Carneiro, 20-Aug-2015.) |
β’ ({β¨β , πβ©, β¨1o, πβ©}:2oβΆπ΄ β (π β π΄ β§ π β π΄)) | ||
Theorem | xpsfval 17382* | The value of the function appearing in xpsval 17386. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ πΉ = (π₯ β π΄, π¦ β π΅ β¦ {β¨β , π₯β©, β¨1o, π¦β©}) β β’ ((π β π΄ β§ π β π΅) β (ππΉπ) = {β¨β , πβ©, β¨1o, πβ©}) | ||
Theorem | xpsff1o 17383* | The function appearing in xpsval 17386 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {β , 1o}. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ πΉ = (π₯ β π΄, π¦ β π΅ β¦ {β¨β , π₯β©, β¨1o, π¦β©}) β β’ πΉ:(π΄ Γ π΅)β1-1-ontoβXπ β 2o if(π = β , π΄, π΅) | ||
Theorem | xpsfrn 17384* | A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ πΉ = (π₯ β π΄, π¦ β π΅ β¦ {β¨β , π₯β©, β¨1o, π¦β©}) β β’ ran πΉ = Xπ β 2o if(π = β , π΄, π΅) | ||
Theorem | xpsff1o2 17385* | The function appearing in xpsval 17386 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {β , 1o}. (Contributed by Mario Carneiro, 24-Jan-2015.) |
β’ πΉ = (π₯ β π΄, π¦ β π΅ β¦ {β¨β , π₯β©, β¨1o, π¦β©}) β β’ πΉ:(π΄ Γ π΅)β1-1-ontoβran πΉ | ||
Theorem | xpsval 17386* | Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ πΉ = (π₯ β π, π¦ β π β¦ {β¨β , π₯β©, β¨1o, π¦β©}) & β’ πΊ = (Scalarβπ ) & β’ π = (πΊXs{β¨β , π β©, β¨1o, πβ©}) β β’ (π β π = (β‘πΉ βs π)) | ||
Theorem | xpsrnbas 17387* | The indexed structure product that appears in xpsval 17386 has the same base as the target of the function πΉ. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ πΉ = (π₯ β π, π¦ β π β¦ {β¨β , π₯β©, β¨1o, π¦β©}) & β’ πΊ = (Scalarβπ ) & β’ π = (πΊXs{β¨β , π β©, β¨1o, πβ©}) β β’ (π β ran πΉ = (Baseβπ)) | ||
Theorem | xpsbas 17388 | The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (π Γ π) = (Baseβπ)) | ||
Theorem | xpsaddlem 17389* | Lemma for xpsadd 17390 and xpsmul 17391. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β (π΄ Β· πΆ) β π) & β’ (π β (π΅ Γ π·) β π) & β’ Β· = (πΈβπ ) & β’ Γ = (πΈβπ) & β’ β = (πΈβπ) & β’ πΉ = (π₯ β π, π¦ β π β¦ {β¨β , π₯β©, β¨1o, π¦β©}) & β’ π = ((Scalarβπ )Xs{β¨β , π β©, β¨1o, πβ©}) & β’ ((π β§ {β¨β , π΄β©, β¨1o, π΅β©} β ran πΉ β§ {β¨β , πΆβ©, β¨1o, π·β©} β ran πΉ) β ((β‘πΉβ{β¨β , π΄β©, β¨1o, π΅β©}) β (β‘πΉβ{β¨β , πΆβ©, β¨1o, π·β©})) = (β‘πΉβ({β¨β , π΄β©, β¨1o, π΅β©} (πΈβπ){β¨β , πΆβ©, β¨1o, π·β©}))) & β’ (({β¨β , π β©, β¨1o, πβ©} Fn 2o β§ {β¨β , π΄β©, β¨1o, π΅β©} β (Baseβπ) β§ {β¨β , πΆβ©, β¨1o, π·β©} β (Baseβπ)) β ({β¨β , π΄β©, β¨1o, π΅β©} (πΈβπ){β¨β , πΆβ©, β¨1o, π·β©}) = (π β 2o β¦ (({β¨β , π΄β©, β¨1o, π΅β©}βπ)(πΈβ({β¨β , π β©, β¨1o, πβ©}βπ))({β¨β , πΆβ©, β¨1o, π·β©}βπ)))) β β’ (π β (β¨π΄, π΅β© β β¨πΆ, π·β©) = β¨(π΄ Β· πΆ), (π΅ Γ π·)β©) | ||
Theorem | xpsadd 17390 | Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β (π΄ Β· πΆ) β π) & β’ (π β (π΅ Γ π·) β π) & β’ Β· = (+gβπ ) & β’ Γ = (+gβπ) & β’ β = (+gβπ) β β’ (π β (β¨π΄, π΅β© β β¨πΆ, π·β©) = β¨(π΄ Β· πΆ), (π΅ Γ π·)β©) | ||
Theorem | xpsmul 17391 | Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β (π΄ Β· πΆ) β π) & β’ (π β (π΅ Γ π·) β π) & β’ Β· = (.rβπ ) & β’ Γ = (.rβπ) & β’ β = (.rβπ) β β’ (π β (β¨π΄, π΅β© β β¨πΆ, π·β©) = β¨(π΄ Β· πΆ), (π΅ Γ π·)β©) | ||
Theorem | xpssca 17392 | Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ πΊ = (Scalarβπ ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β πΊ = (Scalarβπ)) | ||
Theorem | xpsvsca 17393 | Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ πΊ = (Scalarβπ ) & β’ (π β π β π) & β’ (π β π β π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ πΎ = (BaseβπΊ) & β’ Β· = ( Β·π βπ ) & β’ Γ = ( Β·π βπ) & β’ β = ( Β·π βπ) & β’ (π β π΄ β πΎ) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β (π΄ Β· π΅) β π) & β’ (π β (π΄ Γ πΆ) β π) β β’ (π β (π΄ β β¨π΅, πΆβ©) = β¨(π΄ Β· π΅), (π΄ Γ πΆ)β©) | ||
Theorem | xpsless 17394 | Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ β€ = (leβπ) β β’ (π β β€ β ((π Γ π) Γ (π Γ π))) | ||
Theorem | xpsle 17395 | Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
β’ π = (π Γs π) & β’ π = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ β€ = (leβπ) & β’ π = (leβπ ) & β’ π = (leβπ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β (β¨π΄, π΅β© β€ β¨πΆ, π·β© β (π΄ππΆ β§ π΅ππ·))) | ||
Syntax | cmre 17396 | The class of Moore systems. |
class Moore | ||
Syntax | cmrc 17397 | The class function generating Moore closures. |
class mrCls | ||
Syntax | cmri 17398 | mrInd is a class function which takes a Moore system to its set of independent sets. |
class mrInd | ||
Syntax | cacs 17399 | The class of algebraic closure (Moore) systems. |
class ACS | ||
Definition | df-mre 17400* |
Define a Moore collection, which is a family of subsets of a base set
which preserve arbitrary intersection. Elements of a Moore collection
are termed closed; Moore collections generalize the notion of
closedness from topologies (cldmre 22351) and vector spaces (lssmre 20350)
to the most general setting in which such concepts make sense.
Definition of Moore collection of sets in [Schechter] p. 78. A Moore
collection may also be called a closure system (Section 0.6 in
[Gratzer] p. 23.) The name Moore
collection is after Eliakim Hastings
Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.
See ismre 17404, mresspw 17406, mre1cl 17408 and mreintcl 17409 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17414); as such the disjoint union of all Moore collections is sometimes considered as βͺ ran Moore, justified by mreunirn 17415. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.) |
β’ Moore = (π₯ β V β¦ {π β π« π« π₯ β£ (π₯ β π β§ βπ β π« π(π β β β β© π β π))}) |
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