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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwsvscafval 16201 Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑌)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋))

Theorempwsvscaval 16202 Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑌)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))

Theorempwssca 16203 The ring of scalars of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝑆 = (Scalar‘𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑆 = (Scalar‘𝑌))

Theorempwsdiagel 16204 Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑌)       (((𝑅𝑉𝐼𝑊) ∧ 𝐴𝐵) → (𝐼 × {𝐴}) ∈ 𝐶)

Theorempwssnf1o 16205* Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s {𝐼})    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ ({𝐼} × {𝑥}))    &   𝐶 = (Base‘𝑌)       ((𝑅𝑉𝐼𝑊) → 𝐹:𝐵1-1-onto𝐶)

7.1.4  Definition of the structure quotient

Syntaxcordt 16206 Extend class notation with the order topology.
class ordTop

Syntaxcxrs 16207 Extend class notation with the extended real number structure.
class *𝑠

Definitiondf-ordt 16208* Define the order topology, given an order , written as 𝑟 below. A closed subbasis for the order topology is given by the closed rays [𝑦, +∞) = {𝑧𝑋𝑦𝑧} and (-∞, 𝑦] = {𝑧𝑋𝑧𝑦}, along with (-∞, +∞) = 𝑋 itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))

Definitiondf-xrs 16209* The extended real number structure. Unlike df-cnfld 19795, 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 19795. 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 +∞ is infinitely far from all other points. The topology is based on the order and not the extended metric (which would make +∞ an isolated point since there is nothing else in the 1 -ball around it). All components of this structure agree with fld when restricted to . (Contributed by Mario Carneiro, 20-Aug-2015.)
*𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})

Syntaxcqtop 16210 Extend class notation with the quotient topology function.
class qTop

Syntaxcimas 16211 Image structure function.
class s

Syntaxcqus 16212 Quotient structure function.
class /s

Syntaxcxps 16213 Binary product structure function.
class ×s

Definitiondf-qtop 16214* Define the quotient topology given a function 𝑓 and topology 𝑗 on the domain of 𝑓. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})

Definitiondf-imas 16215* 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 𝑓 must either be injective or satisfy the well-definedness condition 𝑓(𝑎) = 𝑓(𝑐) ∧ 𝑓(𝑏) = 𝑓(𝑑) → 𝑓(𝑎 + 𝑏) = 𝑓(𝑐 + 𝑑) for each relevant operation.

Note that although we call this an "image" by association to df-ima 5156, in order to keep the definition simple we consider only the case when the domain of 𝐹 is equal to the base set of 𝑅. Other cases can be achieved by restricting 𝐹 (with df-res 5155) and/or 𝑅 ( with df-ress 15912) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by AV, 6-Oct-2020.)

s = (𝑓 ∈ V, 𝑟 ∈ V ↦ (Base‘𝑟) / 𝑣(({⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(+g𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(.r𝑟)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓𝑞)} ↦ (𝑓‘(𝑝( ·𝑠𝑟)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑝(·𝑖𝑟)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑟) qTop 𝑓)⟩, ⟨(le‘ndx), ((𝑓 ∘ (le‘𝑟)) ∘ 𝑓)⟩, ⟨(dist‘ndx), (𝑥 ∈ ran 𝑓, 𝑦 ∈ ran 𝑓 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑣 × 𝑣) ↑𝑚 (1...𝑛)) ∣ ((𝑓‘(1st ‘(‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(𝑖))) = (𝑓‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑟) ∘ 𝑔))), ℝ*, < ))⟩}))

Definitiondf-qus 16216* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 16215 where the image function is 𝑥 ↦ [𝑥]𝑒. (Contributed by Mario Carneiro, 23-Feb-2015.)
/s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))

Definitiondf-xps 16217* Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.)
×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠}))))

Theoremimasval 16218* 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.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &    × = (.r𝑅)    &   𝐺 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    , = (·𝑖𝑅)    &   𝐽 = (TopOpen‘𝑅)    &   𝐸 = (dist‘𝑅)    &   𝑁 = (le‘𝑅)    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})    &   (𝜑 = 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝐼 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩})    &   (𝜑𝑂 = (𝐽 qTop 𝐹))    &   (𝜑𝐷 = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < )))    &   (𝜑 = ((𝐹𝑁) ∘ 𝐹))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)       (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), ⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(·𝑖‘ndx), 𝐼⟩}) ∪ {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩}))

Theoremimasbas 16219 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.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)       (𝜑𝐵 = (Base‘𝑈))

Theoremimasds 16220* 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.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)       (𝜑𝐷 = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < )))

Theoremimasdsfn 16221 The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)       (𝜑𝐷 Fn (𝐵 × 𝐵))

Theoremimasdsval 16222* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = { ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))}       (𝜑 → (𝑋𝐷𝑌) = inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ))

Theoremimasdsval2 16223* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = { ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))}    &   𝑇 = (𝐸 ↾ (𝑉 × 𝑉))       (𝜑 → (𝑋𝐷𝑌) = inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝑇𝑔))), ℝ*, < ))

Theoremimasplusg 16224* 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.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &    + = (+g𝑅)    &    = (+g𝑈)       (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})

Theoremimasmulr 16225* 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.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &    · = (.r𝑅)    &    = (.r𝑈)       (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})

Theoremimassca 16226 The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐺 = (Scalar‘𝑅)       (𝜑𝐺 = (Scalar‘𝑈))

Theoremimasvsca 16227* The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐺 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑈)       (𝜑 = 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))

Theoremimasip 16228* The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &    , = (·𝑖𝑅)    &   𝐼 = (·𝑖𝑈)       (𝜑𝐼 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩})

Theoremimastset 16229 The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐽 = (TopOpen‘𝑅)    &   𝑂 = (TopSet‘𝑈)       (𝜑𝑂 = (𝐽 qTop 𝐹))

Theoremimasle 16230 The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝑁 = (le‘𝑅)    &    = (le‘𝑈)       (𝜑 = ((𝐹𝑁) ∘ 𝐹))

Theoremf1ocpbllem 16231 Lemma for f1ocpbl 16232. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremf1ocpbl 16232 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Theoremf1ovscpbl 16233 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))

Theoremf1olecpbl 16234 An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐴𝑁𝐵𝐶𝑁𝐷)))

Theoremimasaddfnlem 16235* The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})       (𝜑 Fn (𝐵 × 𝐵))

Theoremimasaddvallem 16236* The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})       ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Theoremimasaddflem 16237* The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)       (𝜑 :(𝐵 × 𝐵)⟶𝐵)

Theoremimasaddfn 16238* The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (+g𝑅)    &    = (+g𝑈)       (𝜑 Fn (𝐵 × 𝐵))

Theoremimasaddval 16239* The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (+g𝑅)    &    = (+g𝑈)       ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Theoremimasaddf 16240* The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (+g𝑅)    &    = (+g𝑈)    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)       (𝜑 :(𝐵 × 𝐵)⟶𝐵)

Theoremimasmulfn 16241* The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (.r𝑅)    &    = (.r𝑈)       (𝜑 Fn (𝐵 × 𝐵))

Theoremimasmulval 16242* The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (.r𝑅)    &    = (.r𝑈)       ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Theoremimasmulf 16243* The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑅𝑍)    &    · = (.r𝑅)    &    = (.r𝑈)    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)       (𝜑 :(𝐵 × 𝐵)⟶𝐵)

Theoremimasvscafn 16244* The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐺 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑈)    &   ((𝜑 ∧ (𝑝𝐾𝑎𝑉𝑞𝑉)) → ((𝐹𝑎) = (𝐹𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))))       (𝜑 Fn (𝐾 × 𝐵))

Theoremimasvscaval 16245* The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐺 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑈)    &   ((𝜑 ∧ (𝑝𝐾𝑎𝑉𝑞𝑉)) → ((𝐹𝑎) = (𝐹𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))))       ((𝜑𝑋𝐾𝑌𝑉) → (𝑋 (𝐹𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Theoremimasvscaf 16246* The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐺 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    = ( ·𝑠𝑈)    &   ((𝜑 ∧ (𝑝𝐾𝑎𝑉𝑞𝑉)) → ((𝐹𝑎) = (𝐹𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))))    &   ((𝜑 ∧ (𝑝𝐾𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)       (𝜑 :(𝐾 × 𝐵)⟶𝐵)

Theoremimasless 16247 The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &    = (le‘𝑈)       (𝜑 ⊆ (𝐵 × 𝐵))

Theoremimasleval 16248* The value of the image structure's ordering when the order is compatible with the mapping function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &    = (le‘𝑈)    &   𝑁 = (le‘𝑅)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → (((𝐹𝑎) = (𝐹𝑐) ∧ (𝐹𝑏) = (𝐹𝑑)) → (𝑎𝑁𝑏𝑐𝑁𝑑)))       ((𝜑𝑋𝑉𝑌𝑉) → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋𝑁𝑌))

Theoremqusval 16249* Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑𝑊)    &   (𝜑𝑅𝑍)       (𝜑𝑈 = (𝐹s 𝑅))

Theoremquslem 16250* The function in qusval 16249 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑𝑊)    &   (𝜑𝑅𝑍)       (𝜑𝐹:𝑉onto→(𝑉 / ))

Theoremqusin 16251 Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑊)    &   (𝜑𝑅𝑍)    &   (𝜑 → ( 𝑉) ⊆ 𝑉)       (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))

Theoremqusbas 16252 Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑊)    &   (𝜑𝑅𝑍)       (𝜑 → (𝑉 / ) = (Base‘𝑈))

Theoremquss 16253 The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝑊)    &   (𝜑𝑅𝑍)    &   𝐾 = (Scalar‘𝑅)       (𝜑𝐾 = (Scalar‘𝑈))

Theoremdivsfval 16254* Value of the function in qusval 16249. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑 Er 𝑉)    &   (𝜑𝑉 ∈ V)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )       (𝜑 → (𝐹𝐴) = [𝐴] )

Theoremercpbllem 16255* Lemma for ercpbl 16256. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑 Er 𝑉)    &   (𝜑𝑉 ∈ V)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑𝐴𝑉)       (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 𝐵))

Theoremercpbl 16256* Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑 Er 𝑉)    &   (𝜑𝑉 ∈ V)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)    &   (𝜑 → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Theoremerlecpbl 16257* Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑 Er 𝑉)    &   (𝜑𝑉 ∈ V)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑 → ((𝐴 𝐶𝐵 𝐷) → (𝐴𝑁𝐵𝐶𝑁𝐷)))       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐴𝑁𝐵𝐶𝑁𝐷)))

Theoremqusaddvallem 16258* Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})       ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )

Theoremqusaddflem 16259* The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})       (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))

Theoremqusaddval 16260* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &    · = (+g𝑅)    &    = (+g𝑈)       ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )

Theoremqusaddf 16261* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &    · = (+g𝑅)    &    = (+g𝑈)       (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))

Theoremqusmulval 16262* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &    · = (.r𝑅)    &    = (.r𝑈)       ((𝜑𝑋𝑉𝑌𝑉) → ([𝑋] [𝑌] ) = [(𝑋 · 𝑌)] )

Theoremqusmulf 16263* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑 Er 𝑉)    &   (𝜑𝑅𝑍)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &    · = (.r𝑅)    &    = (.r𝑈)       (𝜑 :((𝑉 / ) × (𝑉 / ))⟶(𝑉 / ))

Theoremxpsc 16264 A short expression for the pair function mapping 0 to 𝐴 and 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
({𝐴} +𝑐 {𝐵}) = (({∅} × {𝐴}) ∪ ({1𝑜} × {𝐵}))

Theoremxpscg 16265 A short expression for the pair function mapping 0 to 𝐴 and 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} +𝑐 {𝐵}) = {⟨∅, 𝐴⟩, ⟨1𝑜, 𝐵⟩})

Theoremxpscfn 16266 The pair function is a function on 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} +𝑐 {𝐵}) Fn 2𝑜)

Theoremxpsc0 16267 The pair function maps 0 to 𝐴. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐴𝑉 → (({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)

Theoremxpsc1 16268 The pair function maps 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵𝑉 → (({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)

Theoremxpscfv 16269 The value of the pair function at an element of 2𝑜. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊𝐶 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵))

Theoremxpsfrnel 16270* Elementhood in the target space of the function 𝐹 appearing in xpsval 16279. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐺X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))

Theoremxpsfeq 16271 A function on 2𝑜 is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝐺 Fn 2𝑜({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺)

Theoremxpsfrnel2 16272* Elementhood in the target space of the function 𝐹 appearing in xpsval 16279. (Contributed by Mario Carneiro, 15-Aug-2015.)
(({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋𝐴𝑌𝐵))

Theoremxpscf 16273 Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.)
(({𝑋} +𝑐 {𝑌}):2𝑜𝐴 ↔ (𝑋𝐴𝑌𝐴))

Theoremxpsfval 16274* The value of the function appearing in xpsval 16279. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))

Theoremxpsff1o 16275* The function appearing in xpsval 16279 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Theoremxpsfrn 16276* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ran 𝐹 = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Theoremxpsfrn2 16277* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))

Theoremxpsff1o2 16278* The function appearing in xpsval 16279 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹

Theoremxpsval 16279* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))       (𝜑𝑇 = (𝐹s 𝑈))

Theoremxpslem 16280* The indexed structure product that appears in xpsval 16279 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))       (𝜑 → ran 𝐹 = (Base‘𝑈))

Theoremxpsbas 16281 The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)       (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))

Theoremxpsaddlem 16282* Lemma for xpsadd 16283 and xpsmul 16284. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (𝐸𝑅)    &    × = (𝐸𝑆)    &    = (𝐸𝑇)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝑈 = ((Scalar‘𝑅)Xs({𝑅} +𝑐 {𝑆}))    &   ((𝜑({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹) → ((𝐹({𝐴} +𝑐 {𝐵})) (𝐹({𝐶} +𝑐 {𝐷}))) = (𝐹‘(({𝐴} +𝑐 {𝐵})(𝐸𝑈)({𝐶} +𝑐 {𝐷}))))    &   ((({𝑅} +𝑐 {𝑆}) Fn 2𝑜({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈) ∧ ({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈)) → (({𝐴} +𝑐 {𝐵})(𝐸𝑈)({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2𝑜 ↦ ((({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐶} +𝑐 {𝐷})‘𝑘))))       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpsadd 16283 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (+g𝑅)    &    × = (+g𝑆)    &    = (+g𝑇)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpsmul 16284 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (.r𝑅)    &    × = (.r𝑆)    &    = (.r𝑇)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpssca 16285 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.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐺 = (Scalar‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)       (𝜑𝐺 = (Scalar‘𝑇))

Theoremxpsvsca 16286 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐺 = (Scalar‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    × = ( ·𝑠𝑆)    &    = ( ·𝑠𝑇)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑌)    &   (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)    &   (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)       (𝜑 → (𝐴 𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Theoremxpsless 16287 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &    = (le‘𝑇)       (𝜑 ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌)))

Theoremxpsle 16288 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &    = (le‘𝑇)    &   𝑀 = (le‘𝑅)    &   𝑁 = (le‘𝑆)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶𝐵𝑁𝐷)))

7.2  Moore spaces

Syntaxcmre 16289 The class of Moore systems.
class Moore

Syntaxcmrc 16290 The class function generating Moore closures.
class mrCls

Syntaxcmri 16291 mrInd is a class function which takes a Moore system to its set of independent sets.
class mrInd

Syntaxcacs 16292 The class of algebraic closure (Moore) systems.
class ACS

Definitiondf-mre 16293* 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 20930) and vector spaces (lssmre 19014) 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 16297, mresspw 16299, mre1cl 16301 and mreintcl 16302 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16307); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16308. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})

Definitiondf-mrc 16294* Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 20931) and linear span (mrclsp 19037).

A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁𝑥) for all subsets 𝑥 of the base set (mrcssid 16324), (2) isotone, i.e., 𝑥𝑦 implies that (𝑁𝑥) ⊆ (𝑁𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 16323), and (3) idempotent, i.e., (𝑁‘(𝑁𝑥)) = (𝑁𝑥) for all subsets 𝑥 of the base set (mrcidm 16326.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation 𝑁 on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))

Definitiondf-mri 16295* In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})

Definitiondf-acs 16296* An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 8578 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})

Theoremismre 16297* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋𝑋𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → 𝑠𝐶)))

Theoremfnmre 16298 The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 20776 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore Fn V

Theoremmresspw 16299 A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)

Theoremmress 16300 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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