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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrestid 15801 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Theoremtopnval 15802 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽t 𝐵) = (TopOpen‘𝑊)

Theoremtopnid 15803 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽 ⊆ 𝒫 𝐵𝐽 = (TopOpen‘𝑊))

Theoremtopnpropd 15804 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))       (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Syntaxctg 15805 Extend class notation with a function that converts a basis to its corresponding topology.
class topGen

Syntaxcpt 15806 Extend class notation with a function whose value is a product topology.
class t

Syntaxc0g 15807 Extend class notation with group identity element.
class 0g

Syntaxcgsu 15808 Extend class notation to include finitely supported group sums.
class Σg

Definitiondf-0g 15809* Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 15810. The related theorems are provided later, see grpidval 16975. (Contributed by NM, 20-Aug-2011.)
0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))

Definitiondf-gsum 15810* Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or whatever. The variable 𝑘 is normally a free variable in 𝐵 ( i.e. 𝐵 can be thought of as 𝐵(𝑘)). The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺.

1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity.

2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. (𝐵(1) + 𝐵(2)) + 𝐵(3) etc.

3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. See df-tsms 21643. Remark: this definition is required here because the symbol Σg is already used in df-prds 15815 and df-imas 15876. The related theorems are provided later, see gsumvalx 16985. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g𝑤)𝑦) = 𝑦 ∧ (𝑦(+g𝑤)𝑥) = 𝑦)} / 𝑜if(ran 𝑓𝑜, (0g𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝑤), 𝑓)‘𝑛))), (℩𝑥𝑔[(𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(#‘𝑦))–1-1-onto𝑦𝑥 = (seq1((+g𝑤), (𝑓𝑔))‘(#‘𝑦)))))))

Definitiondf-topgen 15811* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78 (see tgval2 20474). See tgval3 20481 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})

Definitiondf-pt 15812* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))

Syntaxcprds 15813 The function constructing structure products.
class Xs

Syntaxcpws 15814 The function constructing structure powers.
class s

Definitiondf-prds 15815* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))

Theoremreldmprds 15816 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Rel dom Xs

Definitiondf-pws 15817* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))

Theoremprdsbasex 15818* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))       𝐵 ∈ V

Theoremimasvalstr 15819 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩})       𝑈 Struct ⟨1, 12⟩

Theoremprdsvalstr 15820 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})) Struct ⟨1, 15⟩

Theoremprdsvallem 15821 Lemma for prdsbas 15824 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))    &   𝐴 = (𝐸𝑈)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑇 ∈ V)    &   {⟨(𝐸‘ndx), 𝑇⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩}))       (𝜑𝐴 = 𝑇)

Theoremprdsval 15822* Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   𝐾 = (Base‘𝑆)    &   (𝜑 → dom 𝑅 = 𝐼)    &   (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))    &   (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑× = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))    &   (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))    &   (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})    &   (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))    &   (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))    &   (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))    &   (𝜑𝑆𝑊)    &   (𝜑𝑅𝑍)       (𝜑𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))

Theoremprdssca 15823 Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑆 = (Scalar‘𝑃))

Theoremprdsbas 15824* Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)       (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))

Theoremprdsplusg 15825* Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    + = (+g𝑃)       (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsmulr 15826* Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    · = (.r𝑃)       (𝜑· = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsvsca 15827* Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠𝑃)       (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsip 15828* Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    , = (·𝑖𝑃)       (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))

Theoremprdsle 15829* Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    = (le‘𝑃)       (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})

Theoremprdsless 15830 Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    = (le‘𝑃)       (𝜑 ⊆ (𝐵 × 𝐵))

Theoremprdsds 15831* Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)       (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))

Theoremprdsdsfn 15832 Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)       (𝜑𝐷 Fn (𝐵 × 𝐵))

Theoremprdstset 15833 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝑂 = (TopSet‘𝑃)       (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Theoremprdshom 15834* Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)       (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))

Theoremprdsco 15835* Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    &    = (comp‘𝑃)       (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))

Theoremprdsbas2 15836* The base set of a structure product is an indexed set product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))

Theoremprdsbasmpt 15837* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → ((𝑥𝐼𝑈) ∈ 𝐵 ↔ ∀𝑥𝐼 𝑈 ∈ (Base‘(𝑅𝑥))))

Theoremprdsbasfn 15838 Points in the structure product are functions; use this with dffn5 6035 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝑇𝐵)       (𝜑𝑇 Fn 𝐼)

Theoremprdsbasprj 15839 Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝑇𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → (𝑇𝐽) ∈ (Base‘(𝑅𝐽)))

Theoremprdsplusgval 15840* Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑌)       (𝜑 → (𝐹 + 𝐺) = (𝑥𝐼 ↦ ((𝐹𝑥)(+g‘(𝑅𝑥))(𝐺𝑥))))

Theoremprdsplusgfval 15841 Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑌)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹𝐽)(+g‘(𝑅𝐽))(𝐺𝐽)))

Theoremprdsmulrval 15842* Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑌)       (𝜑 → (𝐹 · 𝐺) = (𝑥𝐼 ↦ ((𝐹𝑥)(.r‘(𝑅𝑥))(𝐺𝑥))))

Theoremprdsmulrfval 15843 Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑌)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 · 𝐺)‘𝐽) = ((𝐹𝐽)(.r‘(𝑅𝐽))(𝐺𝐽)))

Theoremprdsleval 15844* Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    = (le‘𝑌)       (𝜑 → (𝐹 𝐺 ↔ ∀𝑥𝐼 (𝐹𝑥)(le‘(𝑅𝑥))(𝐺𝑥)))

Theoremprdsdsval 15845* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)(dist‘(𝑅𝑥))(𝐺𝑥))) ∪ {0}), ℝ*, < ))

Theoremprdsvscaval 15846* Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 · 𝐺) = (𝑥𝐼 ↦ (𝐹( ·𝑠 ‘(𝑅𝑥))(𝐺𝑥))))

Theoremprdsvscafval 15847 Scalar multiplication of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 · 𝐺)‘𝐽) = (𝐹( ·𝑠 ‘(𝑅𝐽))(𝐺𝐽)))

Theoremprdsbas3 15848* The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)       (𝜑𝐵 = X𝑥𝐼 𝐾)

Theoremprdsbasmpt2 15849* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)       (𝜑 → ((𝑥𝐼𝑈) ∈ 𝐵 ↔ ∀𝑥𝐼 𝑈𝐾))

Theoremprdsbascl 15850* An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐹𝐵)       (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ 𝐾)

Theoremprdsdsval2 15851* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)𝐸(𝐺𝑥))) ∪ {0}), ℝ*, < ))

Theoremprdsdsval3 15852* Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐾 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝐾 × 𝐾))    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)𝐸(𝐺𝑥))) ∪ {0}), ℝ*, < ))

Theorempwsval 15853 Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐹 = (Scalar‘𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))

Theorempwsbas 15854 Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)       ((𝑅𝑉𝐼𝑊) → (𝐵𝑚 𝐼) = (Base‘𝑌))

Theorempwselbasb 15855 Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)       ((𝑅𝑊𝐼𝑍) → (𝑋𝑉𝑋:𝐼𝐵))

Theorempwselbas 15856 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.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)    &   (𝜑𝑅𝑊)    &   (𝜑𝐼𝑍)    &   (𝜑𝑋𝑉)       (𝜑𝑋:𝐼𝐵)

Theorempwsplusgval 15857 Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑅)    &    = (+g𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 + 𝐺))

Theorempwsmulrval 15858 Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑅)    &    = (.r𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 · 𝐺))

Theorempwsle 15859 Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑂 = (le‘𝑅)    &    = (le‘𝑌)       ((𝑅𝑉𝐼𝑊) → = ( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵)))

Theorempwsleval 15860* Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑂 = (le‘𝑅)    &    = (le‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 𝐺 ↔ ∀𝑥𝐼 (𝐹𝑥)𝑂(𝐺𝑥)))

Theorempwsvscafval 15861 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 15862 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 15863 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 15864 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 15865* 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 15866 Extend class notation with the order topology.
class ordTop

Syntaxcxrs 15867 Extend class notation with the extended real number structure.
class *𝑠

Definitiondf-ordt 15868* 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 15869* The extended real number structure. Unlike df-cnfld 19472, 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 19472. 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 15870 Extend class notation with the quotient topology function.
class qTop

Syntaxcimas 15871 Image structure function.
class s

Syntaxcimasold 15872 Image structure function (old version).
class s

Syntaxcqus 15873 Quotient structure function.
class /s

Syntaxcxps 15874 Binary product structure function.
class ×s

Definitiondf-qtop 15875* 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 15876* 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 4945, 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 4944) and/or 𝑅 ( with df-ress 15586) 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-imasOLD 15877* 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 4945, 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 4944) and/or 𝑅 ( with df-ress 15586) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) Obsolete version of df-imas 15876 as of 6-Oct-2020. (New usage is discouraged.)

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 ↦ sup( 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 15878* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 15876 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 15879* 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 15880* 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⟩}))

TheoremimasvalOLD 15881* 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.) Obsolete version of imasval 15880 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(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 ↦ sup( 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 15882 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 15883* 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 15884 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 15885* 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 15886* 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 15887* 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 15888* 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 15889 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 15890* 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 15891* 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 15892 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 15893 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))

TheoremimasbasOLD 15894 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.) Obsolete version of imasbas 15882 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)       (𝜑𝐵 = (Base‘𝑈))

TheoremimasdsOLD 15895* 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.) Obsolete version of imasds 15883 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)       (𝜑𝐷 = (𝑥𝐵, 𝑦𝐵 ↦ sup( 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*, < )))

TheoremimasdsfnOLD 15896 The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) Obsolete version of imasdsfn 15884 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑈)       (𝜑𝐷 Fn (𝐵 × 𝐵))

TheoremimasdsvalOLD 15897* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) Obsolete version of imasdsval 15885 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(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) = sup( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ))

Theoremimasdsval2OLD 15898* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) Obsolete version of imasdsval2 15886 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(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) = sup( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝑇𝑔))), ℝ*, < ))

Theoremf1ocpbllem 15899 Lemma for f1ocpbl 15900. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremf1ocpbl 15900 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))))

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