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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | slotsdifplendx2 17401 | The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval 48260. (Contributed by AV, 12-Nov-2024.) |
⊢ ((le‘ndx) ≠ (comp‘ndx) ∧ (le‘ndx) ≠ (Hom ‘ndx)) | ||
Theorem | slotsdifocndx 17402 | The index of the slot for the orthocomplementation is not the index of other slots. Formerly part of proof for prstcocval 48263. (Contributed by AV, 12-Nov-2024.) |
⊢ ((oc‘ndx) ≠ (comp‘ndx) ∧ (oc‘ndx) ≠ (Hom ‘ndx)) | ||
Theorem | resshom 17403 | Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ 𝐷 = (𝐶 ↾s 𝐴) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐻 = (Hom ‘𝐷)) | ||
Theorem | ressco 17404 | comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ 𝐷 = (𝐶 ↾s 𝐴) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = (comp‘𝐷)) | ||
Syntax | crest 17405 | Extend class notation with the function returning a subspace topology. |
class ↾t | ||
Syntax | ctopn 17406 | Extend class notation with the topology extractor function. |
class TopOpen | ||
Definition | df-rest 17407* | Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | ||
Definition | df-topn 17408 | Define the topology extractor function. This differs from df-tset 17255 when a structure has been restricted using df-ress 17213; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | ||
Theorem | restfn 17409 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
⊢ ↾t Fn (V × V) | ||
Theorem | topnfn 17410 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ TopOpen Fn V | ||
Theorem | restval 17411* | The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | ||
Theorem | elrest 17412* | The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) | ||
Theorem | elrestr 17413 | Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) | ||
Theorem | 0rest 17414 | Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (∅ ↾t 𝐴) = ∅ | ||
Theorem | restid2 17415 | The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽) | ||
Theorem | restsspw 17416 | The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 | ||
Theorem | firest 17417 | The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (fi‘(𝐽 ↾t 𝐴)) = ((fi‘𝐽) ↾t 𝐴) | ||
Theorem | restid 17418 | 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 ⇒ ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) | ||
Theorem | topnval 17419 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopSet‘𝑊) ⇒ ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊) | ||
Theorem | topnid 17420 | 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 𝐵 → 𝐽 = (TopOpen‘𝑊)) | ||
Theorem | topnpropd 17421 | 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)) | ||
Syntax | ctg 17422 | Extend class notation with a function that converts a basis to its corresponding topology. |
class topGen | ||
Syntax | cpt 17423 | Extend class notation with a function whose value is a product topology. |
class ∏t | ||
Syntax | c0g 17424 | Extend class notation with group identity element. |
class 0g | ||
Syntax | cgsu 17425 | Extend class notation to include finitely supported group sums. |
class Σg | ||
Definition | df-0g 17426* | Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 17427. The related theorems are provided later, see grpidval 18624. (Contributed by NM, 20-Aug-2011.) |
⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) | ||
Definition | df-gsum 17427* |
Define the group sum (also called "iterated 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 any other operation. 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. See gsum0 18647. 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. See gsumval2 18649 and gsumnunsn 34304. 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. See gsumval3 19874. 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 24075. Remark: this definition is required here because the symbol Σg is already used in df-prds 17432 and df-imas 17493. The related theorems are provided later, see gsumvalx 18639. (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(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦))))))) | ||
Definition | df-topgen 17428* | 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 22903). The first use of this definition is tgval 22902 but the token is used in df-pt 17429. See tgval3 22910 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.) |
⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | ||
Definition | df-pt 17429* | 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(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))})) | ||
Syntax | cprds 17430 | The function constructing structure products. |
class Xs | ||
Syntax | cpws 17431 | The function constructing structure powers. |
class ↑s | ||
Definition | df-prds 17432* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx), 𝑣〉, 〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx), 𝑠〉, 〈( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) | ||
Theorem | reldmprds 17433 | The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ Rel dom Xs | ||
Definition | df-pws 17434* | 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}))) | ||
Theorem | prdsbasex 17435* | Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.) |
⊢ 𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ⇒ ⊢ 𝐵 ∈ V | ||
Theorem | imasvalstr 17436 | An image structure 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〉 | ||
Theorem | prdsvalstr 17437 | 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〉 | ||
Theorem | prdsbaslem 17438 | Lemma for prdsbas 17442 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → 𝑈 = (({〈(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 = (𝐸‘𝑈) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝜑 → 𝑇 ∈ 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), ∙ 〉})) ⇒ ⊢ (𝜑 → 𝐴 = 𝑇) | ||
Theorem | prdsvallem 17439* | Lemma for prdsval 17440. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17440, dependency on df-hom 17260 removed. (Revised by AV, 13-Oct-2024.) |
⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V | ||
Theorem | prdsval 17440* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) & ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) & ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅))) & ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) & ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))) & ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) & ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑃 = (({〈(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), ∙ 〉}))) | ||
Theorem | prdssca 17441 | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃)) | ||
Theorem | prdsbas 17442* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) | ||
Theorem | prdsplusg 17443* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ + = (+g‘𝑃) ⇒ ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) | ||
Theorem | prdsmulr 17444* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ · = (.r‘𝑃) ⇒ ⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) | ||
Theorem | prdsvsca 17445* | 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑃) ⇒ ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))) | ||
Theorem | prdsip 17446* | Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ , = (·𝑖‘𝑃) ⇒ ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) | ||
Theorem | prdsle 17447* | Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ ≤ = (le‘𝑃) ⇒ ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) | ||
Theorem | prdsless 17448 | Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ ≤ = (le‘𝑃) ⇒ ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) | ||
Theorem | prdsds 17449* | Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝐷 = (dist‘𝑃) ⇒ ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))) | ||
Theorem | prdsdsfn 17450 | Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝐷 = (dist‘𝑃) ⇒ ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) | ||
Theorem | prdstset 17451 | Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝑂 = (TopSet‘𝑃) ⇒ ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅))) | ||
Theorem | prdshom 17452* | Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝐻 = (Hom ‘𝑃) ⇒ ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) | ||
Theorem | prdsco 17453* | Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ 𝐻 = (Hom ‘𝑃) & ⊢ ∙ = (comp‘𝑃) ⇒ ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) | ||
Theorem | prdsbas2 17454* | 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) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) | ||
Theorem | prdsbasmpt 17455* | 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) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ (Base‘(𝑅‘𝑥)))) | ||
Theorem | prdsbasfn 17456 | Points in the structure product are functions; use this with dffn5 6956 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑇 Fn 𝐼) | ||
Theorem | prdsbasprj 17457 | 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) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))) | ||
Theorem | prdsplusgval 17458* | 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) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) | ||
Theorem | prdsplusgfval 17459 | Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹‘𝐽)(+g‘(𝑅‘𝐽))(𝐺‘𝐽))) | ||
Theorem | prdsmulrval 17460* | Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ · = (.r‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)))) | ||
Theorem | prdsmulrfval 17461 | Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐹 · 𝐺)‘𝐽) = ((𝐹‘𝐽)(.r‘(𝑅‘𝐽))(𝐺‘𝐽))) | ||
Theorem | prdsleval 17462* | Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) | ||
Theorem | prdsdsval 17463* | Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ 𝐷 = (dist‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) | ||
Theorem | prdsvscaval 17464* | Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑥 ∈ 𝐼 ↦ (𝐹( ·𝑠 ‘(𝑅‘𝑥))(𝐺‘𝑥)))) | ||
Theorem | prdsvscafval 17465 | Scalar multiplication of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐹 · 𝐺)‘𝐽) = (𝐹( ·𝑠 ‘(𝑅‘𝐽))(𝐺‘𝐽))) | ||
Theorem | prdsbas3 17466* | The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝐾) | ||
Theorem | prdsbasmpt2 17467* | 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(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾)) | ||
Theorem | prdsbascl 17468* | An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐾) | ||
Theorem | prdsdsval2 17469* | Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ 𝐸 = (dist‘𝑅) & ⊢ 𝐷 = (dist‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)𝐸(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) | ||
Theorem | prdsdsval3 17470* | Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝐾 × 𝐾)) & ⊢ 𝐷 = (dist‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)𝐸(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) | ||
Theorem | pwsval 17471 | Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐹 = (Scalar‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) | ||
Theorem | pwsbas 17472 | Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐵 ↑m 𝐼) = (Base‘𝑌)) | ||
Theorem | pwselbasb 17473 | Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍) → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵)) | ||
Theorem | pwselbas 17474 | 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 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑋:𝐼⟶𝐵) | ||
Theorem | pwsplusgval 17475 | Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) | ||
Theorem | pwsmulrval 17476 | Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ · = (.r‘𝑅) & ⊢ ∙ = (.r‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘f · 𝐺)) | ||
Theorem | pwsle 17477 | Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑂 = (le‘𝑅) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ≤ = ( ∘r 𝑂 ∩ (𝐵 × 𝐵))) | ||
Theorem | pwsleval 17478* | Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑂 = (le‘𝑅) & ⊢ ≤ = (le‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) | ||
Theorem | pwsvscafval 17479 | Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ 𝐹 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) | ||
Theorem | pwsvscaval 17480 | Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ 𝐹 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) | ||
Theorem | pwssca 17481 | The ring of scalars of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝑆 = (Scalar‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑆 = (Scalar‘𝑌)) | ||
Theorem | pwsdiagel 17482 | Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑌) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐼 × {𝐴}) ∈ 𝐶) | ||
Theorem | pwssnf1o 17483* | Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s {𝐼}) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ({𝐼} × {𝑥})) & ⊢ 𝐶 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→𝐶) | ||
Syntax | cordt 17484 | Extend class notation with the order topology. |
class ordTop | ||
Syntax | cxrs 17485 | Extend class notation with the extended real number structure. |
class ℝ*𝑠 | ||
Definition | df-ordt 17486* | 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, +∞) = {𝑧 ∈ 𝑋 ∣ 𝑦 ≤ 𝑧} and (-∞, 𝑦] = {𝑧 ∈ 𝑋 ∣ 𝑧 ≤ 𝑦}, 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})))))) | ||
Definition | df-xrs 17487* | The extended real number structure. Unlike df-cnfld 21297, 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 21297. 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 ℂfld when restricted to ℝ. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ℝ*𝑠 = ({〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∪ {〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))〉}) | ||
Syntax | cqtop 17488 | Extend class notation with the quotient topology function. |
class qTop | ||
Syntax | cimas 17489 | Image structure function. |
class “s | ||
Syntax | cqus 17490 | Quotient structure function. |
class /s | ||
Syntax | cxps 17491 | Binary product structure function. |
class ×s | ||
Definition | df-qtop 17492* | 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) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗}) | ||
Definition | df-imas 17493* |
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 5691, 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 5690) and/or 𝑅 ( with df-ress 17213) 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), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉, 〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉} ∪ {〈(Scalar‘ndx), (Scalar‘𝑟)〉, 〈( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓‘𝑞)} ↦ (𝑓‘(𝑝( ·𝑠 ‘𝑟)𝑞)))〉, 〈(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑝(·𝑖‘𝑟)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 ∈ {ℎ ∈ ((𝑣 × 𝑣) ↑m (1...𝑛)) ∣ ((𝑓‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(ℎ‘𝑖))) = (𝑓‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑟) ∘ 𝑔))), ℝ*, < ))〉})) | ||
Definition | df-qus 17494* | Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 17493 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)) | ||
Definition | df-xps 17495* | Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑟)Xs{〈∅, 𝑟〉, 〈1o, 𝑠〉}))) | ||
Theorem | imasval 17496* | 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.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ + = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝐺 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ , = (·𝑖‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐸 = (dist‘𝑅) & ⊢ 𝑁 = (le‘𝑅) & ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) & ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) & ⊢ (𝜑 → ⊗ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝐼 = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝑝 , 𝑞)〉}) & ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) & ⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))) & ⊢ (𝜑 → ≤ = ((𝐹 ∘ 𝑁) ∘ ◡𝐹)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑈 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), ✚ 〉, 〈(.r‘ndx), ∙ 〉} ∪ {〈(Scalar‘ndx), 𝐺〉, 〈( ·𝑠 ‘ndx), ⊗ 〉, 〈(·𝑖‘ndx), 𝐼〉}) ∪ {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉})) | ||
Theorem | imasbas 17497 | 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.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) | ||
Theorem | imasds 17498* | 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.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = (dist‘𝑅) & ⊢ 𝐷 = (dist‘𝑈) ⇒ ⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))) | ||
Theorem | imasdsfn 17499 | The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = (dist‘𝑅) & ⊢ 𝐷 = (dist‘𝑈) ⇒ ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) | ||
Theorem | imasdsval 17500* | The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) & ⊢ 𝐸 = (dist‘𝑅) & ⊢ 𝐷 = (dist‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ⇒ ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) |
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