Home Metamath Proof ExplorerTheorem List (p. 168 of 451) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28679) Hilbert Space Explorer (28680-30202) Users' Mathboxes (30203-45095)

Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-pws 16701* 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 16702* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))       𝐵 ∈ V

Theoremimasvalstr 16703 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 16704 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 16705 Lemma for prdsbas 16708 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 16706* 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 16707 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 16708* 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 16709* 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 16710* 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 16711* 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 16712* 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 16713* 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 16714 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 16715* 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 16716 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 16717 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 16718* 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 16719* 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 16720* 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 16721* 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 16722 Points in the structure product are functions; use this with dffn5 6697 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 16723 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 16724* 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 16725 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 16726* 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 16727 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 16728* 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 16729* 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 16730* 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 16731 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 16732* 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 16733* 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 16734* 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 16735* 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 16736* 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 16737 Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐹 = (Scalar‘𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))

Theorempwsbas 16738 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 16739 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 16740 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 16741 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 16742 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 16743 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 16744* 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 16745 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 16746 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 16747 The ring of scalars of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝑆 = (Scalar‘𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑆 = (Scalar‘𝑌))

Theorempwsdiagel 16748 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 16749* 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 16750 Extend class notation with the order topology.
class ordTop

Syntaxcxrs 16751 Extend class notation with the extended real number structure.
class *𝑠

Definitiondf-ordt 16752* 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 16753* The extended real number structure. Unlike df-cnfld 20521, 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 20521. 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 16754 Extend class notation with the quotient topology function.
class qTop

Syntaxcimas 16755 Image structure function.
class s

Syntaxcqus 16756 Quotient structure function.
class /s

Syntaxcxps 16757 Binary product structure function.
class ×s

Definitiondf-qtop 16758* 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 16759* 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 5541, 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 5540) and/or 𝑅 ( with df-ress 16469) 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 16760* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 16759 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 16761* 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⟩})))

Theoremimasval 16762* 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 16763 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 16764* 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 16765 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 16766* 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 16767* 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 16768* 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 16769* 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 16770 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 16771* 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 16772* 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 16773 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 16774 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 16775 Lemma for f1ocpbl 16776. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑉1-1-onto𝑋)       ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremf1ocpbl 16776 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 16777 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 16778 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 16779* 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 16780* 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 16781* 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 16782* 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 16783* 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 16784* 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 16785* 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 16786* 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 16787* 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 16788* 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 16789* 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 16790* 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 16791 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 16792* 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 16793* 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 16794* The function in qusval 16793 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 16795 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 16796 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 16797 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 16798* Value of the function in qusval 16793. (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 16799* Lemma for ercpbl 16800. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑 Er 𝑉)    &   (𝜑𝑉 ∈ V)    &   𝐹 = (𝑥𝑉 ↦ [𝑥] )    &   (𝜑𝐴𝑉)       (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 𝐵))

Theoremercpbl 16800* 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))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45095
 Copyright terms: Public domain < Previous  Next >