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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | strfv2d 16301 | Deduction version of strfv2 16302. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → Fun ◡◡𝑆) & ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strfv2 16302 | A variation on strfv 16303 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑆 ∈ V & ⊢ Fun ◡◡𝑆 & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strfv 16303 | Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 17332) with a component extractor 𝐸 (such as the base set extractor df-base 16261). By virtue of ndxid 16281, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝑆 Struct 𝑋 & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strfv3 16304 | Variant on strfv 16303 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) |
⊢ (𝜑 → 𝑈 = 𝑆) & ⊢ 𝑆 Struct 𝑋 & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ 𝐴 = (𝐸‘𝑈) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | strssd 16305 | Deduction version of strss 16306. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑇) & ⊢ (𝜑 → 𝑆 ⊆ 𝑇) & ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) | ||
Theorem | strss 16306 | Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.) |
⊢ 𝑇 ∈ V & ⊢ Fun 𝑇 & ⊢ 𝑆 ⊆ 𝑇 & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ⇒ ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) | ||
Theorem | str0 16307 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐹 = Slot 𝐼 ⇒ ⊢ ∅ = (𝐹‘∅) | ||
Theorem | base0 16308 | The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ ∅ = (Base‘∅) | ||
Theorem | strfvi 16309 | Structure slot extractors cannot distinguish between proper classes and ∅, so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑋 = (𝐸‘𝑆) ⇒ ⊢ 𝑋 = (𝐸‘( I ‘𝑆)) | ||
Theorem | setsid 16310 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐸 = Slot (𝐸‘ndx) ⇒ ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝐸‘(𝑊 sSet 〈(𝐸‘ndx), 𝐶〉))) | ||
Theorem | setsnid 16311 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ 𝐷 ⇒ ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) | ||
Theorem | sbcie2s 16312* | A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
⊢ 𝐴 = (𝐸‘𝑊) & ⊢ 𝐵 = (𝐹‘𝑊) & ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑)) | ||
Theorem | sbcie3s 16313* | A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
⊢ 𝐴 = (𝐸‘𝑊) & ⊢ 𝐵 = (𝐹‘𝑊) & ⊢ 𝐶 = (𝐺‘𝑊) & ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑)) | ||
Theorem | baseval 16314 | Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐾 ∈ V ⇒ ⊢ (Base‘𝐾) = (𝐾‘1) | ||
Theorem | baseid 16315 | Utility theorem: index-independent form of df-base 16261. (Contributed by NM, 20-Oct-2012.) |
⊢ Base = Slot (Base‘ndx) | ||
Theorem | elbasfv 16316 | Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
⊢ 𝑆 = (𝐹‘𝑍) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V) | ||
Theorem | elbasov 16317 | Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ Rel dom 𝑂 & ⊢ 𝑆 = (𝑋𝑂𝑌) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) | ||
Theorem | strov2rcl 16318 | Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.) |
⊢ 𝑆 = (𝐼𝐹𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ Rel dom 𝐹 ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) | ||
Theorem | basendx 16319 | Index value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.) |
⊢ (Base‘ndx) = 1 | ||
Theorem | basendxnn 16320 | The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) |
⊢ (Base‘ndx) ∈ ℕ | ||
Theorem | basprssdmsets 16321 | The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet 〈𝐼, 𝐸〉)) | ||
Theorem | reldmress 16322 | The structure restriction is a proper operator, so it can be used with ovprc1 6960. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ Rel dom ↾s | ||
Theorem | ressval 16323 | Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) | ||
Theorem | ressid2 16324 | General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) | ||
Theorem | ressval2 16325 | Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) | ||
Theorem | ressbas 16326 | Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) | ||
Theorem | ressbas2 16327 | Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑅)) | ||
Theorem | ressbasss 16328 | The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (Base‘𝑅) ⊆ 𝐵 | ||
Theorem | resslem 16329 | Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐶 = (𝐸‘𝑊) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 1 < 𝑁 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) | ||
Theorem | ress0 16330 | All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (∅ ↾s 𝐴) = ∅ | ||
Theorem | ressid 16331 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) | ||
Theorem | ressinbas 16332 | Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵))) | ||
Theorem | ressval3d 16333 | Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) |
⊢ 𝑅 = (𝑆 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐸 = (Base‘ndx) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑆) & ⊢ (𝜑 → 𝐸 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉)) | ||
Theorem | ressval3dOLD 16334 | Obsolete version of ressval3d 16333 as of 3-Jul-2022. (Contributed by AV, 14-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑅 = (𝑆 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐸 = (Base‘ndx) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑆) & ⊢ (𝜑 → 𝐸 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝑌) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉)) | ||
Theorem | ressress 16335 | Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) | ||
Theorem | ressabs 16336 | Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) | ||
Theorem | wunress 16337 | Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈) | ||
Syntax | cplusg 16338 | Extend class notation with group (addition) operation. |
class +g | ||
Syntax | cmulr 16339 | Extend class notation with ring multiplication. |
class .r | ||
Syntax | cstv 16340 | Extend class notation with involution. |
class *𝑟 | ||
Syntax | csca 16341 | Extend class notation with scalar field. |
class Scalar | ||
Syntax | cvsca 16342 | Extend class notation with scalar product. |
class ·𝑠 | ||
Syntax | cip 16343 | Extend class notation with Hermitian form (inner product). |
class ·𝑖 | ||
Syntax | cts 16344 | Extend class notation with the topology component of a topological space. |
class TopSet | ||
Syntax | cple 16345 | Extend class notation with "less than or equal to" for posets. |
class le | ||
Syntax | coc 16346 | Extend class notation with the class of orthocomplementation extractors. |
class oc | ||
Syntax | cds 16347 | Extend class notation with the metric space distance function. |
class dist | ||
Syntax | cunif 16348 | Extend class notation with the uniform structure. |
class UnifSet | ||
Syntax | chom 16349 | Extend class notation with the hom-set structure. |
class Hom | ||
Syntax | cco 16350 | Extend class notation with the composition operation. |
class comp | ||
Definition | df-plusg 16351 | Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ +g = Slot 2 | ||
Definition | df-mulr 16352 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ .r = Slot 3 | ||
Definition | df-starv 16353 | Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ *𝑟 = Slot 4 | ||
Definition | df-sca 16354 | Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ Scalar = Slot 5 | ||
Definition | df-vsca 16355 | Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ·𝑠 = Slot 6 | ||
Definition | df-ip 16356 | Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ·𝑖 = Slot 8 | ||
Definition | df-tset 16357 | Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ TopSet = Slot 9 | ||
Definition | df-ple 16358 | Define "less than or equal to" ordering extractor for posets and related structures. We use ;10 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ le = Slot ;10 | ||
Definition | df-ocomp 16359 | Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ oc = Slot ;11 | ||
Definition | df-ds 16360 | Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ dist = Slot ;12 | ||
Definition | df-unif 16361 | Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ UnifSet = Slot ;13 | ||
Definition | df-hom 16362 | Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ Hom = Slot ;14 | ||
Definition | df-cco 16363 | Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ comp = Slot ;15 | ||
Theorem | strleun 16364 | Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐹 Struct 〈𝐴, 𝐵〉 & ⊢ 𝐺 Struct 〈𝐶, 𝐷〉 & ⊢ 𝐵 < 𝐶 ⇒ ⊢ (𝐹 ∪ 𝐺) Struct 〈𝐴, 𝐷〉 | ||
Theorem | strle1 16365 | Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼 ⇒ ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 | ||
Theorem | strle2 16366 | Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼 & ⊢ 𝐼 < 𝐽 & ⊢ 𝐽 ∈ ℕ & ⊢ 𝐵 = 𝐽 ⇒ ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉 | ||
Theorem | strle3 16367 | Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼 & ⊢ 𝐼 < 𝐽 & ⊢ 𝐽 ∈ ℕ & ⊢ 𝐵 = 𝐽 & ⊢ 𝐽 < 𝐾 & ⊢ 𝐾 ∈ ℕ & ⊢ 𝐶 = 𝐾 ⇒ ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉} Struct 〈𝐼, 𝐾〉 | ||
Theorem | plusgndx 16368 | Index value of the df-plusg 16351 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (+g‘ndx) = 2 | ||
Theorem | plusgid 16369 | Utility theorem: index-independent form of df-plusg 16351. (Contributed by NM, 20-Oct-2012.) |
⊢ +g = Slot (+g‘ndx) | ||
Theorem | opelstrbas 16370 | The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.) |
⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑉 = (Base‘𝑆)) | ||
Theorem | 1strstr 16371 | A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} ⇒ ⊢ 𝐺 Struct 〈1, 1〉 | ||
Theorem | 1strbas 16372 | The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) | ||
Theorem | 1strwunbndx 16373 | A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) | ||
Theorem | 1strwun 16374 | A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈) | ||
Theorem | 2strstr 16375 | A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁 & ⊢ 1 < 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐺 Struct 〈1, 𝑁〉 | ||
Theorem | 2strbas 16376 | The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁 & ⊢ 1 < 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) | ||
Theorem | 2strop 16377 | The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁 & ⊢ 1 < 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺)) | ||
Theorem | 2strstr1 16378 | A constructed two-slot structure. Version of 2strstr 16375 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢ (Base‘ndx) < 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑁〉 | ||
Theorem | 2strbas1 16379 | The base set of a constructed two-slot structure. Version of 2strbas 16376 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢ (Base‘ndx) < 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) | ||
Theorem | 2strop1 16380 | The other slot of a constructed two-slot structure. Version of 2strop 16377 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢ (Base‘ndx) < 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐸 = Slot 𝑁 ⇒ ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺)) | ||
Theorem | basendxnplusgndx 16381 | The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) |
⊢ (Base‘ndx) ≠ (+g‘ndx) | ||
Theorem | grpstr 16382 | A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝐺 Struct 〈1, 2〉 | ||
Theorem | grpbase 16383 | The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) | ||
Theorem | grpplusg 16384 | The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ ( + ∈ 𝑉 → + = (+g‘𝐺)) | ||
Theorem | ressplusg 16385 | +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) | ||
Theorem | grpbasex 16386 | The base of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpbase 16383 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {〈1, 𝐵〉, 〈2, + 〉} ⇒ ⊢ 𝐵 = (Base‘𝐺) | ||
Theorem | grpplusgx 16387 | The operation of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpplusg 16384 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {〈1, 𝐵〉, 〈2, + 〉} ⇒ ⊢ + = (+g‘𝐺) | ||
Theorem | mulrndx 16388 | Index value of the df-mulr 16352 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (.r‘ndx) = 3 | ||
Theorem | mulrid 16389 | Utility theorem: index-independent form of df-mulr 16352. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ .r = Slot (.r‘ndx) | ||
Theorem | plusgndxnmulrndx 16390 | The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
⊢ (+g‘ndx) ≠ (.r‘ndx) | ||
Theorem | basendxnmulrndx 16391 | The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
⊢ (Base‘ndx) ≠ (.r‘ndx) | ||
Theorem | rngstr 16392 | A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⇒ ⊢ 𝑅 Struct 〈1, 3〉 | ||
Theorem | rngbase 16393 | The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝑅)) | ||
Theorem | rngplusg 16394 | The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⇒ ⊢ ( + ∈ 𝑉 → + = (+g‘𝑅)) | ||
Theorem | rngmulr 16395 | The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ⇒ ⊢ ( · ∈ 𝑉 → · = (.r‘𝑅)) | ||
Theorem | starvndx 16396 | Index value of the df-starv 16353 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (*𝑟‘ndx) = 4 | ||
Theorem | starvid 16397 | Utility theorem: index-independent form of df-starv 16353. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ *𝑟 = Slot (*𝑟‘ndx) | ||
Theorem | ressmulr 16398 | .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆)) | ||
Theorem | ressstarv 16399 | *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ ∗ = (*𝑟‘𝑅) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆)) | ||
Theorem | srngstr 16400 | A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗ 〉}) ⇒ ⊢ 𝑅 Struct 〈1, 4〉 |
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