P. 172 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	strfvi 17101	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	fveqprc 17102	Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like zlmlem 21454. (Contributed by AV, 31-Oct-2024.)
⊢ (𝐸‘∅) = ∅    &   ⊢ 𝑌 = (𝐹‘𝑋)    ⇒   ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
Theorem	oveqprc 17103	Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 33296. (Contributed by AV, 31-Oct-2024.)
⊢ (𝐸‘∅) = ∅    &   ⊢ 𝑍 = (𝑋𝑂𝑌)    &   ⊢ Rel dom 𝑂    ⇒   ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))
7.1.1.4  Structure component indices The structure component index extractor ndx, defined in this subsection, is used to get the numeric argument from a defined structure component extractor such as df-base 17121 (see ndxarg 17107). For each defined structure component extractor, there should be a corresponding specific theorem providing its index, like basendx 17129. The usage of these theorems, however, is discouraged since the particular value for the index is an implementation detail. It is generally sufficient to work with (Base‘ndx) instead of the hard-coded index value, and use theorems such as baseid 17123 and basendxnplusgndx 17191. The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint (for example in proofs such as cznabel 48297, based on setsnid 17119) or even ordered (in proofs such as lmodstr 17229). The requirement that the indices are distinct is necessary for sets of ordered pairs to be extensible structures, whereas the ordering allows for proofs avoiding the usage of quadradically many inequalities (compare cnfldfun 21306 with cnfldfunALT 21307). As for the inequalities, it is recommended to provide them explicitly as theorems like basendxnplusgndx 17191, whenever they are required. Since these theorems use discouraged slot theorems, they should be placed near the definition of a slot (within the same subsection), so that the range of usages of discouraged theorems is tightly limited. Although there could be quadradically many of them in the total number of indices, much less are actually available (and not much more are expected). As for the ordering, there are some theorems like basendxltplusgndx 17190 providing the less-than relationship between two indices. These theorems are also proved by discouraged theorems, so they should be placed near the definition of a slot (within the same subsection), too. However, since such theorems are rarely used (in structure building theorems *str like rngstr 17202), it is not recommended to provide explicit theorems for all of them, but to use the (discouraged) *ndx theorems as in lmodstr 17229. Therefore, *str theorems generally depend on the hard-coded values of the indices.
Syntax	cnx 17104	Extend class notation with the structure component index extractor.
class ndx
Definition	df-ndx 17105	Define the structure component index extractor. See Theorem ndxarg 17107 to understand its purpose. The restriction to ℕ ensures that ndx is a set. The restriction to some set is necessary since I is a proper class. In principle, we could have chosen ℂ or (if we revise all structure component definitions such as df-base 17121) another set such as the set of finite ordinals ω (df-om 7797). (Contributed by NM, 4-Sep-2011.)
⊢ ndx = ( I ↾ ℕ)
Theorem	wunndx 17106	Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → ndx ∈ 𝑈)
Theorem	ndxarg 17107	Get the numeric argument from a defined structure component extractor such as df-base 17121. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	ndxid 17108	A structure component extractor is defined by its own index. This theorem, together with strfv 17114 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17121 and the ;10 in df-ple 17181, making it easier to change should the need arise. For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨;10, 𝐿⟩}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐸 = Slot (𝐸‘ndx)
Theorem	strndxid 17109	The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) (New usage is discouraged.) Use strfvnd 17096 directly with 𝑁 set to (𝐸‘ndx) if possible.
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆))
Theorem	setsidvald 17110	Value of the structure replacement function, deduction version. Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))
Theorem	strfvd 17111	Deduction version of strfv 17114. (Contributed by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv2d 17112	Deduction version of strfv2 17113. (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 17113	A variation on strfv 17114 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 17114	Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 18219) with a component extractor 𝐸 (such as the base set extractor df-base 17121). By virtue of ndxid 17108, 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 17115	Variant on strfv 17114 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 17116	Deduction version of strss 17117. (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 17117	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	setsid 17118	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 17119	Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ 𝐷    ⇒   ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
7.1.1.5  Base sets
Syntax	cbs 17120	Extend class notation with the class of all base set extractors.
class Base
Definition	df-base 17121	Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form baseid 17123 instead. (New usage is discouraged.)
⊢ Base = Slot 1
Theorem	baseval 17122	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 17123	Utility theorem: index-independent form of df-base 17121. (Contributed by NM, 20-Oct-2012.)
⊢ Base = Slot (Base‘ndx)
Theorem	basfn 17124	The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ Base Fn V
Theorem	base0 17125	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ ∅ = (Base‘∅)
Theorem	elbasfv 17126	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 17127	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 17128	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 17129	Index value of the base set extractor. (Contributed by Mario Carneiro, 2-Aug-2013.) Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail, see section header comment mmtheorems.html#cnx for more information. (New usage is discouraged.)
⊢ (Base‘ndx) = 1
Theorem	basendxnn 17130	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.) (Proof shortened by AV, 13-Oct-2024.)
⊢ (Base‘ndx) ∈ ℕ
Theorem	basndxelwund 17131	The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 17137. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
Theorem	basprssdmsets 17132	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	opelstrbas 17133	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 17134	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
Theorem	1strbas 17135	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	1strwunbndx 17136	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 17137	A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	2strstr 17138	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
Theorem	2strbas 17139	The base set of a constructed two-slot structure 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	2strop 17140	The other slot of a constructed two-slot structure 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))
7.1.1.6  Base set restrictions
Syntax	cress 17141	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-ress 17142*	Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp). (Credit for this operator goes to Mario Carneiro.) See ressbas 17147 for the altered base set, and resseqnbas 17153 (subrg0 20495, ressplusg 17195, subrg1 20498, ressmulr 17211) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Theorem	reldmress 17143	The structure restriction is a proper operator, so it can be used with ovprc1 7385. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ Rel dom ↾s
Theorem	ressval 17144	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 17145	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 17146	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 17147	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbasssg 17148	The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)
Theorem	ressbas2 17149	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 17150	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.) (Proof shortened by SN, 25-Feb-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasssOLD 17151	Obsolete version of ressbas 17147 as of 25-Feb-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasss2 17152	The base set of a restriction to 𝐴 is a subset of 𝐴. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐴
Theorem	resseqnbas 17153	The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐶 = (𝐸‘𝑊)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Base‘ndx)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
Theorem	ress0 17154	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 17155	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)
Theorem	ressinbas 17156	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 17157	Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 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	ressress 17158	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 17159	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
Theorem	wunress 17160	Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈)
7.1.2  Slot definitions
Syntax	cplusg 17161	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 17162	Extend class notation with ring multiplication.
class .r
Syntax	cstv 17163	Extend class notation with involution.
class *𝑟
Syntax	csca 17164	Extend class notation with scalar field.
class Scalar
Syntax	cvsca 17165	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 17166	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 17167	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 17168	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 17169	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 17170	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 17171	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 17172	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 17173	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17174	Define group operation. In the context of less restrictive structures, this operation is also called magma, semigroup or monoid operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form plusgid 17188 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17175	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11110 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17176	Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form starvid 17207 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17177	Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form scaid 17219 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17178	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17224 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17179	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17235 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17180	Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form tsetid 17257 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17181	Define "less than or equal to" ordering extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) Use its index-independent form pleid 17271 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17182	Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ocid 17286 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17183	Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form dsid 17290 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17184	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17300 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17185	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17316 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17186	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17318 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17187	Index value of the df-plusg 17174 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17188	Utility theorem: index-independent form of df-plusg 17174. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17189	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
⊢ (+g‘ndx) ∈ ℕ
Theorem	basendxltplusgndx 17190	The index of the slot for the base set is less than the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
⊢ (Base‘ndx) < (+g‘ndx)
Theorem	basendxnplusgndx 17191	The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Oct-2024.)
⊢ (Base‘ndx) ≠ (+g‘ndx)
Theorem	grpstr 17192	A constructed group is a structure. Version not depending on the implementation of the indices. (Contributed by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (+g‘ndx)⟩
Theorem	grpbase 17193	The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	grpplusg 17194	The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐺))
Theorem	ressplusg 17195	+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 17196	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 17193 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17197	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 17194 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17198	Index value of the df-mulr 17175 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17199	Utility theorem: index-independent form of df-mulr 17175. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17200	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.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (Base‘ndx) ≠ (.r‘ndx)