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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-sets 15701* Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 15702 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 18313, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the +g slot instead of the .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

Definitiondf-ress 15702* 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 15757 for the altered base set, and resslem 15760 (subrg0 18610, ressplusg 15818, subrg1 18613, ressmulr 15829) 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))⟩)))

Theorembrstruct 15703 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Rel Struct

Theoremisstruct2 15704 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Theoremisstruct 15705 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))

Theoremstructcnvcnv 15706 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋𝐹 = (𝐹 ∖ {∅}))

Theoremstructfun 15707 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct 𝑋       Fun 𝐹

Theoremstructfn 15708 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))

Theoremslotfn 15709 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐸 = Slot 𝑁       𝐸 Fn V

Theoremstrfvnd 15710 Deduction version of strfvn 15712. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))

Theoremwunndx 15711 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       (𝜑 → ndx ∈ 𝑈)

Theoremstrfvn 15712 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 15700) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 16769) where 𝑆 ∈ Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 15735. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁       (𝐸𝑆) = (𝑆𝑁)

Theoremstrfvss 15713 A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐸 = Slot 𝑁       (𝐸𝑆) ⊆ ran 𝑆

Theoremwunstr 15714 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = Slot 𝑁    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)       (𝜑 → (𝐸𝑆) ∈ 𝑈)

Theoremndxarg 15715 Get the numeric argument from a defined structure component extractor such as df-base 15700. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theoremndxid 15716 A structure component extractor is defined by its own index. This theorem, together with strfv 15735 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 15700 and the 10 in df-ple 15788, 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.)

𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)

Theoremstrndxid 15717 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))

Theoremreldmsets 15718 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet

Theoremsetsvalg 15719 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))

Theoremsetsval 15720 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))

Theoremsetsidvald 15721 Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))

Theoremfvsetsid 15722 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)

Theoremfsets 15723 The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
(((𝐹𝑉𝐹:𝐴𝐵) ∧ 𝑋𝐴𝑌𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴𝐵)

Theoremsetsdm 15724 The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))

Theoremsetsfun 15725 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))

Theoremsetsfun0 15726 A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This varaint of setsfun 15725 is useful for proofs based on isstruct 15705 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsstruct 15727 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.)
((𝐺𝑈𝐸𝑉𝐼 ∈ (ℤ𝑀)) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩))

Theoremwunsets 15728 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)    &   (𝜑𝐴𝑈)       (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈)

Theoremsetsres 15729 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑆𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))

Theoremsetsabs 15730 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
((𝑆𝑉𝐶𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Theoremsetscom 15731 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝑆𝑉𝐴𝐵) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))

Theoremstrfvd 15732 Deduction version of strfv 15735. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrfv2d 15733 Deduction version of strfv 15735. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrfv2 15734 A variation on strfv 15735 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       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrfv 15735 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 16769) with a component extractor 𝐸 (such as the base set extractor df-base 15700). By virtue of ndxid 15716, 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       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrfv3 15736 Variant on strfv 15735 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)

Theoremstrssd 15737 Deduction version of strss 15738. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))

Theoremstrss 15738 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    &   𝑆𝑇    &   𝐸 = Slot (𝐸‘ndx)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)

Theoremstr0 15739 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‘∅)

Theorembase0 15740 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)

Theoremstrfvi 15741 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 = (𝐸𝑆)       𝑋 = (𝐸‘( I ‘𝑆))

Theoremsetsid 15742 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)       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))

Theoremsetsnid 15743 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       (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))

Theoremsbcie2s 15744* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))

Theoremsbcie3s 15745* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   𝐶 = (𝐺𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))

Theorembaseval 15746 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)

Theorembaseid 15747 Utility theorem: index-independent form of df-base 15700. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)

Theoremelbasfv 15748 Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝑆 = (𝐹𝑍)    &   𝐵 = (Base‘𝑆)       (𝑋𝐵𝑍 ∈ V)

Theoremelbasov 15749 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)       (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Theoremstrov2rcl 15750 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       (𝑋𝐵𝐼 ∈ V)

Theorembasendx 15751 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

Theorembasendxnn 15752 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

Theoremreldmress 15753 The structure restriction is a proper operator, so it can be used with ovprc1 6582. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s

Theoremressval 15754 Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))

Theoremressid2 15755 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremressval2 15756 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))

Theoremressbas 15757 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝑉 → (𝐴𝐵) = (Base‘𝑅))

Theoremressbas2 15758 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝐵𝐴 = (Base‘𝑅))

Theoremressbasss 15759 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 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (Base‘𝑅) ⊆ 𝐵

Theoremresslem 15760 Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   1 < 𝑁       (𝐴𝑉𝐶 = (𝐸𝑅))

Theoremress0 15761 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) = ∅

Theoremressid 15762 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)

Theoremressinbas 15763 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)       (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))

Theoremressval3d 15764 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.)
𝑅 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑𝐸 ∈ dom 𝑆)    &   (𝜑𝐴𝑌)    &   (𝜑𝐴𝐵)       (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Theoremressress 15765 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))

Theoremressabs 15766 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐴𝑋𝐵𝐴) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))

Theoremwunress 15767 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑊𝑈)       (𝜑 → (𝑊s 𝐴) ∈ 𝑈)

7.1.2  Slot definitions

Syntaxcplusg 15768 Extend class notation with group (addition) operation.
class +g

Syntaxcmulr 15769 Extend class notation with ring multiplication.
class .r

Syntaxcstv 15770 Extend class notation with involution.
class *𝑟

Syntaxcsca 15771 Extend class notation with scalar field.
class Scalar

Syntaxcvsca 15772 Extend class notation with scalar product.
class ·𝑠

Syntaxcip 15773 Extend class notation with Hermitian form (inner product).
class ·𝑖

Syntaxcts 15774 Extend class notation with the topology component of a topological space.
class TopSet

Syntaxcple 15775 Extend class notation with less-than-or-equal for posets.
class le

Syntaxcoc 15776 Extend class notation with the class of orthocomplementation extractors.
class oc

Syntaxcds 15777 Extend class notation with the metric space distance function.
class dist

Syntaxcunif 15778 Extend class notation with the uniform structure.
class UnifSet

Syntaxchom 15779 Extend class notation with the hom-set structure.
class Hom

Syntaxcco 15780 Extend class notation with the composition operation.
class comp

Definitiondf-plusg 15781 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2

Definitiondf-mulr 15782 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3

Definitiondf-starv 15783 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
*𝑟 = Slot 4

Definitiondf-sca 15784 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

Definitiondf-vsca 15785 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6

Definitiondf-ip 15786 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8

Definitiondf-tset 15787 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

Definitiondf-ple 15788 Define less-than-or-equal 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

TheoremdfpleOLD 15789 Obsolete version of df-ple 15788 as of 9-Sep-2021. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
le = Slot 10

Definitiondf-ocomp 15790 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

Definitiondf-ds 15791 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

Definitiondf-unif 15792 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13

Definitiondf-hom 15793 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14

Definitiondf-cco 15794 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15

Theoremstrlemor0 15795 Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(Fun ∅ ∧ dom ∅ ⊆ (1...0))

Theoremstrlemor1 15796 Add one element to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐽))

Theoremstrlemor2 15797 Add two elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐽 < 𝐾    &   𝐾 ∈ ℕ    &   𝐵 = 𝐾    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐾))

Theoremstrlemor3 15798 Add three elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐽 < 𝐾    &   𝐾 ∈ ℕ    &   𝐵 = 𝐾    &   𝐾 < 𝐿    &   𝐿 ∈ ℕ    &   𝐶 = 𝐿    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐿))

Theoremstrleun 15799 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝐴, 𝐵    &   𝐺 Struct ⟨𝐶, 𝐷    &   𝐵 < 𝐶       (𝐹𝐺) Struct ⟨𝐴, 𝐷

Theoremstrle1 15800 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼       {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼

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