P. 128 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elrestr 12701	Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
Theorem	restid2 12702	The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽)
Theorem	restsspw 12703	The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴
Theorem	restid 12704	The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
Theorem	topnvalg 12705	Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊))
Theorem	topnidg 12706	Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐵) → 𝐽 = (TopOpen‘𝑊))
Theorem	topnpropgd 12707	The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Syntax	ctg 12708	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 12709	Extend class notation with a function whose value is a product topology.
class ∏t
Syntax	c0g 12710	Extend class notation with group identity element.
class 0g
Syntax	cgsu 12711	Extend class notation to include finitely supported group sums.
class Σg
Definition	df-0g 12712*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 12713. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
Definition	df-gsum 12713*	Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or any other operation. The variable 𝑘 is normally a free variable in 𝐵 (i.e., 𝐵 can be thought of as 𝐵(𝑘)). The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺. 1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity. 2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., (𝐵(1) + 𝐵(2)) + 𝐵(3), etc. 3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. 4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦)))))))
Definition	df-topgen 12714*	Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
Definition	df-pt 12715*	Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ ∏t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))}))
Theorem	tgval 12716*	The topology generated by a basis. See also tgval2 13636 and tgval3 13643. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
Theorem	tgvalex 12717	The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
Theorem	ptex 12718	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
⊢ (𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)
Syntax	cprds 12719	The function constructing structure products.
class Xs
Syntax	cpws 12720	The function constructing structure powers.
class ↑s
Definition	df-prds 12721*	Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
Theorem	reldmprds 12722	The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
⊢ Rel dom Xs
Theorem	prdsex 12723	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
Definition	df-pws 12724*	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})))
6.1.4  Definition of the structure quotient
Syntax	cimas 12725	Image structure function.
class “s
Syntax	cqus 12726	Quotient structure function.
class /s
Syntax	cxps 12727	Binary product structure function.
class ×s
Definition	df-iimas 12728*	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 4641, 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 4640) and/or 𝑅 ( with df-iress 12472) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by AV, 6-Oct-2020.)
⊢ “s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩})
Definition	df-qus 12729*	Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 12728 where the image function is 𝑥 ↦ [𝑥]𝑒. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
Definition	df-xps 12730*	Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
Theorem	imasex 12731	Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)
Theorem	imasival 12732*	Value of an image structure. The is a lemma for the theorems imasbas 12733, imasplusg 12734, and imasmulr 12735 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ + = (+g‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 × 𝑞))⟩})    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), ∙ ⟩})
Theorem	imasbas 12733	The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
Theorem	imasplusg 12734*	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.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 + 𝑞))⟩})
Theorem	imasmulr 12735*	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.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
Theorem	f1ocpbllem 12736	Lemma for f1ocpbl 12737. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	f1ocpbl 12737	An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
Theorem	f1ovscpbl 12738	An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))
Theorem	f1olecpbl 12739	An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))
Theorem	imasaddfnlemg 12740*	The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
Theorem	imasaddvallemg 12741*	The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Theorem	imasaddflemg 12742*	The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵)
Theorem	imasaddfn 12743*	The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
Theorem	imasaddval 12744*	The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Theorem	imasaddf 12745*	The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵)
Theorem	imasmulfn 12746*	The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
Theorem	imasmulval 12747*	The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
Theorem	imasmulf 12748*	The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵)
Theorem	qusval 12749*	Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∼ ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
Theorem	quslem 12750*	The function in qusval 12749 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∼ ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))
Theorem	qusin 12751	Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉)    ⇒   ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))
Theorem	qusbas 12752	Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → (𝑉 / ∼ ) = (Base‘𝑈))
Theorem	divsfvalg 12753*	Value of the function in qusval 12749. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ )
Theorem	ercpbllemg 12754*	Lemma for ercpbl 12755. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵))
Theorem	ercpbl 12755*	Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)    &   ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
Theorem	erlecpbl 12756*	Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))
Theorem	qusaddvallemg 12757*	Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ⊢ (𝜑 → · ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusaddflemg 12758*	The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )    &   ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})    &   ⊢ (𝜑 → · ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	qusaddval 12759*	The addition in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusaddf 12760*	The addition in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (+g‘𝑅)    &   ⊢ ∙ = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	qusmulval 12761*	The multiplication in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	qusmulf 12762*	The multiplication in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑈)    ⇒   ⊢ (𝜑 → ∙ :((𝑉 / ∼ ) × (𝑉 / ∼ ))⟶(𝑉 / ∼ ))
Theorem	fnpr2o 12763	Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Theorem	fnpr2ob 12764	Biconditional version of fnpr2o 12763. (Contributed by Jim Kingdon, 27-Sep-2023.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
Theorem	fvpr0o 12765	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ (𝐴 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
Theorem	fvpr1o 12766	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
⊢ (𝐵 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
Theorem	fvprif 12767	The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵))
Theorem	xpsfrnel 12768*	Elementhood in the target space of the function 𝐹 appearing in xpsval 12776. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
Theorem	xpsfeq 12769	A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝐺 Fn 2o → {⟨∅, (𝐺‘∅)⟩, ⟨1o, (𝐺‘1o)⟩} = 𝐺)
Theorem	xpsfrnel2 12770*	Elementhood in the target space of the function 𝐹 appearing in xpsval 12776. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ ({⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
Theorem	xpscf 12771	Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ({⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
Theorem	xpsfval 12772*	The value of the function appearing in xpsval 12776. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
Theorem	xpsff1o 12773*	The function appearing in xpsval 12776 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Theorem	xpsfrn 12774*	A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Theorem	xpsff1o2 12775*	The function appearing in xpsval 12776 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹
Theorem	xpsval 12776*	Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ 𝐺 = (Scalar‘𝑅)    &   ⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))
PART 7  BASIC ALGEBRAIC STRUCTURES
7.1  Monoids
7.1.1  Magmas According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.". Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following: With df-mpo 5882, binary operations are defined by a rule, and with df-ov 5880, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation 5880 (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆⟶𝑆. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 are more precisely called internal binary operations. If, in addition, the result is also contained in the set 𝑆, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set 𝑆" according to Wikipedia is a "closed internal binary operation" in a more precise terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 5880). The definition of magmas (Mgm, see df-mgm 12780) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.
Syntax	cplusf 12777	Extend class notation with group addition as a function.
class +𝑓
Syntax	cmgm 12778	Extend class notation with class of all magmas.
class Mgm
Definition	df-plusf 12779*	Define group addition function. Usually we will use +g directly instead of +𝑓, and they have the same behavior in most cases. The main advantage of +𝑓 for any magma is that it is a guaranteed function (mgmplusf 12790), while +g only has closure (mgmcl 12783). (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
Definition	df-mgm 12780*	A magma is a set equipped with an everywhere defined internal operation. Definition 1 in [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
Theorem	ismgm 12781*	The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
Theorem	ismgmn0 12782*	The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
Theorem	mgmcl 12783	Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
Theorem	isnmgm 12784	A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Theorem	mgmsscl 12785	If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐻)    ⇒   ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
Theorem	plusffvalg 12786*	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
Theorem	plusfvalg 12787	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))
Theorem	plusfeqg 12788	If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ + Fn (𝐵 × 𝐵)) → ⨣ = + )
Theorem	plusffng 12789	The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → ⨣ Fn (𝐵 × 𝐵))
Theorem	mgmplusf 12790	The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⨣ = (+𝑓‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
Theorem	intopsn 12791	The internal operation for a set is the trivial operation iff the set is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgmb1mgm1 12792	The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgm0 12793	Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
Theorem	mgm1 12794	The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm)
Theorem	opifismgmdc 12795*	A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → DECID 𝜓)    &   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mgm)
7.1.2  Identity elements According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 12796) is an important property of monoids, and therefore also for groups, but also for magmas not required to be associative. Magmas with an identity element are called "unital magmas" (see Definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15). In the context of extensible structures, the identity element (of any magma 𝑀) is defined as "group identity element" (0g‘𝑀), see df-0g 12712. Related theorems which are already valid for magmas are provided in the following.
Theorem	mgmidmo 12796*	A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Theorem	grpidvalg 12797*	The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
Theorem	grpidpropdg 12798*	If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
Theorem	fn0g 12799	The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 0g Fn V
Theorem	0g0 12800	The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ ∅ = (0g‘∅)