P. 143 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 14201-14300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	abl32 14201	Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theorem	cmnmndd 14202	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ CMnd) ⇒ ⊢ (𝜑 → 𝐺 ∈ Mnd)

Theorem	cmnsubm 14203	A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)
⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ 𝐻 = (𝐺 ↾_s 𝑆) ⇒ ⊢ (𝜑 → 𝐻 ∈ CMnd)

Theorem	rinvmod 14204*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6286. (Contributed by AV, 31-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )

Theorem	ablinvadd 14205	The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌)))

Theorem	ablsub2inv 14206	Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋))

Theorem	ablsubadd 14207	Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))

Theorem	ablsub4 14208	Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑍 + 𝑊)) = ((𝑋 − 𝑍) + (𝑌 − 𝑊)))

Theorem	abladdsub4 14209	Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 − 𝑍) = (𝑊 − 𝑌)))

Theorem	abladdsub 14210	Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 − 𝑍) + 𝑌))

Theorem	ablpncan2 14211	Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌)

Theorem	ablpncan3 14212	A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + (𝑌 − 𝑋)) = 𝑌)

Theorem	ablsubsub 14213	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍))

Theorem	ablsubsub4 14214	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍)))

Theorem	ablpnpcan 14215	Cancellation law for mixed addition and subtraction. (pnpcan 8573 analog.) (Contributed by NM, 29-May-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))

Theorem	ablnncan 14216	Cancellation law for group subtraction. (nncan 8563 analog.) (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)

Theorem	ablsub32 14217	Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌))

Theorem	ablnnncan 14218	Cancellation law for group subtraction. (nnncan 8569 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌))

Theorem	ablnnncan1 14219	Cancellation law for group subtraction. (nnncan1 8570 analog.) (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌))

Theorem	ablsubsub23 14220	Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵))

Theorem	ghmfghm 14221*	The function fulfilling the conditions of ghmgrp 13981 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+_g‘𝐺) & ⊢ ⨣ = (+_g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Theorem	ghmcmn 14222*	The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+_g‘𝐺) & ⊢ ⨣ = (+_g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ CMnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ CMnd)

Theorem	ghmabl 14223*	The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑌 = (Base‘𝐻) & ⊢ + = (+_g‘𝐺) & ⊢ ⨣ = (+_g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Abel) ⇒ ⊢ (𝜑 → 𝐻 ∈ Abel)

Theorem	invghm 14224	The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (inv_g‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Theorem	eqgabl 14225	Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑋 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ ∼ = (𝐺 ~_QG 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐵 − 𝐴) ∈ 𝑆)))

Theorem	qusecsub 14226	Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-_g‘𝐺) & ⊢ ∼ = (𝐺 ~_QG 𝑆) ⇒ ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))

Theorem	subgabl 14227	A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾_s 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Theorem	subcmnd 14228	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ (𝜑 → 𝐻 = (𝐺 ↾_s 𝑆)) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐻 ∈ CMnd)

Theorem	ablnsg 14229	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Theorem	ablressid 14230	A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13485. (Contributed by Jim Kingdon, 5-May-2025.)
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel → (𝐺 ↾_s 𝐵) ∈ Abel)

Theorem	imasabl 14231*	The image structure of an abelian group is an abelian group (imasgrp 13974 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+_g‘𝑅)) & ⊢ (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 → 𝑅 ∈ Abel) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0_g‘𝑈)))

7.2.6.2 Group sum operation

Theorem	gzsumreidx 14232	Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) & ⊢ (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_gz (𝐹 ∘ 𝐻)))

Theorem	gzsumsubmcl 14233	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝑆) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) ∈ 𝑆)

Theorem	gzsumconst 14234*	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (._g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σ_gz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))

Theorem	gzsumconstf 14235*	Sum of a constant series. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ Ⅎ𝑘𝑋 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (._g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σ_gz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))

Theorem	gzsummhm 14236	Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) ⇒ ⊢ (𝜑 → (𝐻 Σ_gz (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_gz 𝐹)))

Theorem	gzsummhm2 14237*	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑋 ∈ 𝐵) & ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) & ⊢ (𝑥 = (𝐺 Σ_gz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → (𝐻 Σ_gz (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)

Theorem	gzsumsnfd 14238*	Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) & ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐶 ⇒ ⊢ (𝜑 → (𝐺 Σ_gz (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)

Theorem	gzsumsplit0 14239	Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13775 except that 𝑁 can equal 𝑀 − 1. (Contributed by Jim Kingdon, 4-Apr-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 − 1))) & ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = ((𝐺 Σ_gz (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))

Theorem	gzsumshift 14240*	Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) & ⊢ 𝑆 = (𝑗 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑗 − (1 − 𝑀))) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_gz (𝐹 ∘ 𝑆)))

7.2.7 Finite group sum over unordered finite set

Syntax	cgsu 14241	Extend class notation to include group sums over finite sets.
class Σ_g

Definition	df-gsumfi 14242*	Define the finite group sum (iterated sum) over an unordered finite set. Given 𝐺 Σ_g 𝐹 where 𝐹:𝐴⟶(Base‘𝐺), the set of indices is 𝐴 and the values are given by 𝐹 at each index. For this notation, 𝐴 is a finite set and 𝐺 is a commutative monoid, and the sum adds up these elements in some order (the sum does not depend on the order). For a sum indexed by consecutive integers (and thus defining an order for the sum), see df-gzsum 13673. (Contributed by Jim Kingdon, 23-Mar-2026.)
⊢ Σ_g = (𝑤 ∈ CMnd, 𝑓 ∈ V ↦ (℩𝑥(dom 𝑓 ∈ Fin ∧ ∃𝑔(𝑔:(1...(♯‘dom 𝑓))–1-1-onto→dom 𝑓 ∧ 𝑥 = (𝑤 Σ_gz (𝑓 ∘ 𝑔))))))

Theorem	gsumvalfi 14243	Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐺:(1...(♯‘𝐴))–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝑊 Σ_g 𝐹) = (𝑊 Σ_gz (𝐹 ∘ 𝐺)))

Theorem	gzsumgsum1 14244	On an integer range starting at one, Σ_gz and Σ_g agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐹:(1...𝑁)⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_g 𝐹))

Theorem	gsum0cmn 14245	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
⊢ (𝐺 ∈ CMnd → (𝐺 Σ_g ∅) = (0_g‘𝐺))

Theorem	gzsumgsum 14246	On an integer range, Σ_gz and Σ_g agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σ_gz 𝐹) = (𝐺 Σ_g 𝐹))

Theorem	gsumsncmn 14247*	Group sum of a singleton. (Contributed by Jim Kingdon, 2-Apr-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)

Theorem	gsump1 14248	Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = ((𝐺 Σ_g (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))

Theorem	gsumzfi 14249*	Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
⊢ 0 = (0_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

Theorem	gsumclfi 14250	Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) ∈ 𝐵)

Theorem	gsumf1ofi 14251	Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))

Theorem	gsummptfidmadd 14252*	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Theorem	gsummptfidmadd2 14253*	The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ∘_𝑓 + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻)))

Theorem	gsumsubmclfi 14254	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) ∈ 𝑆)

Theorem	gsummptfidmsplit 14255*	Split a group sum expressed as mapping with a finite domain into two parts. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σ_g (𝑘 ∈ 𝐶 ↦ 𝑌)) + (𝐺 Σ_g (𝑘 ∈ 𝐷 ↦ 𝑌))))

Theorem	gsummptfidmsplitres 14256*	Split a group sum expressed as mapping with a finite domain into two parts using restrictions. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) & ⊢ 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = ((𝐺 Σ_g (𝐹 ↾ 𝐶)) + (𝐺 Σ_g (𝐹 ↾ 𝐷))))

Theorem	gsumsplitfi 14257	Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = ((𝐺 Σ_g (𝐹 ↾ 𝐶)) + (𝐺 Σ_g (𝐹 ↾ 𝐷))))

Theorem	gsumsnfdcmn 14258*	Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) & ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐶 ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)

Theorem	gsumsndcmn 14259*	Group sum of a singleton, deduction form. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)

Theorem	gsumunsnfd 14260*	Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) & ⊢ Ⅎ𝑘𝑌 ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))

Theorem	gsumunsnd 14261*	Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 2-Jan-2019.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))

Theorem	gsumunsnf 14262*	Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
⊢ Ⅎ𝑘𝑌 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))

Theorem	gsumunsn 14263*	Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Proof shortened by AV, 8-Mar-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))

Theorem	gsumresfi 14264*	Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 24-Sep-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑥) = 0 ) & ⊢ (𝜑 → 𝑊 ∈ Fin) & ⊢ (𝜑 → 𝑊 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))

Theorem	gsummhmfi 14265	Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))

Theorem	gsummhm2fi 14266*	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) & ⊢ (𝑥 = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸)

Theorem	gsumconstcmn 14267*	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (._g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))

Theorem	gsumressfi 14268*	The group sum in a substructure is the same as the group sum in the original structure. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾_s 𝑆) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 0 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐻 Σ_g 𝐹))

Theorem	gsumsubmfi 14269	Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ 𝐻 = (𝐺 ↾_s 𝑆) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐻 Σ_g 𝐹))

Theorem	idjufi 14270*	A finite indexed disjoint union of finite sets is finite. (Contributed by Jim Kingdon, 8-Oct-2026.)
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ Fin) ⇒ ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∈ Fin)

Theorem	idjusnfi 14271*	The image of a singleton through an indexed disjoint union, where it is of finite sets. (Contributed by Jim Kingdon, 2-Oct-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ Fin) & ⊢ 𝑈 = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ 𝐴) → (𝑈 “ {𝐾}) ∈ Fin)

Theorem	gsum2dfilem1 14272*	Lemma 1 for gsum2dfi 14274. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 2-Oct-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝐴 “ {𝑗}) ∈ Fin) ⇒ ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝐺 Σ_g (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)

Theorem	gsum2dfilem2 14273*	Lemma for gsum2dfi 14274. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 4-Oct-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷) & ⊢ (𝜑 → dom 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝐴 “ {𝑗}) ∈ Fin) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑗 ∈ dom 𝐴 ↦ (𝐺 Σ_g (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

Theorem	gsum2dfi 14274*	Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) (Revised by Jim Kingdon, 6-Oct-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷) & ⊢ (𝜑 → dom 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝐴 “ {𝑗}) ∈ Fin) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑗 ∈ 𝐷 ↦ (𝐺 Σ_g (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

Theorem	gsum2d2fi 14275*	Write a group sum over a two-dimensional region as a double sum. Note that 𝐶(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σ_g (𝑗 ∈ 𝐴 ↦ (𝐺 Σ_g (𝑘 ∈ 𝐶 ↦ 𝑋)))))

Theorem	gsumcom2fi 14276*	Two-dimensional commutation of a group sum. Note that while 𝐴 and 𝐷 are constants w.r.t. 𝑗, 𝑘, 𝐶(𝑗) and 𝐸(𝑘) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ Fin) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 ) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸))) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σ_g (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))

7.2.8 Structure product

Syntax	cprds 14277	The function constructing structure products.
class X_s

Definition	df-prds 14278*	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.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ X_s = (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 Σ_g (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 ∈ ((2^nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))

Theorem	reldmprds 14279	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 X_s

Theorem	prdsex 14280	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆X_s𝑅) ∈ V)

Theorem	prdsval 14281*	Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) & ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·_𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))) & ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) & ⊢ (𝜑 → 𝑂 = (∏_t‘(TopOpen ∘ 𝑅))) & ⊢ (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) & ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))) & ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) & ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2^nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), + ⟩, ⟨(._r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·_𝑠 ‘ndx), · ⟩, ⟨(·_𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))

Theorem	prdsbaslemss 14282	Lemma for prdsbas 14284 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐴 = (𝐸‘𝑃) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ∈ ℕ & ⊢ (𝜑 → 𝑇 ∈ 𝑋) & ⊢ (𝜑 → {⟨(𝐸‘ndx), 𝑇⟩} ⊆ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 = 𝑇)

Theorem	prdssca 14283	Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))

Theorem	prdsbas 14284*	Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Theorem	prdsplusg 14285*	Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ + = (+_g‘𝑃) ⇒ ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+_g‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Theorem	prdsmulr 14286*	Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → dom 𝑅 = 𝐼) & ⊢ · = (._r‘𝑃) ⇒ ⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(._r‘(𝑅‘𝑥))(𝑔‘𝑥)))))

Theorem	prdsbas2 14287*	The base set of a structure product is an indexed set product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))

Theorem	prdsbasmpt 14288*	A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ (Base‘(𝑅‘𝑥))))

Theorem	prdsbasfn 14289	Points in the structure product are functions; use this with dffn5im 5751 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑇 Fn 𝐼)

Theorem	prdsbasprj 14290	Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽)))

Theorem	prdsplusgval 14291*	Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+_g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+_g‘(𝑅‘𝑥))(𝐺‘𝑥))))

Theorem	prdsplusgfval 14292	Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+_g‘𝑌) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹‘𝐽)(+_g‘(𝑅‘𝐽))(𝐺‘𝐽)))

Theorem	prdsmulrval 14293*	Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ · = (._r‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(._r‘(𝑅‘𝑥))(𝐺‘𝑥))))

Theorem	prdsmulrfval 14294	Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ · = (._r‘𝑌) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝐹 · 𝐺)‘𝐽) = ((𝐹‘𝐽)(._r‘(𝑅‘𝐽))(𝐺‘𝐽)))

Theorem	prdsbas3 14295*	The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝐾)

Theorem	prdsbasmpt2 14296*	A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 𝑈 ∈ 𝐾))

Theorem	prdsbascl 14297*	An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ 𝑅)) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ 𝑋) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ 𝐾)

Theorem	prdsplusgsgrpcl 14298	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+_g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)

Theorem	prdssgrpd 14299	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp) ⇒ ⊢ (𝜑 → 𝑌 ∈ Smgrp)

Theorem	prdsplusgcl 14300	Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆X_s𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+_g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)

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