P. 142 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	abladdsub4 14101	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 14102	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 14103	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 14104	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 14105	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 14106	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 14107	Cancellation law for mixed addition and subtraction. (pnpcan 8559 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 14108	Cancellation law for group subtraction. (nncan 8549 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 14109	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 14110	Cancellation law for group subtraction. (nnncan 8555 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 14111	Cancellation law for group subtraction. (nnncan1 8556 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 14112	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 14113*	The function fulfilling the conditions of ghmgrp 13904 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 14114*	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 14115*	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 14116	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 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Theorem	eqgabl 14117	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 14118	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 14119	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 14120	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 14121	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Theorem	ablressid 14122	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 13408. (Contributed by Jim Kingdon, 5-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
Theorem	imasabl 14123*	The image structure of an abelian group is an abelian group (imasgrp 13897 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 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
7.2.5.2  Group sum operation
Theorem	gzsumreidx 14124	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 = (0g‘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 Σgz 𝐹) = (𝐺 Σgz (𝐹 ∘ 𝐻)))
Theorem	gzsumsubmcl 14125	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 Σgz 𝐹) ∈ 𝑆)
Theorem	gzsumconst 14126*	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 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
Theorem	gzsumconstf 14127*	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 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
Theorem	gzsummhm 14128	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 = (0g‘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 Σgz (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σgz 𝐹)))
Theorem	gzsummhm2 14129*	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 = (0g‘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 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → (𝐻 Σgz (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)
Theorem	gzsumsnfd 14130*	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 Σgz (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gzsumsplit0 14131	Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13698 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 Σgz 𝐹) = ((𝐺 Σgz (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
Theorem	gzsumshift 14132*	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 Σgz 𝐹) = (𝐺 Σgz (𝐹 ∘ 𝑆)))
7.2.6  Finite group sum over unordered finite set
Syntax	cgsu 14133	Extend class notation to include group sums over finite sets.
class Σg
Definition	df-gsumfi 14134*	Define the finite group sum (iterated sum) over an unordered finite set. Given 𝐺 Σg 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 13596. (Contributed by Jim Kingdon, 23-Mar-2026.)
⊢ Σg = (𝑤 ∈ CMnd, 𝑓 ∈ V ↦ (℩𝑥(dom 𝑓 ∈ Fin ∧ ∃𝑔(𝑔:(1...(♯‘dom 𝑓))–1-1-onto→dom 𝑓 ∧ 𝑥 = (𝑤 Σgz (𝑓 ∘ 𝑔))))))
Theorem	gsumvalfi 14135	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 Σg 𝐹) = (𝑊 Σgz (𝐹 ∘ 𝐺)))
Theorem	gzsumgsum1 14136	On an integer range starting at one, Σgz and Σg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹:(1...𝑁)⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σgz 𝐹) = (𝐺 Σg 𝐹))
Theorem	gsum0cmn 14137	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
⊢ (𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
Theorem	gzsumgsum 14138	On an integer range, Σgz and Σg 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 Σgz 𝐹) = (𝐺 Σg 𝐹))
Theorem	gsumsncmn 14139*	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 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gsump1 14140	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 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
Theorem	gsumzfi 14141*	Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
Theorem	gsumclfi 14142	Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Theorem	gsumf1ofi 14143	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsummptfidmadd 14144*	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 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsummptfidmadd2 14145*	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 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumsubmclfi 14146	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
Theorem	gsummhmfi 14147	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Theorem	gsummhm2fi 14148*	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 = (0g‘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 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸)
Theorem	gsumconstcmn 14149*	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 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
Theorem	gsumressfi 14150*	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 Σg 𝐹) = (𝐻 Σg 𝐹))
Theorem	gsumsubmfi 14151	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 Σg 𝐹) = (𝐻 Σg 𝐹))
7.2.7  Structure product
Syntax	cprds 14152	The function constructing structure products.
class Xs
Definition	df-prds 14153*	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.)
⊢ 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 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
Theorem	reldmprds 14154	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 14155	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑆Xs𝑅) ∈ V)
Theorem	prdsval 14156*	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 = (𝑆Xs𝑅)    &   ⊢ 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 Σg (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 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘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 14157	Lemma for prdsbas 14159 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (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 14158	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 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))
Theorem	prdsbas 14159*	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 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    ⇒   ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
Theorem	prdsplusg 14160*	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 = (𝑆Xs𝑅)    &   ⊢ (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 14161*	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 = (𝑆Xs𝑅)    &   ⊢ (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 14162*	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 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    ⇒   ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
Theorem	prdsbasmpt 14163*	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 = (𝑆Xs𝑅)    &   ⊢ 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 14164	Points in the structure product are functions; use this with dffn5im 5745 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Fn 𝐼)
Theorem	prdsbasprj 14165	Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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 14166*	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 = (𝑆Xs𝑅)    &   ⊢ 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 14167	Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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 14168*	Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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 14169	Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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 14170*	The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑌 = (𝑆Xs(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 14171*	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 = (𝑆Xs(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 14172*	An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑌 = (𝑆Xs(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 14173	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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 14174	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Smgrp)
Theorem	prdsplusgcl 14175	Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 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)
Theorem	prdsidlem 14176*	Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ 0 = (0g ∘ 𝑅)    ⇒   ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
Theorem	prdsmndd 14177	The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Mnd)
Theorem	prds0g 14178	The identity in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    ⇒   ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌))
Theorem	prdsinvlem 14179*	Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 0 = (0g ∘ 𝑅)    &   ⊢ 𝑁 = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ (𝑁 + 𝐹) = 0 ))
Theorem	prdsgrpd 14180	The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Grp)
Theorem	prdsinvgd 14181*	Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑁 = (invg‘𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))))
7.2.8  Binary product on structures
Syntax	cxps 14182	Binary product structure function.
class ×s
Definition	df-xps 14183*	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	xpsval 14184*	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))
7.2.9  Structure power
Syntax	cpws 14185	The function constructing structure powers.
class ↑s
Definition	df-pws 14186*	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})))
Theorem	pwsval 14187	Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐹 = (Scalar‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
Theorem	pwsbas 14188	Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐵 ↑𝑚 𝐼) = (Base‘𝑌))
Theorem	pwselbasb 14189	Membership in the base set of a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍) → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵))
Theorem	pwselbas 14190	An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑋:𝐼⟶𝐵)
Theorem	pwsplusgval 14191	Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺))
Theorem	pwsmulrval 14192	Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝐹 ∘𝑓 · 𝐺))
Theorem	pwsdiagel 14193	Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑌)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐼 × {𝐴}) ∈ 𝐶)
Theorem	pwssnf1o 14194*	Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s {𝐼})    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ({𝐼} × {𝑥}))    &   ⊢ 𝐶 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	pwsmnd 14195	The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Mnd)
Theorem	pws0g 14196	The identity in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 0 }) = (0g‘𝑌))
Theorem	pwsgrp 14197	A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Grp)
Theorem	pwsinvg 14198	Negation in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑀 = (invg‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑀 ∘ 𝑋))
Theorem	pwssub 14199	Subtraction in a structure power. (Contributed by Mario Carneiro, 12-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑀 = (-g‘𝑅)    &   ⊢ − = (-g‘𝑌)    ⇒   ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑉) ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝐹 − 𝐺) = (𝐹 ∘𝑓 𝑀𝐺))
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 14200	Multiplicative group.
class mulGrp