P. 223 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumsmonply1 22201*	A finite group sum of scaled monomials is a univariate polynomial. (Contributed by AV, 8-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))) ∈ 𝐵)
Theorem	gsummoncoe1 22202*	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by AV, 13-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) finSupp 0 )    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = ⦋𝐿 / 𝑘⦌𝐴)
Theorem	gsumply1eq 22203*	Two univariate polynomials given as (finitely supported) sum of scaled monomials are equal iff the corresponding coefficients are equal. (Contributed by AV, 21-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐴) finSupp 0 )    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐵) finSupp 0 )    &   ⊢ (𝜑 → 𝑂 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))    &   ⊢ (𝜑 → 𝑄 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐵 ∗ (𝑘 ↑ 𝑋)))))    ⇒   ⊢ (𝜑 → (𝑂 = 𝑄 ↔ ∀𝑘 ∈ ℕ0 𝐴 = 𝐵))
Theorem	lply1binom 22204*	The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings: (𝑋 + 𝐴)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝑋↑𝑘)). (Contributed by AV, 25-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ + = (+g‘𝑃)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = (.g‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵) → (𝑁 ↑ (𝑋 + 𝐴)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝑋))))))
Theorem	lply1binomsc 22205*	The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings, expressed by an element of this ring: (𝑋 + 𝐴)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝑋↑𝑘)). (Contributed by AV, 25-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ + = (+g‘𝑃)    &   ⊢ × = (.r‘𝑃)    &   ⊢ · = (.g‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝐾) → (𝑁 ↑ (𝑋 + (𝑆‘𝐴))) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · ((𝑆‘((𝑁 − 𝑘)𝐸𝐴)) × (𝑘 ↑ 𝑋))))))
Theorem	ply1fermltlchr 22206	Fermat's little theorem for polynomials in a commutative ring 𝐹 of characteristic 𝑃 prime: we have the polynomial equation (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴). (Contributed by Thierry Arnoux, 9-Jan-2025.)
⊢ 𝑊 = (Poly1‘𝐹)    &   ⊢ 𝑋 = (var1‘𝐹)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸))    &   ⊢ 𝑃 = (chr‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝐸 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
11.3.5  Univariate polynomial evaluation
Syntax	ces1 22207	Evaluation of a univariate polynomial in a subring.
class evalSub1
Syntax	ce1 22208	Evaluation of a univariate polynomial.
class eval1
Definition	df-evls1 22209*	Define the evaluation map for the univariate polynomial algebra. The function (𝑆 evalSub1 𝑅):𝑉⟶(𝑆 ↑m 𝑆) makes sense when 𝑆 is a ring and 𝑅 is a subring of 𝑆, and where 𝑉 is the set of polynomials in (Poly1‘𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments to the variable from 𝑆 into an element of 𝑆 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
Definition	df-evl1 22210*	Define the evaluation map for the univariate polynomial algebra. The function (eval1‘𝑅):𝑉⟶(𝑅 ↑m 𝑅) makes sense when 𝑅 is a ring, and 𝑉 is the set of polynomials in (Poly1‘𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments to the variable from 𝑅 into an element of 𝑅 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
Theorem	reldmevls1 22211	Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
⊢ Rel dom evalSub1
Theorem	ply1frcl 22212	Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
⊢ 𝑄 = ran (𝑆 evalSub1 𝑅)    ⇒   ⊢ (𝑋 ∈ 𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))
Theorem	evls1fval 22213*	Value of the univariate polynomial evaluation map function. (Contributed by AV, 7-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐸 = (1o evalSub 𝑆)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))
Theorem	evls1val 22214*	Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐸 = (1o evalSub 𝑆)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑀 = (1o mPoly (𝑆 ↾s 𝑅))    &   ⊢ 𝐾 = (Base‘𝑀)    ⇒   ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
Theorem	evls1rhmlem 22215*	Lemma for evl1rhm 22226 and evls1rhm 22216 (formerly part of the proof of evl1rhm 22226): The first function of the composition forming the univariate polynomial evaluation map function for a (sub)ring is a ring homomorphism. (Contributed by AV, 11-Sep-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑇 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))    ⇒   ⊢ (𝑅 ∈ CRing → 𝐹 ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇))
Theorem	evls1rhm 22216	Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑇 = (𝑆 ↑s 𝐵)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    ⇒   ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))
Theorem	evls1sca 22217	Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))
Theorem	evls1gsumadd 22218*	Univariate polynomial evaluation maps (additive) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (𝑆 ↑s 𝐾)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evls1gsummul 22219*	Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (𝑆 ↑s 𝐾)    &   ⊢ 𝐻 = (mulGrp‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evls1pw 22220	Univariate polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐾)))(𝑄‘𝑋)))
Theorem	evls1varpw 22221	Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐵)))(𝑄‘𝑋)))
Theorem	evl1fval 22222*	Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑄 = (1o eval 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)
Theorem	evl1val 22223*	Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑄 = (1o eval 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (1o mPoly 𝑅)    &   ⊢ 𝐾 = (Base‘𝑀)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐾) → (𝑂‘𝐴) = ((𝑄‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
Theorem	evl1fval1lem 22224	Lemma for evl1fval1 22225. (Contributed by AV, 11-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))
Theorem	evl1fval1 22225	Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝑄 = (𝑅 evalSub1 𝐵)
Theorem	evl1rhm 22226	Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑇 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇))
Theorem	fveval1fvcl 22227	The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵)
Theorem	evl1sca 22228	Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝑂‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))
Theorem	evl1scad 22229	Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴‘𝑋))‘𝑌) = 𝑋))
Theorem	evl1var 22230	Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐵))
Theorem	evl1vard 22231	Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∧ ((𝑂‘𝑋)‘𝑌) = 𝑌))
Theorem	evls1var 22232	Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑋 = (var1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵))
Theorem	evls1scasrng 22233	The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑂 = (eval1‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐶 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋)))
Theorem	evls1varsrng 22234	The evaluation of the variable of univariate polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑂 = (eval1‘𝑆)    &   ⊢ 𝑉 = (var1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → (𝑄‘𝑉) = (𝑂‘𝑉))
Theorem	evl1addd 22235	Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))    &   ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))    &   ⊢ ✚ = (+g‘𝑃)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))
Theorem	evl1subd 22236	Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))    &   ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐷 = (-g‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑀 − 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊)))
Theorem	evl1muld 22237	Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))    &   ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊)))
Theorem	evl1vsd 22238	Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ ∙ = ( ·𝑠 ‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))
Theorem	evl1expd 22239	Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))    &   ⊢ ∙ = (.g‘(mulGrp‘𝑃))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 ↑ 𝑉)))
Theorem	pf1const 22240	Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = ran (eval1‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)
Theorem	pf1id 22241	The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = ran (eval1‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄)
Theorem	pf1subrg 22242	Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = ran (eval1‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅 ↑s 𝐵)))
Theorem	pf1rcl 22243	Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)
Theorem	pf1f 22244	Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑄 → 𝐹:𝐵⟶𝐵)
Theorem	mpfpf1 22245*	Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = ran (1o eval 𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐸 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)
Theorem	pf1mpf 22246*	Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = ran (1o eval 𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵 ↑m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
Theorem	pf1addcl 22247	The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f + 𝐺) ∈ 𝑄)
Theorem	pf1mulcl 22248	The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f · 𝐺) ∈ 𝑄)
Theorem	pf1ind 22249*	Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑄 = ran (eval1‘𝑅)    &   ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)    &   ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)    &   ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = ( I ↾ 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))    &   ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))    &   ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑄)    ⇒   ⊢ (𝜑 → 𝜌)
Theorem	evl1gsumdlem 22250*	Lemma for evl1gsumd 22251 (induction step). (Contributed by AV, 17-Sep-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))))
Theorem	evl1gsumd 22251*	Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    ⇒   ⊢ (𝜑 → ((𝑂‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))
Theorem	evl1gsumadd 22252*	Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 22218. (Contributed by AV, 15-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝑃 = (𝑅 ↑s 𝐾)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evl1gsumaddval 22253*	Value of a univariate polynomial evaluation mapping an additive group sum to a group sum of the evaluated variable. (Contributed by AV, 17-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝑃 = (𝑅 ↑s 𝐾)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((𝑄‘𝑌)‘𝐶))))
Theorem	evl1gsummul 22254*	Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝑃 = (𝑅 ↑s 𝐾)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝐻 = (mulGrp‘𝑃)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )    ⇒   ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
Theorem	evl1varpw 22255	Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 22252, the proof is shorter using evls1varpw 22221 instead of proving it directly. (Contributed by AV, 15-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅 ↑s 𝐵)))(𝑄‘𝑋)))
Theorem	evl1varpwval 22256	Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑁 ↑ 𝑋))‘𝐶) = (𝑁𝐸𝐶))
Theorem	evl1scvarpw 22257	Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑆 = (𝑅 ↑s 𝐵)    &   ⊢ ∙ = (.r‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ 𝐹 = (.g‘𝑀)    ⇒   ⊢ (𝜑 → (𝑄‘(𝐴 × (𝑁 ↑ 𝑋))) = ((𝐵 × {𝐴}) ∙ (𝑁𝐹(𝑄‘𝑋))))
Theorem	evl1scvarpwval 22258	Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))
Theorem	evl1gsummon 22259*	Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ 𝐺 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑀 ⊆ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))))
11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism
Theorem	evls1scafv 22260	Value of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐶) = 𝑋)
Theorem	evls1expd 22261	Univariate polynomial evaluation builder for an exponential. See also evl1expd 22239. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ ∧ = (.g‘(mulGrp‘𝑊))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑀))‘𝐶) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐶)))
Theorem	evls1varpwval 22262	Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22256. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑋 = (var1‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ ∧ = (.g‘(mulGrp‘𝑊))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ 𝐶))
Theorem	evls1fpws 22263*	Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ 𝐴 = (coe1‘𝑀)    ⇒   ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
Theorem	ressply1evl 22264	Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐸 = (eval1‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵))
Theorem	evls1addd 22265	Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ⨣ = (+g‘𝑊)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))
Theorem	evls1muld 22266	Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))
Theorem	evls1vsca 22267	Univariate polynomial evaluation of a scalar product of polynomials. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴 × 𝑁))‘𝐶) = (𝐴 · ((𝑄‘𝑁)‘𝐶)))
Theorem	asclply1subcl 22268	Closure of the algebra scalar injection function in a polynomial on a subring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝐴 = (algSc‘𝑉)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝑉 = (Poly1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑃 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑃)
Theorem	evls1fvcl 22269	Variant of fveval1fvcl 22227 for the subring evaluation function evalSub1 (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵)
Theorem	evls1maprhm 22270*	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. (Contributed by Thierry Arnoux, 8-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
Theorem	evls1maplmhm 22271*	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴 is a module homomorphism, when considering the subring algebra. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    &   ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑆)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom 𝐴))
Theorem	evls1maprnss 22272*	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴 takes all values in the subring 𝑆. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    ⇒   ⊢ (𝜑 → 𝑆 ⊆ ran 𝐹)
Theorem	evl1maprhm 22273*	The function 𝐹 mapping polynomials 𝑝 to their evaluation at a given point 𝑋 is a ring homomorphism. (Contributed by metakunt, 19-May-2025.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
Theorem	mhmcompl 22274	The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑄 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
Theorem	mhmcoaddmpl 22275	Show that the ring homomorphism in rhmmpl 22277 preserves addition. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑄 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ + = (+g‘𝑃)    &   ⊢ ✚ = (+g‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))
Theorem	rhmcomulmpl 22276	Show that the ring homomorphism in rhmmpl 22277 preserves multiplication. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑄 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ∙ = (.r‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 · 𝐺)) = ((𝐻 ∘ 𝐹) ∙ (𝐻 ∘ 𝐺)))
Theorem	rhmmpl 22277*	Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20419. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑄 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	ply1vscl 22278	Closure of scalar multiplication for univariate polynomials. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
Theorem	mhmcoply1 22279	The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
Theorem	rhmply1 22280*	Provide a ring homomorphism between two univariate polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	rhmply1vr1 22281*	A ring homomorphism between two univariate polynomial algebras sends one variable to the other. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑌 = (var1‘𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)
Theorem	rhmply1vsca 22282*	Apply a ring homomorphism between two univariate polynomial algebras to a scaled polynomial. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))
Theorem	rhmply1mon 22283*	Apply a ring homomorphism between two univariate polynomial algebras to a scaled monomial, as in ply1coe 22192. (Contributed by SN, 20-May-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑌 = (var1‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝑄)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑄)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ ∧ = (.g‘𝑁)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐶 · (𝐸 ↑ 𝑋))) = ((𝐻‘𝐶) ∙ (𝐸 ∧ 𝑌)))
11.4  Matrices According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 21663) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 21680) and scalar multiplication (see frlmvscafval 21682) for free modules. Actually, there is no definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 22285. Thus, a statement like "Then the set of m x n matrices in R is a module (i.e., an R-module)" as in [Lang] p. 504 follows immediately from frlmlmod 21665. However, for square matrices there is Definition df-mat 22302, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication. A "usual" matrix (aij), (i = 1,..., m and j = 1,... n) would be represented as an element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))) and a square matrix (aij), (i = 1,..., n and j = 1,... n) would be represented as an element of (the base set of) ((1...𝑛) Mat 𝑅). Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which are excluded from the definitions of many authors, e.g., in [Lang] p. 503. It is shown in mat0dimbas0 22360 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 22360 Its determinant is the ring unity, see mdet0fv0 22488.
11.4.1  The matrix multiplication This section is about the multiplication of m x n matrices.
Syntax	cmmul 22284	Syntax for the matrix multiplication operator.
class maMul
Definition	df-mamu 22285*	The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd ‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘)))))))
Theorem	mamufval 22286*	Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
Theorem	mamuval 22287*	Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
Theorem	mamufv 22288*	A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
Theorem	mamudm 22289	The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))
Theorem	mamufacex 22290	Every solution of the equation 𝐴∗𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑀 × 𝑃))    &   ⊢ 𝐷 = (Base‘𝐺)    ⇒   ⊢ (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌 → 𝑍 ∈ 𝐶))
Theorem	mamures 22291	Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐺 = (𝑅 maMul ⟨𝐼, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ⊆ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌))
Theorem	grpvlinv 22292	Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 }))
Theorem	grpvrinv 22293	Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 }))
Theorem	ringvcl 22294	Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f · 𝑌) ∈ (𝐵 ↑m 𝐼))
Theorem	mamucl 22295	Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑃)))
Theorem	mamuass 22296	Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑂 × 𝑃)))    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ 𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   ⊢ 𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))
Theorem	mamudi 22297	Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → ((𝑋 ∘f + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘f + (𝑌𝐹𝑍)))
Theorem	mamudir 22298	Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)))
Theorem	mamuvs1 22299	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)))
Theorem	mamuvs2 22300	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂)))    ⇒   ⊢ (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘f · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘f · (𝑋𝐹𝑍)))