P. 227 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ply1sclf1 22601	The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵)
Theorem	ply1scl0 22602	The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑌 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = 𝑌)
Theorem	ply1scln0 22603	Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑌 = (0g‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘𝑋) ≠ 𝑌)
Theorem	ply1scl1 22604	The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.) (Proof shortened by SN, 12-Mar-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 1 ) = 𝑁)
Theorem	coe1id 22605*	Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐼 = (1r‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (coe1‘𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))
Theorem	ply1idvr1 22606	The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.) (Proof shortened by SN, 3-Jul-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃))
Theorem	cply1mul 22607*	The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ × = (.r‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))
Theorem	ply1coefsupp 22608*	The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 22609. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 𝐴 = (coe1‘𝐾)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
Theorem	ply1coe 22609*	Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 𝐴 = (coe1‘𝐾)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
Theorem	eqcoe1ply1eq 22610*	Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) → 𝐾 = 𝐿))
Theorem	ply1coe1eq 22611*	Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) ↔ 𝐾 = 𝐿))
Theorem	cply1coe0 22612*	All but the first coefficient of a constant polynomial ( i.e. a "lifted scalar") are zero. (Contributed by AV, 16-Nov-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑆))‘𝑛) = 0 )
Theorem	cply1coe0bi 22613*	A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))
Theorem	coe1fzgsumdlem 22614*	Lemma for coe1fzgsumd 22615 (induction step). (Contributed by AV, 8-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
Theorem	coe1fzgsumd 22615*	Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    ⇒   ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))
Theorem	ply1scleq 22616	Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹))
Theorem	ply1chr 22617	The characteristic of a polynomial ring is the characteristic of the underlying ring. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (chr‘𝑃) = (chr‘𝑅))
Theorem	gsumsmonply1 22618*	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 22619*	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 22620*	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 22621*	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 22622*	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 22623	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))
11.3.6  Univariate polynomial evaluation
Syntax	ces1 22624	Evaluation of a univariate polynomial in a subring.
class evalSub1
Syntax	ce1 22625	Evaluation of a univariate polynomial.
class eval1
Definition	df-evls1 22626*	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 22627*	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 22628	Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
⊢ Rel dom evalSub1
Theorem	ply1frcl 22629	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 22630*	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 22631*	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 22632*	Lemma for evl1rhm 22643 and evls1rhm 22633 (formerly part of the proof of evl1rhm 22643): 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 22633	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 22634	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 22635*	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 22636*	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 22637	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 22638	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 22639*	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 22640*	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 22641	Lemma for evl1fval1 22642. (Contributed by AV, 11-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))
Theorem	evl1fval1 22642	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 22643	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 22644	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 22645	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 22646	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 22647	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 22648	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 22649	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 22650	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 22651	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 22652	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 22653	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 22654	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 22655	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 22656	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 22657	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 22658	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 22659	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 22660	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 22661	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 22662*	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 22663*	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 22664	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 22665	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 22666*	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 22667*	Lemma for evl1gsumd 22668 (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 22668*	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 22669*	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 22635. (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 22670*	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 22671*	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 22672	Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 22669, the proof is shorter using evls1varpw 22638 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 22673	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 22674	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 22675	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 22676*	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)))))
11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism
Theorem	evls1scafv 22677	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 22678	Univariate polynomial evaluation builder for an exponential. See also evl1expd 22656. (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 22679	Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22673. (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 22680*	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 22681	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 22682	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 22683	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 22684	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 22685	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 22686	Variant of fveval1fvcl 22644 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 22687*	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 22688*	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 22689*	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 22690*	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	rhmmpl 22691*	Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20735. (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 22692	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 22693	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 22694*	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 22695*	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 22696*	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 22697*	Apply a ring homomorphism between two univariate polynomial algebras to a scaled monomial, as in ply1coe 22609. (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 22046) 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 22063) and scalar multiplication (see frlmvscafval 22065) 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 22699. 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 22048. However, for square matrices there is Definition df-mat 22716, 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 22774 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). 22774 Its determinant is the ring unity, see mdet0fv0 22902.
11.4.1  The matrix multiplication This section is about the multiplication of m x n matrices.
Syntax	cmmul 22698	Syntax for the matrix multiplication operator.
class maMul
Definition	df-mamu 22699*	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 22700*	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)))))))