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
Syntax	ce1 22201	Evaluation of a univariate polynomial.
class eval1
Definition	df-evls1 22202*	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 22203*	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 22204	Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
⊢ Rel dom evalSub1
Theorem	ply1frcl 22205	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 22206*	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 22207*	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 22208*	Lemma for evl1rhm 22219 and evls1rhm 22209 (formerly part of the proof of evl1rhm 22219): 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 22209	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 22210	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 22211*	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 22212*	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 22213	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 22214	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 22215*	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 22216*	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 22217	Lemma for evl1fval1 22218. (Contributed by AV, 11-Sep-2019.)
⊢ 𝑄 = (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵))
Theorem	evl1fval1 22218	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 22219	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 22220	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 22221	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 22222	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 22223	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 22224	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 22225	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 22226	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 22227	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 22228	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 22229	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 22230	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 22231	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 22232	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 22233	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 22234	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 22235	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 22236	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 22237	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 22238*	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 22239*	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 22240	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 22241	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 22242*	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 22243*	Lemma for evl1gsumd 22244 (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 22244*	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 22245*	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 22211. (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 22246*	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 22247*	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 22248	Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 22245, the proof is shorter using evls1varpw 22214 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 22249	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 22250	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 22251	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 22252*	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 22253	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 22254	Univariate polynomial evaluation builder for an exponential. See also evl1expd 22232. (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 22255	Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22249. (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 22256*	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 22257	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 22258	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 22259	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 22260	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 22261	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 22262	Variant of fveval1fvcl 22220 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 22263*	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 22264*	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 22265*	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 22266*	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 22267	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 22268	Show that the ring homomorphism in rhmmpl 22270 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 22269	Show that the ring homomorphism in rhmmpl 22270 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 22270*	Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20412. (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 22271	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 22272	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 22273*	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 22274*	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 22275*	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 22276*	Apply a ring homomorphism between two univariate polynomial algebras to a scaled monomial, as in ply1coe 22185. (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 21656) 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 21673) and scalar multiplication (see frlmvscafval 21675) 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 22278. 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 21658. However, for square matrices there is Definition df-mat 22295, 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 22353 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). 22353 Its determinant is the ring unity, see mdet0fv0 22481.
11.4.1  The matrix multiplication This section is about the multiplication of m x n matrices.
Syntax	cmmul 22277	Syntax for the matrix multiplication operator.
class maMul
Definition	df-mamu 22278*	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 22279*	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 22280*	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 22281*	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 22282	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 22283	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 22284	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 22285	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 22286	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 22287	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 22288	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 22289	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 22290	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 22291	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 22292	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 22293	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)))
11.4.2  Square matrices In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 22316. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.
Syntax	cmat 22294	Syntax for the square matrix algebra.
class Mat
Definition	df-mat 22295*	Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
Theorem	matbas0pc 22296	There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
Theorem	matbas0 22297	There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
Theorem	matval 22298	Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &   ⊢ · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
Theorem	matrcl 22299	Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
Theorem	matbas 22300	The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴))