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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcoe1mul 20001* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑌 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))))

Theoremply1moncl 20002 Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → (𝐷 𝑋) ∈ 𝐵)

Theoremply1tmcl 20003 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 25-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)

Theoremcoe1tm 20004* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))

Theoremcoe1tmfv1 20005 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)

Theoremcoe1tmfv2 20006 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐷𝐹)       (𝜑 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐹) = 0 )

Theoremcoe1tmmul2 20007* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))

Theoremcoe1tmmul 20008* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))

Theoremcoe1tmmul2fv 20009 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → ((coe1‘(𝐴 (𝐶 · (𝐷 𝑋))))‘(𝐷 + 𝑌)) = (((coe1𝐴)‘𝑌) × 𝐶))

Theoremcoe1pwmul 20010* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘((𝐷 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, ((coe1𝐴)‘(𝑥𝐷)), 0 )))

Theoremcoe1pwmulfv 20011 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → ((coe1‘((𝐷 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1𝐴)‘𝑌))

Theoremply1scltm 20012 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾) → (𝐴𝐹) = (𝐹 · (0 𝑋)))

Theoremcoe1sclmul 20013 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾𝑌𝐵) → (coe1‘((𝐴𝑋) 𝑌)) = ((ℕ0 × {𝑋}) ∘𝑓 · (coe1𝑌)))

Theoremcoe1sclmulfv 20014 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ 0 ∈ ℕ0) → ((coe1‘((𝐴𝑋) 𝑌))‘ 0 ) = (𝑋 · ((coe1𝑌)‘ 0 )))

Theoremcoe1sclmul2 20015 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾𝑌𝐵) → (coe1‘(𝑌 (𝐴𝑋))) = ((coe1𝑌) ∘𝑓 · (ℕ0 × {𝑋})))

Theoremply1sclf 20016 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝐴:𝐾𝐵)

Theoremply1sclcl 20017 The value of the algebra scalars function for (univariate) polynomials applied to a scalar results in a constant polynomial. (Contributed by AV, 27-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑆𝐾) → (𝐴𝑆) ∈ 𝐵)

Theoremcoe1scl 20018* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾) → (coe1‘(𝐴𝑋)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑋, 0 )))

Theoremply1sclid 20019 Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾) → 𝑋 = ((coe1‘(𝐴𝑋))‘0))

Theoremply1sclf1 20020 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝐴:𝐾1-1𝐵)

Theoremply1scl0 20021 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &   𝑌 = (0g𝑃)       (𝑅 ∈ Ring → (𝐴0 ) = 𝑌)

Theoremply1scln0 20022 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &   𝑌 = (0g𝑃)    &   𝐾 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾𝑋0 ) → (𝐴𝑋) ≠ 𝑌)

Theoremply1scl1 20023 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    1 = (1r𝑅)    &   𝑁 = (1r𝑃)       (𝑅 ∈ Ring → (𝐴1 ) = 𝑁)

Theoremply1idvr1 20024 The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       (𝑅 ∈ Ring → (0 𝑋) = (1r𝑃))

Theoremcply1mul 20025* The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    × = (.r𝑃)       ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))

Theoremply1coefsupp 20026* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 20027. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)    &   𝐴 = (coe1𝐾)       ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))

Theoremply1coe 20027* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)    &   𝐴 = (coe1𝐾)       ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))

Theoremeqcoe1ply1eq 20028* Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑘 ∈ ℕ0 (𝐴𝑘) = (𝐶𝑘) → 𝐾 = 𝐿))

Theoremply1coe1eq 20029* Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑘 ∈ ℕ0 (𝐴𝑘) = (𝐶𝑘) ↔ 𝐾 = 𝐿))

Theoremcply1coe0 20030* 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 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑆𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑆))‘𝑛) = 0 )

Theoremcply1coe0bi 20031* 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 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))

Theoremcoe1fzgsumdlem 20032* Lemma for coe1fzgsumd 20033 (induction step). (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾 ∈ ℕ0)       ((𝑚 ∈ Fin ∧ ¬ 𝑎𝑚𝜑) → ((∀𝑥𝑚 𝑀𝐵 → ((coe1‘(𝑃 Σg (𝑥𝑚𝑀)))‘𝐾) = (𝑅 Σg (𝑥𝑚 ↦ ((coe1𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1𝑀)‘𝐾))))))

Theoremcoe1fzgsumd 20033* Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → ∀𝑥𝑁 𝑀𝐵)    &   (𝜑𝑁 ∈ Fin)       (𝜑 → ((coe1‘(𝑃 Σg (𝑥𝑁𝑀)))‘𝐾) = (𝑅 Σg (𝑥𝑁 ↦ ((coe1𝑀)‘𝐾))))

Theoremgsumsmonply1 20034* A finite group sum of scaled monomials is a univariate polynomial. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )       (𝜑 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))) ∈ 𝐵)

Theoremgsummoncoe1 20035* 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 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )    &   (𝜑𝐿 ∈ ℕ0)       (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))))‘𝐿) = 𝐿 / 𝑘𝐴)

Theoremgsumply1eq 20036* 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 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐵𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐵) finSupp 0 )    &   (𝜑𝑂 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))))    &   (𝜑𝑄 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐵 (𝑘 𝑋)))))       (𝜑 → (𝑂 = 𝑄 ↔ ∀𝑘 ∈ ℕ0 𝐴 = 𝐵))

Theoremlply1binom 20037* 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↑(𝑁𝑘)) · (𝑋𝑘)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    + = (+g𝑃)    &    × = (.r𝑃)    &    · = (.g𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0𝐴𝐵) → (𝑁 (𝑋 + 𝐴)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝑋))))))

Theoremlply1binomsc 20038* 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↑(𝑁𝑘)) · (𝑋𝑘)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    + = (+g𝑃)    &    × = (.r𝑃)    &    · = (.g𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐾 = (Base‘𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0𝐴𝐾) → (𝑁 (𝑋 + (𝑆𝐴))) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · ((𝑆‘((𝑁𝑘)𝐸𝐴)) × (𝑘 𝑋))))))

10.10.5  Univariate polynomial evaluation

Syntaxces1 20039 Evaluation of a univariate polynomial in a subring.
class evalSub1

Syntaxce1 20040 Evaluation of a univariate polynomial.
class eval1

Definitiondf-evls1 20041* Define the evaluation map for the univariate polynomial algebra. The function (𝑆 evalSub1 𝑅):𝑉⟶(𝑆𝑚 𝑆) makes sense when 𝑆 is a ring and 𝑅 is a subring of 𝑆, and where 𝑉 is the set of polynomials in (Poly1𝑅). 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 ∈ (𝑏𝑚 (𝑏𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))

Definitiondf-evl1 20042* Define the evaluation map for the univariate polynomial algebra. The function (eval1𝑅):𝑉⟶(𝑅𝑚 𝑅) makes sense when 𝑅 is a ring, and 𝑉 is the set of polynomials in (Poly1𝑅). 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 ∈ (𝑏𝑚 (𝑏𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))

Theoremreldmevls1 20043 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Rel dom evalSub1

Theoremply1frcl 20044 Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
𝑄 = ran (𝑆 evalSub1 𝑅)       (𝑋𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))

Theoremevls1fval 20045* 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)       ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))

Theoremevls1val 20046* 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 (𝑆s 𝑅))    &   𝐾 = (Base‘𝑀)       ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))

Theoremevls1rhmlem 20047* Lemma for evl1rhm 20057 and evls1rhm 20048 (formerly part of the proof of evl1rhm 20057): 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 = (𝑅s 𝐵)    &   𝐹 = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))       (𝑅 ∈ CRing → 𝐹 ∈ ((𝑅s (𝐵𝑚 1o)) RingHom 𝑇))

Theoremevls1rhm 20048 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑇 = (𝑆s 𝐵)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)       ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevls1sca 20049 Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Theoremevls1gsumadd 20050* Univariate polynomial evaluation maps (additive) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &    0 = (0g𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevls1gsummul 20051* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    1 = (1r𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s 𝐾)    &   𝐻 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevls1varpw 20052 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 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑈)    &   𝐵 = (Base‘𝑆)    &    = (.g𝐺)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆s 𝐵)))(𝑄𝑋)))

Theoremevl1fval 20053* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1o eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Theoremevl1val 20054* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1o eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑀 = (1o mPoly 𝑅)    &   𝐾 = (Base‘𝑀)       ((𝑅 ∈ CRing ∧ 𝐴𝐾) → (𝑂𝐴) = ((𝑄𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))

Theoremevl1fval1lem 20055 Lemma for evl1fval1 20056. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))

Theoremevl1fval1 20056 Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = (𝑅 evalSub1 𝐵)

Theoremevl1rhm 20057 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑅s 𝐵)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇))

Theoremfveval1fvcl 20058 The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀𝑈)       (𝜑 → ((𝑂𝑀)‘𝑌) ∈ 𝐵)

Theoremevl1sca 20059 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝑂‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Theoremevl1scad 20060 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐴𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴𝑋))‘𝑌) = 𝑋))

Theoremevl1var 20061 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → (𝑂𝑋) = ( I ↾ 𝐵))

Theoremevl1vard 20062 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑈 ∧ ((𝑂𝑋)‘𝑌) = 𝑌))

Theoremevls1var 20063 Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑋 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑋) = ( I ↾ 𝐵))

Theoremevls1scasrng 20064 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 = (eval1𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (Poly1𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))

Theoremevls1varsrng 20065 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 = (eval1𝑆)    &   𝑉 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑉) = (𝑂𝑉))

Theoremevl1addd 20066 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (+g𝑃)    &    + = (+g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Theoremevl1subd 20067 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (-g𝑃)    &   𝐷 = (-g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉𝐷𝑊)))

Theoremevl1muld 20068 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (.r𝑃)    &    · = (.r𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉 · 𝑊)))

Theoremevl1vsd 20069 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑𝑁𝐵)    &    = ( ·𝑠𝑃)    &    · = (.r𝑅)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 · 𝑉)))

Theoremevl1expd 20070 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &    = (.g‘(mulGrp‘𝑃))    &    = (.g‘(mulGrp‘𝑅))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 𝑉)))

Theorempf1const 20071 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)

Theorempf1id 20072 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄)

Theorempf1subrg 20073 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅s 𝐵)))

Theorempf1rcl 20074 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)       (𝑋𝑄𝑅 ∈ CRing)

Theorempf1f 20075 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝑄𝐹:𝐵𝐵)

Theoremmpfpf1 20076* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝐸 → (𝐹 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)

Theorempf1mpf 20077* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵𝑚 1o) ↦ (𝑥‘∅))) ∈ 𝐸)

Theorempf1addcl 20078 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    + = (+g𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 + 𝐺) ∈ 𝑄)

Theorempf1mulcl 20079 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    · = (.r𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 · 𝐺) ∈ 𝑄)

Theorempf1ind 20080* 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)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝑄 = ran (eval1𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))    &   (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝐵) → 𝜒)    &   (𝜑𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)

Theoremevl1gsumdlem 20081* Lemma for evl1gsumd 20082 (induction step). (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       ((𝑚 ∈ Fin ∧ ¬ 𝑎𝑚𝜑) → ((∀𝑥𝑚 𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥𝑚𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑚 ↦ ((𝑂𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂𝑀)‘𝑌))))))

Theoremevl1gsumd 20082* Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → ∀𝑥𝑁 𝑀𝑈)    &   (𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑂‘(𝑃 Σg (𝑥𝑁𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑂𝑀)‘𝑌))))

Theoremevl1gsumadd 20083* 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 20050. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    0 = (0g𝑊)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevl1gsumaddval 20084* 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 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑁𝑌)))‘𝐶) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑄𝑌)‘𝐶))))

Theoremevl1gsummul 20085* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    1 = (1r𝑊)    &   𝐺 = (mulGrp‘𝑊)    &   𝐻 = (mulGrp‘𝑃)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevl1varpw 20086 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 20083, the proof is shorter using evls1varpw 20052 instead of proving it directly. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅s 𝐵)))(𝑄𝑋)))

Theoremevl1varpwval 20087 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 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)       (𝜑 → ((𝑄‘(𝑁 𝑋))‘𝐶) = (𝑁𝐸𝐶))

Theoremevl1scvarpw 20088 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 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   𝑆 = (𝑅s 𝐵)    &    = (.r𝑆)    &   𝑀 = (mulGrp‘𝑆)    &   𝐹 = (.g𝑀)       (𝜑 → (𝑄‘(𝐴 × (𝑁 𝑋))) = ((𝐵 × {𝐴}) (𝑁𝐹(𝑄𝑋))))

Theoremevl1scvarpwval 20089 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 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &    · = (.r𝑅)       (𝜑 → ((𝑄‘(𝐴 × (𝑁 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))

Theoremevl1gsummon 20090* 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 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐵 = (Base‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &    × = ( ·𝑠𝑊)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑 → ∀𝑥𝑀 𝐴𝐾)    &   (𝜑𝑀 ⊆ ℕ0)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑 → ∀𝑥𝑀 𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑀 ↦ (𝐴 × (𝑁 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))))

10.11  The complex numbers as an algebraic extensible structure

10.11.1  Definition and basic properties

Syntaxcpsmet 20091 Extend class notation with the class of all pseudometric spaces.
class PsMet

Syntaxcxmet 20092 Extend class notation with the class of all extended metric spaces.
class ∞Met

Syntaxcmet 20093 Extend class notation with the class of all metrics.
class Met

Syntaxcbl 20094 Extend class notation with the metric space ball function.
class ball

Syntaxcfbas 20095 Extend class definition to include the class of filter bases.
class fBas

Syntaxcfg 20096 Extend class definition to include the filter generating function.
class filGen

Syntaxcmopn 20097 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen

Syntaxcmetu 20098 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif

Definitiondf-psmet 20099* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Definitiondf-xmet 20100* Define the set of all extended metrics on a given base set. The definition is similar to df-met 20101, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43448
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