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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-evls1 19501* 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 19502* 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 19503 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Rel dom evalSub1
 
Theoremply1frcl 19504 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 19505* 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 19506* 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 19507* Lemma for evl1rhm 19517 and evls1rhm 19508 (formerly part of the proof of evl1rhm 19517): 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 19508 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 19509 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 19510* 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 19511* 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 19512 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 19513* 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 19514* 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 19515 Lemma for evl1fval1 19516. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
 
Theoremevl1fval1 19516 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 19517 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 19518 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 19519 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 19520 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 19521 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 19522 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 19523 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 19524 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 19525 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 19526 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 19527 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 19528 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 19529 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 19530 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 19531 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)
 
Theorempf1id 19532 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 19533 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 19534 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)       (𝑋𝑄𝑅 ∈ CRing)
 
Theorempf1f 19535 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝑄𝐹:𝐵𝐵)
 
Theoremmpfpf1 19536* 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 19537* 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 19538 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    + = (+g𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 + 𝐺) ∈ 𝑄)
 
Theorempf1mulcl 19539 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    · = (.r𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 · 𝐺) ∈ 𝑄)
 
Theorempf1ind 19540* 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 19541* Lemma for evl1gsumd 19542 (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 19542* 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 19543* 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 19510. (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 19544* 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 19545* 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 19546 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 19543, the proof is shorter using evls1varpw 19512 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 19547 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 19548 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 19549 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 19550* 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 19551 Extend class notation with the class of all pseudometric spaces.
class PsMet
 
Syntaxcxmt 19552 Extend class notation with the class of all extended metric spaces.
class ∞Met
 
Syntaxcme 19553 Extend class notation with the class of all metrics.
class Met
 
Syntaxcbl 19554 Extend class notation with the metric space ball function.
class ball
 
Syntaxcfbas 19555 Extend class definition to include the class of filter bases.
class fBas
 
Syntaxcfg 19556 Extend class definition to include the filter generating function.
class filGen
 
Syntaxcmopn 19557 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
 
Syntaxcmetu 19558 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
 
Definitiondf-psmet 19559* 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 19560* Define the set of all extended metrics on a given base set. The definition is similar to df-met 19561, 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)))})
 
Definitiondf-met 19561* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 21936. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 21958, metgt0 21974, metsym 21965, and mettri 21967. (Contributed by NM, 25-Aug-2006.)
Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
 
Definitiondf-bl 19562* Define the metric space ball function. See blval 22001 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
 
Definitiondf-mopn 19563 Define a function whose value is the family of open sets of a metric space. See elmopn 22057 for its main property. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
 
Definitiondf-fbas 19564* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑥 ∩ 𝒫 (𝑦𝑧)) ≠ ∅)})
 
Definitiondf-fg 19565* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
 
Definitiondf-metu 19566* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
 
Syntaxccnfld 19567 Extend class notation with the field of complex numbers.
class fld
 
Definitiondf-cnfld 19568 The field of complex numbers. Other number fields and rings can be constructed by applying the s restriction operator, for instance (ℂfld ↾ 𝔸) is the field of algebraic numbers.

The contract of this set is defined entirely by cnfldex 19570, cnfldadd 19572, cnfldmul 19573, cnfldcj 19574, cnfldtset 19575, cnfldle 19576, cnfldds 19577, and cnfldbas 19571. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (New usage is discouraged.)

fld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
 
Theoremcnfldstr 19569 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
fld Struct ⟨1, 13⟩
 
Theoremcnfldex 19570 The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
fld ∈ V
 
Theoremcnfldbas 19571 The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
ℂ = (Base‘ℂfld)
 
Theoremcnfldadd 19572 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
+ = (+g‘ℂfld)
 
Theoremcnfldmul 19573 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
· = (.r‘ℂfld)
 
Theoremcnfldcj 19574 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
∗ = (*𝑟‘ℂfld)
 
Theoremcnfldtset 19575 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
(MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
 
Theoremcnfldle 19576 The ordering of the field of complex numbers. (Note that this is not actually an ordering on , but we put it in the structure anyway because restricting to does not affect this component, so that (ℂflds ℝ) is an ordered field even though fld itself is not.) (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
≤ = (le‘ℂfld)
 
Theoremcnfldds 19577 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
(abs ∘ − ) = (dist‘ℂfld)
 
Theoremcnfldunif 19578 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
 
Theoremxrsstr 19579 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 Struct ⟨1, 12⟩
 
Theoremxrsex 19580 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ V
 
Theoremxrsbas 19581 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
* = (Base‘ℝ*𝑠)
 
Theoremxrsadd 19582 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 = (+g‘ℝ*𝑠)
 
Theoremxrsmul 19583 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e = (.r‘ℝ*𝑠)
 
Theoremxrstset 19584 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
 
Theoremxrsle 19585 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
≤ = (le‘ℝ*𝑠)
 
Theoremcncrng 19586 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
fld ∈ CRing
 
Theoremcnring 19587 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ Ring
 
Theoremxrsmcmn 19588 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrsmgmdifsgrp 19602.) (Contributed by Mario Carneiro, 21-Aug-2015.)
(mulGrp‘ℝ*𝑠) ∈ CMnd
 
Theoremcnfld0 19589 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0g‘ℂfld)
 
Theoremcnfld1 19590 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r‘ℂfld)
 
Theoremcnfldneg 19591 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
 
Theoremcnfldplusf 19592 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+𝑓‘ℂfld)
 
Theoremcnfldsub 19593 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
− = (-g‘ℂfld)
 
Theoremcndrng 19594 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ DivRing
 
Theoremcnflddiv 19595 The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
/ = (/r‘ℂfld)
 
Theoremcnfldinv 19596 The multiplicative inverse in the field of complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋))
 
Theoremcnfldmulg 19597 The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℂ) → (𝐴(.g‘ℂfld)𝐵) = (𝐴 · 𝐵))
 
Theoremcnfldexp 19598 The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴𝐵))
 
Theoremcnsrng 19599 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
fld ∈ *-Ring
 
Theoremxrsmgm 19600 The (additive group of the) extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ Mgm
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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 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