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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlply1binom 19401* 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 19402* 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 19403 Evaluation of a univariate polynomial in a subring.
class evalSub1
 
Syntaxce1 19404 Evaluation of a univariate polynomial.
class eval1
 
Definitiondf-evls1 19405* 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 19406* 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 19407 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Rel dom evalSub1
 
Theoremply1frcl 19408 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 19409* 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 19410* 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 19411* Lemma for evl1rhm 19421 and evls1rhm 19412 (formerly part of the proof of evl1rhm 19421): 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 19412 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 19413 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 19414* 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 19415* 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 19416 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 19417* 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 19418* 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 19419 Lemma for evl1fval1 19420. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
 
Theoremevl1fval1 19420 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 19421 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 19422 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 19423 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 19424 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 19425 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 19426 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 19427 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 19428 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 19429 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 19430 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 19431 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 19432 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 19433 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 19434 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 19435 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)
 
Theorempf1id 19436 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 19437 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 19438 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)       (𝑋𝑄𝑅 ∈ CRing)
 
Theorempf1f 19439 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝑄𝐹:𝐵𝐵)
 
Theoremmpfpf1 19440* 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 19441* 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 19442 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    + = (+g𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 + 𝐺) ∈ 𝑄)
 
Theorempf1mulcl 19443 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    · = (.r𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 · 𝐺) ∈ 𝑄)
 
Theorempf1ind 19444* 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 19445* Lemma for evl1gsumd 19446 (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 19446* 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 19447* 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 19414. (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 19448* 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 19449* 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 19450 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 19447, the proof is shorter using evls1varpw 19416 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 19451 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 19452 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 19453 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 19454* 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 19455 Extend class notation with the class of all pseudometric spaces.
class PsMet
 
Syntaxcxmt 19456 Extend class notation with the class of all extended metric spaces.
class ∞Met
 
Syntaxcme 19457 Extend class notation with the class of all metrics.
class Met
 
Syntaxcbl 19458 Extend class notation with the metric space ball function.
class ball
 
Syntaxcfbas 19459 Extend class definition to include the class of filter bases.
class fBas
 
Syntaxcfg 19460 Extend class definition to include the filter generating function.
class filGen
 
Syntaxcmopn 19461 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
 
Syntaxcmetu 19462 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
 
Definitiondf-psmet 19463* 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 19464* Define the set of all extended metrics on a given base set. The definition is similar to df-met 19465, 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 19465* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 21838. 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 21860, metgt0 21876, metsym 21867, and mettri 21869. (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 19466* Define the metric space ball function. See blval 21903 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 19467 Define a function whose value is the family of open sets of a metric space. See elmopn 21959 for its main property. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
 
Definitiondf-fbas 19468* 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 19469* 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 19470* 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 19471 Extend class notation with the field of complex numbers.
class fld
 
Definitiondf-cnfld 19472 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 19474, cnfldadd 19476, cnfldmul 19477, cnfldcj 19478, cnfldtset 19479, cnfldle 19480, cnfldds 19481, and cnfldbas 19475. 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 19473 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 19474 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 19475 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 19476 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 19477 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 19478 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 19479 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 19480 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 19481 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 19482 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
 
Theoremxrsstr 19483 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 Struct ⟨1, 12⟩
 
Theoremxrsex 19484 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ V
 
Theoremxrsbas 19485 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
* = (Base‘ℝ*𝑠)
 
Theoremxrsadd 19486 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 = (+g‘ℝ*𝑠)
 
Theoremxrsmul 19487 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e = (.r‘ℝ*𝑠)
 
Theoremxrstset 19488 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
 
Theoremxrsle 19489 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
≤ = (le‘ℝ*𝑠)
 
Theoremcncrng 19490 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
fld ∈ CRing
 
Theoremcnring 19491 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ Ring
 
Theoremxrsmcmn 19492 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrsmgmdifsgrp 19506.) (Contributed by Mario Carneiro, 21-Aug-2015.)
(mulGrp‘ℝ*𝑠) ∈ CMnd
 
Theoremcnfld0 19493 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0g‘ℂfld)
 
Theoremcnfld1 19494 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r‘ℂfld)
 
Theoremcnfldneg 19495 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
 
Theoremcnfldplusf 19496 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+𝑓‘ℂfld)
 
Theoremcnfldsub 19497 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
− = (-g‘ℂfld)
 
Theoremcndrng 19498 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ DivRing
 
Theoremcnflddiv 19499 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 19500 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))
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