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Type | Label | Description |
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Statement | ||
Definition | df-evls1 19501* | 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 19502* | 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 19503 | Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.) |
⊢ Rel dom evalSub1 | ||
Theorem | ply1frcl 19504 | 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 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) ⇒ ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))) | ||
Theorem | evls1val 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 (𝑆 ↾s 𝑅)) & ⊢ 𝐾 = (Base‘𝑀) ⇒ ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) | ||
Theorem | evls1rhmlem 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 = (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 19508 | 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 19509 | 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 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 = (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 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 = (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 | evls1varpw 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 = (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 19513* | 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 19514* | 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 19515 | Lemma for evl1fval1 19516. (Contributed by AV, 11-Sep-2019.) |
⊢ 𝑄 = (eval1‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑄 = (𝑅 evalSub1 𝐵)) | ||
Theorem | evl1fval1 19516 | 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 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 = (eval1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑇 = (𝑅 ↑s 𝐵) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) | ||
Theorem | fveval1fvcl 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 = (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 19519 | 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 19520 | 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 19521 | 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 19522 | 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 19523 | 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 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 = (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 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 = (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 19526 | 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 19527 | 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 19528 | 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 19529 | 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 19530 | 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 19531 | 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 19532 | 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 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 (eval1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅 ↑s 𝐵))) | ||
Theorem | pf1rcl 19534 | 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 19535 | 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 19536* | 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 19537* | 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 19538 | 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 19539 | 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 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) & ⊢ + = (+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 19541* | Lemma for evl1gsumd 19542 (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 19542* | 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 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 = (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 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 = (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 19545* | 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 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 = (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 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 = (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 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 = (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 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 = (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 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 = (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))))) | ||
Syntax | cpsmet 19551 | Extend class notation with the class of all pseudometric spaces. |
class PsMet | ||
Syntax | cxmt 19552 | Extend class notation with the class of all extended metric spaces. |
class ∞Met | ||
Syntax | cme 19553 | Extend class notation with the class of all metrics. |
class Met | ||
Syntax | cbl 19554 | Extend class notation with the metric space ball function. |
class ball | ||
Syntax | cfbas 19555 | Extend class definition to include the class of filter bases. |
class fBas | ||
Syntax | cfg 19556 | Extend class definition to include the filter generating function. |
class filGen | ||
Syntax | cmopn 19557 | Extend class notation with a function mapping each metric space to the family of its open sets. |
class MetOpen | ||
Syntax | cmetu 19558 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
class metUnif | ||
Definition | df-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)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-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)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-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)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}) | ||
Definition | df-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})) | ||
Definition | df-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))) | ||
Definition | df-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 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)}) | ||
Definition | df-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) ≠ ∅}) | ||
Definition | df-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))))) | ||
Syntax | ccnfld 19567 | Extend class notation with the field of complex numbers. |
class ℂfld | ||
Definition | df-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.) |
⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | ||
Theorem | cnfldstr 19569 | The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
⊢ ℂfld Struct 〈1, ;13〉 | ||
Theorem | cnfldex 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.) |
⊢ ℂfld ∈ V | ||
Theorem | cnfldbas 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) | ||
Theorem | cnfldadd 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) | ||
Theorem | cnfldmul 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) | ||
Theorem | cnfldcj 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) | ||
Theorem | cnfldtset 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) | ||
Theorem | cnfldle 19576 | The ordering of the field of complex numbers. (Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld 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) | ||
Theorem | cnfldds 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) | ||
Theorem | cnfldunif 19578 | The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | ||
Theorem | xrsstr 19579 | The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 Struct 〈1, ;12〉 | ||
Theorem | xrsex 19580 | The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 ∈ V | ||
Theorem | xrsbas 19581 | The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ* = (Base‘ℝ*𝑠) | ||
Theorem | xrsadd 19582 | The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ +𝑒 = (+g‘ℝ*𝑠) | ||
Theorem | xrsmul 19583 | The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ·e = (.r‘ℝ*𝑠) | ||
Theorem | xrstset 19584 | The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠) | ||
Theorem | xrsle 19585 | The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ≤ = (le‘ℝ*𝑠) | ||
Theorem | cncrng 19586 | The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) |
⊢ ℂfld ∈ CRing | ||
Theorem | cnring 19587 | The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ Ring | ||
Theorem | xrsmcmn 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 | ||
Theorem | cnfld0 19589 | The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 0 = (0g‘ℂfld) | ||
Theorem | cnfld1 19590 | The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 1 = (1r‘ℂfld) | ||
Theorem | cnfldneg 19591 | The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ (𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋) | ||
Theorem | cnfldplusf 19592 | The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ + = (+𝑓‘ℂfld) | ||
Theorem | cnfldsub 19593 | The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ − = (-g‘ℂfld) | ||
Theorem | cndrng 19594 | The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ DivRing | ||
Theorem | cnflddiv 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) | ||
Theorem | cnfldinv 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)) | ||
Theorem | cnfldmulg 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)) | ||
Theorem | cnfldexp 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) = (𝐴↑𝐵)) | ||
Theorem | cnsrng 19599 | The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ℂfld ∈ *-Ring | ||
Theorem | xrsmgm 19600 | The (additive group of the) extended reals is a magma. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∈ Mgm |
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