P. 261 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvfsum2 26001*	The reverse of dvfsumrlim 25998, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.)
⊢ 𝑆 = (𝑇(,)+∞)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))    &   ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐵 ≤ 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ≤ 𝑋)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑈)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐸)    ⇒   ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ 𝐸)
Theorem	ftc1lem1 26002*	Lemma for ftc1a 26004 and ftc1 26009. (Contributed by Mario Carneiro, 31-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
Theorem	ftc1lem2 26003*	Lemma for ftc1 26009. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
Theorem	ftc1a 26004*	The Fundamental Theorem of Calculus, part one. The function 𝐺 formed by varying the right endpoint of an integral of 𝐹 is continuous if 𝐹 is integrable. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	ftc1lem3 26005*	Lemma for ftc1 26009. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   ⊢ 𝐽 = (𝐿 ↾t ℝ)    &   ⊢ 𝐾 = (𝐿 ↾t 𝐷)    &   ⊢ 𝐿 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
Theorem	ftc1lem4 26006*	Lemma for ftc1 26009. (Contributed by Mario Carneiro, 31-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   ⊢ 𝐽 = (𝐿 ↾t ℝ)    &   ⊢ 𝐾 = (𝐿 ↾t 𝐷)    &   ⊢ 𝐿 = (TopOpen‘ℂfld)    &   ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑌 − 𝐶)) < 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶))) < 𝐸)
Theorem	ftc1lem5 26007*	Lemma for ftc1 26009. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   ⊢ 𝐽 = (𝐿 ↾t ℝ)    &   ⊢ 𝐾 = (𝐿 ↾t 𝐷)    &   ⊢ 𝐿 = (TopOpen‘ℂfld)    &   ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸)
Theorem	ftc1lem6 26008*	Lemma for ftc1 26009. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   ⊢ 𝐽 = (𝐿 ↾t ℝ)    &   ⊢ 𝐾 = (𝐿 ↾t 𝐷)    &   ⊢ 𝐿 = (TopOpen‘ℂfld)    &   ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))    ⇒   ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 limℂ 𝐶))
Theorem	ftc1 26009*	The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at 𝐶 with derivative 𝐹(𝐶) if the original function is continuous at 𝐶. This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   ⊢ 𝐽 = (𝐿 ↾t ℝ)    &   ⊢ 𝐾 = (𝐿 ↾t 𝐷)    &   ⊢ 𝐿 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))
Theorem	ftc1cn 26010*	Strengthen the assumptions of ftc1 26009 to when the function 𝐹 is continuous on the entire interval (𝐴, 𝐵); in this case we can calculate D 𝐺 exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)
Theorem	ftc2 26011*	The Fundamental Theorem of Calculus, part two. If 𝐹 is a function continuous on [𝐴, 𝐵] and continuously differentiable on (𝐴, 𝐵), then the integral of the derivative of 𝐹 is equal to 𝐹(𝐵) − 𝐹(𝐴). This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
Theorem	ftc2ditglem 26012*	Lemma for ftc2ditg 26013. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
Theorem	ftc2ditg 26013*	Directed integral analogue of ftc2 26011. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
Theorem	itgparts 26014*	Integration by parts. If 𝐵(𝑥) is the derivative of 𝐴(𝑥) and 𝐷(𝑥) is the derivative of 𝐶(𝑥), and 𝐸 = (𝐴 · 𝐵)(𝑋) and 𝐹 = (𝐴 · 𝐵)(𝑌), then under suitable integrability and differentiability assumptions, the integral of 𝐴 · 𝐷 from 𝑋 to 𝑌 is equal to 𝐹 − 𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿1)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹)    ⇒   ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))
Theorem	itgsubstlem 26015*	Lemma for itgsubst 26016. (Contributed by Mario Carneiro, 12-Sep-2014.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑊 ∈ ℝ*)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑍(,)𝑊))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑍(,)𝑊))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑀(,)𝑁))    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgsubst 26016*	Integration by 𝑢-substitution. If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. In this part of the proof we discharge the assumptions in itgsubstlem 26015, which use the fact that (𝑍, 𝑊) is open to shrink the interval a little to (𝑀, 𝑁) where 𝑍 < 𝑀 < 𝑁 < 𝑊- this is possible because 𝐴(𝑥) is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑊 ∈ ℝ*)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgpowd 26017*	The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1)))
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
14.1  Polynomials
14.1.1  Polynomial degrees
Syntax	cmdg 26018	Multivariate polynomial degree.
class mDeg
Syntax	cdg1 26019	Univariate polynomial degree.
class deg1
Definition	df-mdeg 26020*	Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial -∞, contrary to the convention used in df-dgr 26156. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
Definition	df-deg1 26021	Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
Theorem	reldmmdeg 26022	Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Rel dom mDeg
Theorem	tdeglem1 26023*	Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ 𝐻:𝐴⟶ℕ0
Theorem	tdeglem3 26024*	Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌)))
Theorem	tdeglem4 26025*	There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ (𝑋 ∈ 𝐴 → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))
Theorem	tdeglem2 26026	Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ))
Theorem	mdegfval 26027*	Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
Theorem	mdegval 26028*	Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ))
Theorem	mdegleb 26029*	Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐺 < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )))
Theorem	mdeglt 26030*	If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋))    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) = 0 )
Theorem	mdegldg 26031*	A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    &   ⊢ 𝑌 = (0g‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))
Theorem	mdegxrcl 26032	Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*)
Theorem	mdegxrf 26033	Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ 𝐷:𝐵⟶ℝ*
Theorem	mdegcl 26034	Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞}))
Theorem	mdeg0 26035	Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞)
Theorem	mdegnn0cl 26036	Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0)
Theorem	degltlem1 26037	Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)))
Theorem	degltp1le 26038	Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < (𝑌 + 1) ↔ 𝑋 ≤ 𝑌))
Theorem	mdegaddle 26039	The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)))
Theorem	mdegvscale 26040	The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺))
Theorem	mdegvsca 26041	The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a nonzero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐸)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺))
Theorem	mdegle0 26042	A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐴 = (algSc‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))
Theorem	mdegmullem 26043*	Lemma for mdegmulle2 26044. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = (.r‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾)    &   ⊢ 𝐴 = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (𝑏 ∈ 𝐴 ↦ (ℂfld Σg 𝑏))    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))
Theorem	mdegmulle2 26044	The multivariate degree of a product of polynomials is at most the sum of the degrees of the polynomials. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐷 = (𝐼 mDeg 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = (.r‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))
Theorem	deg1fval 26045	Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐷 = (deg1‘𝑅)    ⇒   ⊢ 𝐷 = (1o mDeg 𝑅)
Theorem	deg1xrf 26046	Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ 𝐷:𝐵⟶ℝ*
Theorem	deg1xrcl 26047	Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*)
Theorem	deg1cl 26048	Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞}))
Theorem	mdegpropd 26049*	Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆))
Theorem	deg1fvi 26050	Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ (deg1‘𝑅) = (deg1‘( I ‘𝑅))
Theorem	deg1propd 26051*	Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (deg1‘𝑅) = (deg1‘𝑆))
Theorem	deg1z 26052	Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞)
Theorem	deg1nn0cl 26053	Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0)
Theorem	deg1n0ima 26054	Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0)
Theorem	deg1nn0clb 26055	A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0))
Theorem	deg1lt0 26056	A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 ))
Theorem	deg1ldg 26057	A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑌 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌)
Theorem	deg1ldgn 26058	An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑌 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋)
Theorem	deg1ldgdomn 26059	A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸)
Theorem	deg1leb 26060*	Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 )))
Theorem	deg1val 26061	Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < ))
Theorem	deg1lt 26062	If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 )
Theorem	deg1ge 26063	Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐴 = (coe1‘𝐹)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹))
Theorem	coe1mul3 26064	The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ ∙ = (.r‘𝑌)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽)    ⇒   ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe1‘𝐹)‘𝐼) · ((coe1‘𝐺)‘𝐽)))
Theorem	coe1mul4 26065	Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ ∙ = (.r‘𝑌)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    ⇒   ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹)) · ((coe1‘𝐺)‘(𝐷‘𝐺))))
Theorem	deg1addle 26066	The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)))
Theorem	deg1addle2 26067	If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ*)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿)
Theorem	deg1add 26068	Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹))    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷‘𝐹))
Theorem	deg1vscale 26069	The degree of a scalar times a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺))
Theorem	deg1vsca 26070	The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐸)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺))
Theorem	deg1invg 26071	The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑁 = (invg‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹))
Theorem	deg1suble 26072	The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)))
Theorem	deg1sub 26073	Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹))    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘𝐹))
Theorem	deg1mulle2 26074	Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = (.r‘𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))
Theorem	deg1sublt 26075	Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿)    &   ⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐶 = (coe1‘𝐺)    &   ⊢ (𝜑 → ((coe1‘𝐹)‘𝐿) = ((coe1‘𝐺)‘𝐿))    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿)
Theorem	deg1le0 26076	A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0))))
Theorem	deg1sclle 26077	A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐷‘(𝐴‘𝐹)) ≤ 0)
Theorem	deg1scl 26078	A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0)
Theorem	deg1mul2 26079	Degree of multiplication of two nonzero polynomials when the first leads with a nonzero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    &   ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺)))
Theorem	deg1mul 26080	Degree of multiplication of two nonzero polynomials in a domain. (Contributed by metakunt, 6-May-2025.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺)))
Theorem	deg1mul3 26081	Degree of multiplication of a polynomial on the left by a nonzero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺))
Theorem	deg1mul3le 26082	Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) ≤ (𝐷‘𝐺))
Theorem	deg1tmle 26083	Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹)
Theorem	deg1tm 26084	Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹)
Theorem	deg1pwle 26085	Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹)
Theorem	deg1pw 26086	Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹)
Theorem	ply1nz 26087	Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing)
Theorem	ply1nzb 26088	Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing))
Theorem	ply1domn 26089	Corollary of deg1mul2 26079: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn → 𝑃 ∈ Domn)
Theorem	ply1idom 26090	The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ IDomn → 𝑃 ∈ IDomn)
14.1.2  The division algorithm for univariate polynomials
Syntax	cmn1 26091	Monic polynomials.
class Monic1p
Syntax	cuc1p 26092	Unitic polynomials.
class Unic1p
Syntax	cq1p 26093	Univariate polynomial quotient.
class quot1p
Syntax	cr1p 26094	Univariate polynomial remainder.
class rem1p
Syntax	cig1p 26095	Univariate polynomial ideal generator.
class idlGen1p
Definition	df-mon1 26096*	Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
Definition	df-uc1p 26097*	Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 26103. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
Definition	df-q1p 26098*	Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 26103. We actually use the reversed version for better harmony with our divisibility df-dvdsr 20337. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ quot1p = (𝑟 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 ((deg1‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < ((deg1‘𝑟)‘𝑔))))
Definition	df-r1p 26099*	Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
Definition	df-ig1p 26100*	Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )))))