P. 416 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41501-41600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lcmineqlem2 41501*	Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.)
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥))
Theorem	lcmineqlem3 41502*	Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 30-Apr-2024.)
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘))))
Theorem	lcmineqlem4 41503	Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. F is found in lcmineqlem6 41505. (Contributed by metakunt, 10-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(𝑁 − 𝑀)))    ⇒   ⊢ (𝜑 → ((lcm‘(1...𝑁)) / (𝑀 + 𝐾)) ∈ ℤ)
Theorem	lcmineqlem5 41504	Technical lemma for reciprocal multiplication in deduction form. (Contributed by metakunt, 10-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 · (𝐵 · (1 / 𝐶))) = (𝐵 · (𝐴 / 𝐶)))
Theorem	lcmineqlem6 41505*	Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 10-May-2024.)
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ)
Theorem	lcmineqlem7 41506	Derivative of 1-x for chain rule application. (Contributed by metakunt, 12-May-2024.)
⊢ (ℂ D (𝑥 ∈ ℂ ↦ (1 − 𝑥))) = (𝑥 ∈ ℂ ↦ -1)
Theorem	lcmineqlem8 41507*	Derivative of (1-x)^(N-M). (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1)))))
Theorem	lcmineqlem9 41508*	(1-x)^(N-M) is continuous. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀))) ∈ (ℂ–cn→ℂ))
Theorem	lcmineqlem10 41509*	Induction step of lcmineqlem13 41512 (deduction form). (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    ⇒   ⊢ (𝜑 → ∫(0[,]1)((𝑥↑((𝑀 + 1) − 1)) · ((1 − 𝑥)↑(𝑁 − (𝑀 + 1)))) d𝑥 = ((𝑀 / (𝑁 − 𝑀)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥))
Theorem	lcmineqlem11 41510	Induction step, continuation for binomial coefficients. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    ⇒   ⊢ (𝜑 → (1 / ((𝑀 + 1) · (𝑁C(𝑀 + 1)))) = ((𝑀 / (𝑁 − 𝑀)) · (1 / (𝑀 · (𝑁C𝑀)))))
Theorem	lcmineqlem12 41511*	Base case for induction. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∫(0[,]1)((𝑡↑(1 − 1)) · ((1 − 𝑡)↑(𝑁 − 1))) d𝑡 = (1 / (1 · (𝑁C1))))
Theorem	lcmineqlem13 41512*	Induction proof for lcm integral. (Contributed by metakunt, 12-May-2024.)
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 = (1 / (𝑀 · (𝑁C𝑀))))
Theorem	lcmineqlem14 41513	Technical lemma for inequality estimate. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐷)    &   ⊢ (𝜑 → (𝐵 · 𝐶) ∥ 𝐸)    &   ⊢ (𝜑 → 𝐷 ∥ 𝐸)    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸)
Theorem	lcmineqlem15 41514*	F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.)
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ)
Theorem	lcmineqlem16 41515	Technical divisibility lemma. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∥ (lcm‘(1...𝑁)))
Theorem	lcmineqlem17 41516	Inequality of 2^{2n}. (Contributed by metakunt, 29-Apr-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
Theorem	lcmineqlem18 41517	Technical lemma to shift factors in binomial coefficient. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
Theorem	lcmineqlem19 41518	Dividing implies inequality for lcm inequality lemma. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1))))
Theorem	lcmineqlem20 41519	Inequality for lcm lemma. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
Theorem	lcmineqlem21 41520	The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 4 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1))))
Theorem	lcmineqlem22 41521	The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 4 ≤ 𝑁)    ⇒   ⊢ (𝜑 → ((2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))) ∧ (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2)))))
Theorem	lcmineqlem23 41522	Penultimate step to the lcm inequality lemma. (Contributed by metakunt, 12-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 9 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁)))
Theorem	lcmineqlem 41523	The least common multiple inequality lemma, a central result for future use. Theorem 3.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 16-May-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 7 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁)))
21.27.5  Logarithm inequalities
Theorem	3exp7 41524	3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024.)
⊢ (3↑7) = ;;;2187
Theorem	3lexlogpow5ineq1 41525	First inequality in inequality chain, proposed by Mario Carneiro (Contributed by metakunt, 22-May-2024.)
⊢ 9 < ((;11 / 7)↑5)
Theorem	3lexlogpow5ineq2 41526	Second inequality in inequality chain, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 3 ≤ 𝑋)    ⇒   ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5))
Theorem	3lexlogpow5ineq4 41527	Sharper logarithm inequality chain. (Contributed by metakunt, 21-Aug-2024.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 3 ≤ 𝑋)    ⇒   ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5))
Theorem	3lexlogpow5ineq3 41528	Combined inequality chain for a specific power of the binary logarithm, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 3 ≤ 𝑋)    ⇒   ⊢ (𝜑 → 7 < ((2 logb 𝑋)↑5))
Theorem	3lexlogpow2ineq1 41529	Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
⊢ ((3 / 2) < (2 logb 3) ∧ (2 logb 3) < (5 / 3))
Theorem	3lexlogpow2ineq2 41530	Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
⊢ (2 < ((2 logb 3)↑2) ∧ ((2 logb 3)↑2) < 3)
Theorem	3lexlogpow5ineq5 41531	Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
⊢ ((2 logb 3)↑5) ≤ ;15
21.27.6  Miscellaneous results for AKS formalisation
Theorem	intlewftc 41532*	Inequality inference by invoking fundamental theorem of calculus. (Contributed by metakunt, 22-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐷 = (ℝ D 𝐹))    &   ⊢ (𝜑 → 𝐸 = (ℝ D 𝐺))    &   ⊢ (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐷 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃))    &   ⊢ (𝜑 → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ≤ 𝑄)    &   ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐺‘𝐴))    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐺‘𝐵))
Theorem	aks4d1lem1 41533	Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
Theorem	aks4d1p1p1 41534*	Exponential law for finite products, special case. (Contributed by metakunt, 22-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ (1...𝑁)(𝐴↑𝑐𝑘) = (𝐴↑𝑐Σ𝑘 ∈ (1...𝑁)𝑘))
Theorem	dvrelog2 41535*	The derivative of the logarithm, ftc2 25992 version. (Contributed by metakunt, 11-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))    ⇒   ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
Theorem	dvrelog3 41536*	The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))    ⇒   ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
Theorem	dvrelog2b 41537*	Derivative of the binary logarithm. (Contributed by metakunt, 11-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))    ⇒   ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
Theorem	0nonelalab 41538	Technical lemma for open interval. (Contributed by metakunt, 12-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 0 ≠ 𝐶)
Theorem	dvrelogpow2b 41539*	Derivative of the power of the binary logarithm. (Contributed by metakunt, 12-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))    &   ⊢ 𝐶 = (𝑁 / ((log‘2)↑𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
Theorem	aks4d1p1p3 41540	Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ (𝜑 → 3 ≤ 𝑁)    ⇒   ⊢ (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1))))
Theorem	aks4d1p1p2 41541*	Rewrite 𝐴 in more suitable form. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ (𝜑 → 3 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐴 < (𝑁↑𝑐(((2 logb (((2 logb 𝑁)↑5) + 1)) + (((2 logb 𝑁)↑2) / 2)) + (((2 logb 𝑁)↑4) / 2))))
Theorem	aks4d1p1p4 41542*	Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ (𝜑 → 3 ≤ 𝑁)    &   ⊢ 𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   ⊢ 𝐷 = ((2 logb 𝑁)↑2)    &   ⊢ 𝐸 = ((2 logb 𝑁)↑4)    &   ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)    ⇒   ⊢ (𝜑 → 𝐴 < (2↑𝐵))
Theorem	dvle2 41543*	Collapsed dvle 25953. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻)    &   ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃)    &   ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄)    &   ⊢ (𝑥 = 𝐵 → 𝐸 = 𝑅)    &   ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆)    &   ⊢ (𝜑 → 𝑃 ≤ 𝑄)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝑅 ≤ 𝑆)
Theorem	aks4d1p1p6 41544*	Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 3 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 logb (((2 logb 𝑥)↑5) + 1))) + ((2 logb 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 logb 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))
Theorem	aks4d1p1p7 41545	Bound of intermediary of inequality step. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 4 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((2 · ((1 / ((((2 logb 𝐴)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝐴)↑4)) · (1 / (𝐴 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝐴)↑(2 − 1)) / 𝐴))) ≤ ((4 / ((log‘2)↑4)) · (((log‘𝐴)↑3) / 𝐴)))
Theorem	aks4d1p1p5 41546*	Show inequality for existence of a non-divisor. (Contributed by metakunt, 19-Aug-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ (𝜑 → 4 ≤ 𝑁)    &   ⊢ 𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   ⊢ 𝐷 = ((2 logb 𝑁)↑2)    &   ⊢ 𝐸 = ((2 logb 𝑁)↑4)    ⇒   ⊢ (𝜑 → 𝐴 < (2↑𝐵))
Theorem	aks4d1p1 41547*	Show inequality for existence of a non-divisor. (Contributed by metakunt, 21-Aug-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   ⊢ (𝜑 → 𝐴 < (2↑𝐵))
Theorem	aks4d1p2 41548	Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   ⊢ (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
Theorem	aks4d1p3 41549*	There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
Theorem	aks4d1p4 41550*	There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 28-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
Theorem	aks4d1p5 41551*	Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)    ⇒   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
Theorem	aks4d1p6 41552*	The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑅)    &   ⊢ 𝐾 = (𝑃 pCnt 𝑅)    ⇒   ⊢ (𝜑 → 𝐾 ≤ (⌊‘(2 logb 𝐵)))
Theorem	aks4d1p7d1 41553*	Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁))    ⇒   ⊢ (𝜑 → 𝑅 ∥ (𝑁↑(⌊‘(2 logb 𝐵))))
Theorem	aks4d1p7 41554*	Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁))
Theorem	aks4d1p8d1 41555	If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.)
⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑀)    &   ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁)    ⇒   ⊢ (𝜑 → 𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁)))
Theorem	aks4d1p8d2 41556	Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑄 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑅)    &   ⊢ (𝜑 → 𝑄 ∥ 𝑅)    &   ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝑄 ∥ 𝑁)    ⇒   ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)
Theorem	aks4d1p8d3 41557	The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    ⇒   ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1)
Theorem	aks4d1p8 41558*	Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
Theorem	aks4d1p9 41559*	Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))
Theorem	aks4d1 41560*	Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 logb 𝑁)↑2) < ((odℤ‘𝑟)‘𝑁)))
Theorem	fldhmf1 41561	A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝐿 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿))    &   ⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐵 = (Base‘𝐿)    ⇒   ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
Syntax	cprimroots 41562	Define the class of primitive roots. (Contributed by metakunt, 25-Apr-2025.)
class PrimRoots
Definition	df-primroots 41563*	A 𝑟-th primitive root is a root of unity such that the exponent divides 𝑟. (Contributed by metakunt, 25-Apr-2025.)
⊢ PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑎 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑎) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑎) = (0g‘𝑟) → 𝑘 ∥ 𝑙))})
Theorem	isprimroot 41564*	The value of a primitive root. (Contributed by metakunt, 25-Apr-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ ↑ = (.g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
Theorem	mndmolinv 41565*	An element of a monoid that has a right inverse has at most one left inverse. (Contributed by metakunt, 25-Apr-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀))    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))
Theorem	linvh 41566*	If an element has a unique left inverse, then the value satisfies the left inverse value equation. (Contributed by metakunt, 25-Apr-2025.)
⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))    &   ⊢ (𝜑 → ∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅))    ⇒   ⊢ (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))
Theorem	primrootsunit1 41567*	Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel))
Theorem	primrootsunit 41568*	Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel))
Theorem	ressmulgnnd 41569	Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋))
Theorem	primrootscoprmpow 41570*	Coprime powers of primitive roots are primitive roots. (Contributed by metakunt, 25-Apr-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ)    &   ⊢ (𝜑 → (𝐸 gcd 𝐾) = 1)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   ⊢ (𝜑 → (𝐸(.g‘𝑅)𝑀) ∈ (𝑅 PrimRoots 𝐾))
Theorem	posbezout 41571*	Bezout's identity restricted on positive integers in all but one variable. (Contributed by metakunt, 26-Apr-2025.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℤ (𝐴 gcd 𝐵) = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
Theorem	primrootscoprf 41572*	Coprime powers of primitive roots are primitive roots, as a function. (Contributed by metakunt, 26-Apr-2025.)
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐸(.g‘𝑅)𝑚))    &   ⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ)    &   ⊢ (𝜑 → (𝐸 gcd 𝐾) = 1)    ⇒   ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)⟶(𝑅 PrimRoots 𝐾))
Theorem	primrootscoprbij 41573*	A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.)
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚))    &   ⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → 𝑍 ∈ ℤ)    &   ⊢ (𝜑 → 1 = ((𝐼 · 𝐽) + (𝐾 · 𝑍)))    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾))
Theorem	primrootscoprbij2 41574*	A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.)
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚))    &   ⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    &   ⊢ (𝜑 → (𝐼 gcd 𝐾) = 1)    ⇒   ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾))
Theorem	remexz 41575*	Division with rest. (Contributed by metakunt, 15-May-2025.)
⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦))
Theorem	primrootlekpowne0 41576	There is no smaller power of a primitive root that sends it to the neutral element. (Contributed by metakunt, 15-May-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   ⊢ (𝜑 → 𝑁 ∈ (1...(𝐾 − 1)))    ⇒   ⊢ (𝜑 → (𝑁(.g‘𝑅)𝑀) ≠ (0g‘𝑅))
Theorem	primrootspoweq0 41577*	The power of a 𝑅-th primitive root is zero if and only if it divides 𝑅. (Contributed by metakunt, 15-May-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((𝑁(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ↔ 𝐾 ∥ 𝑁))
Theorem	aks6d1c1p1 41578*	Definition of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
Theorem	aks6d1c1p1rcl 41579*	Reverse closure of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}    &   ⊢ (𝜑 → 𝐸 ∼ 𝐹)    ⇒   ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
Theorem	aks6d1c1p2 41580*	𝑃 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑃 ∼ 𝐹)
Theorem	aks6d1c1p3 41581*	In a field with a Frobenius isomorphism (read: algebraic closure or finite field), 𝑁 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∼ 𝐹)    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   ⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹)
Theorem	aks6d1c1p4 41582*	The product of polynomials is introspective. (Contributed by metakunt, 25-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐹)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐺)    ⇒   ⊢ (𝜑 → 𝐸 ∼ (𝐹(+g‘𝑊)𝐺))
Theorem	aks6d1c1p5 41583*	The product of exponents is introspective. (Contributed by metakunt, 26-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝐸 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝐷 ∼ 𝐹)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐹)    ⇒   ⊢ (𝜑 → (𝐷 · 𝐸) ∼ 𝐹)
Theorem	aks6d1c1p7 41584*	𝑋 is introspective to all positive integers. (Contributed by metakunt, 30-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐿 ∼ 𝑋)
Theorem	aks6d1c1p6 41585*	If a polynomials 𝐹 is introspective to 𝐸, then so are its powers. (Contributed by metakunt, 30-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐹)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))
Theorem	aks6d1c1p8 41586*	If a number 𝐸 is introspective to 𝐹, then so are its powers. (Contributed by metakunt, 30-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐹)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐸 gcd 𝑅) = 1)    ⇒   ⊢ (𝜑 → (𝐸↑𝐿) ∼ 𝐹)
Theorem	aks6d1c1 41587*	Claim 1 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 30-Apr-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   ⊢ 𝑆 = (Poly1‘𝐾)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑊 = (mulGrp‘𝑆)    &   ⊢ 𝑉 = (mulGrp‘𝐾)    &   ⊢ ↑ = (.g‘𝑉)    &   ⊢ 𝐶 = (algSc‘𝑆)    &   ⊢ 𝐷 = (.g‘𝑊)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ 𝑂 = (eval1‘𝐾)    &   ⊢ + = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ 𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹))
Theorem	evl1gprodd 41588*	Polynomial evaluation builder for a finite group product of polynomials. (Contributed by metakunt, 29-Apr-2025.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑄 = (mulGrp‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑆 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    ⇒   ⊢ (𝜑 → ((𝑂‘(𝑄 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑆 Σg (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))
Theorem	aks6d1c2p1 41589*	In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    ⇒   ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
Theorem	aks6d1c2p2 41590*	Injective condition for countability argument assuming that 𝑁 is not a prime power. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ (𝜑 → 𝑄 ∈ ℙ)    &   ⊢ (𝜑 → 𝑄 ∥ 𝑁)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)–1-1→ℕ)
Theorem	hashscontpowcl 41591	Closure of E for https://www3.nd.edu/%7eandyp/notes/AKS.pdf Theorem 6.1. (Contributed by metakunt, 28-Apr-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑅)    ⇒   ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
Theorem	hashscontpow1 41592	Helper lemma for to prove inequality in Zr. (Contributed by metakunt, 28-Apr-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (1...((odℤ‘𝑅)‘𝑁)))    &   ⊢ (𝜑 → 𝐵 ∈ (1...((odℤ‘𝑅)‘𝑁)))    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑅)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐿‘(𝑁↑𝐴)) ≠ (𝐿‘(𝑁↑𝐵)))
Theorem	hashscontpow 41593*	If a set contains all 𝑁-th powers, then the size of the image under the ZR homomorphism is greater than the 𝑅-th order of 𝑁. (Contributed by metakunt, 28-Apr-2025.)
⊢ (𝜑 → 𝐸 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑁↑𝑘) ∈ 𝐸)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑅)    ⇒   ⊢ (𝜑 → ((odℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))
Theorem	aks6d1c3 41594*	Claim 3 of Theorem 6.1 of the AKS inequality lemma. https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 28-Apr-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑅)    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    ⇒   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
Theorem	aks6d1c4 41595*	Claim 4 of Theorem 6.1 of the AKS inequality lemma. https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 12-May-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    ⇒   ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))
Theorem	aks6d1c1rh 41596*	Claim 1 of AKS primality proof with collapsed definitions since their ease of use is no longer needed. (Contributed by metakunt, 1-May-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐿 ∈ ℕ0)    &   ⊢ 𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹))
Theorem	aks6d1c2lem3 41597*	Lemma for aks6d1c2 41601 to simplify context. (Contributed by metakunt, 1-May-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   ⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   ⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐼 < 𝐽)    &   ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑆 = ((𝐽 ↑ 𝑋)(-g‘(Poly1‘𝐾))(𝐼 ↑ 𝑋))    &   ⊢ (𝜑 → 𝑈 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))    &   ⊢ (𝜑 → 𝑠 ∈ (ℕ0 ↑m (0...𝐴)))    &   ⊢ (𝜑 → 𝑟 ∈ (0...𝐵))    &   ⊢ (𝜑 → 𝑜 ∈ (0...𝐵))    &   ⊢ (𝜑 → 𝐽 = (𝑟𝐸𝑜))    &   ⊢ (𝜑 → 𝑝 ∈ (0...𝐵))    &   ⊢ (𝜑 → 𝑞 ∈ (0...𝐵))    &   ⊢ (𝜑 → 𝐼 = (𝑝𝐸𝑞))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
Theorem	aks6d1c2lem4 41598*	Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof. (Contributed by metakunt, 1-May-2025.)
⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   ⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   ⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐼 < 𝐽)    &   ⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   ⊢ 𝑋 = (var1‘𝐾)    &   ⊢ 𝑆 = ((𝐽 ↑ 𝑋)(-g‘(Poly1‘𝐾))(𝐼 ↑ 𝑋))    &   ⊢ (𝜑 → 𝑈 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))    ⇒   ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
Theorem	hashnexinj 41599*	If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. (Contributed by metakunt, 2-May-2025.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))
Theorem	hashnexinjle 41600*	If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. Also we introduce a one sided inequality to simplify a duplicateable proof. (Contributed by metakunt, 2-May-2025.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (♯‘𝐵) < (♯‘𝐴))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))