P. 429 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	aks6d1c6lem3 42801*	Claim 6 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf TODO, eliminate hypothesis. (Contributed by metakunt, 8-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 → 𝐴 < 𝑃)    &   ⊢ 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))    &   ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   ⊢ 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))    &   ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))    ⇒   ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
Theorem	aks6d1c6lem4 42802*	Claim 6 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf Add hypothesis on coprimality, lift function to the integers so that group operations may be applied. Inline definition. (Contributed by metakunt, 14-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 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 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)))))))    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ 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))    &   ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   ⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))    &   ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))    &   ⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}    ⇒   ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
Theorem	aks6d1c6isolem1 42803*	Lemma to construct the map out of the quotient for AKS. (Contributed by metakunt, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    ⇒   ⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) ∈ Grp)
Theorem	aks6d1c6isolem2 42804*	Lemma to construct the group homomorphism for the AKS Theorem. (Contributed by metakunt, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℤring GrpHom ((𝑅 ↾s 𝑈) ↾s ran 𝐹)))
Theorem	aks6d1c6isolem3 42805*	The preimage of a map sending a primitive root to its powers of zero is equal to the set of integers that divide 𝑅. (Contributed by metakunt, 15-May-2025.)
⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   ⊢ 𝑆 = (RSpan‘ℤring)    ⇒   ⊢ (𝜑 → (𝑆‘{𝐾}) = (◡𝐹 “ {(0g‘(𝑅 ↾s 𝑈))}))
Theorem	aks6d1c6lem5 42806*	Eliminate the size hypothesis. Claim 6. (Contributed by metakunt, 15-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 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 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)))))))    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ 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))    &   ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   ⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))    &   ⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}    &   ⊢ 𝑋 = (𝑏 ∈ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ↦ ∪ (𝐽 “ 𝑏))    ⇒   ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
Theorem	bcled 42807	Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))
Theorem	bcle2d 42808	Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐷 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶)C(𝐴 + 𝐷)) ≤ ((𝐵 + 𝐶)C(𝐵 + 𝐷)))
Theorem	aks6d1c7lem1 42809*	The last set of inequalities of Claim 7 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 12-May-2025.)
⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    ⇒   ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
Theorem	aks6d1c7lem2 42810*	Contradiction to Claim 2 and Claim 7. We assumed in Claim 2 that there are two different prime numbers 𝑃 and 𝑄. (Contributed by metakunt, 16-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 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (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 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 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 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    ⇒   ⊢ (𝜑 → 𝑃 = 𝑄)
Theorem	aks6d1c7lem3 42811*	Remove lots of hypotheses now that we have the AKS contradiction. (Contributed by metakunt, 16-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 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   ⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))    ⇒   ⊢ (𝜑 → 𝑃 = 𝑄)
Theorem	aks6d1c7lem4 42812*	In the AKS algorithm there exists a unique prime number 𝑝 that divides 𝑁. (Contributed by metakunt, 16-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 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    ⇒   ⊢ (𝜑 → ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
Theorem	aks6d1c7 42813*	𝑁 is a prime power if the hypotheses of the AKS algorithm hold. Claim 7 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 16-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 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    ⇒   ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
Theorem	rhmqusspan 42814*	Ring homomorphism out of a quotient given an ideal spanned by a singleton. (Contributed by metakunt, 7-Jun-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    &   ⊢ 𝑁 = ((RSpan‘𝐺)‘{𝑋})    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))    &   ⊢ (𝜑 → (𝐹‘𝑋) = 0 )    ⇒   ⊢ (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))
Theorem	aks5lem1 42815*	Section 5 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf. Construction of a ring homomorphism out of Zn X to K. (Contributed by metakunt, 7-Jun-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   ⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   ⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   ⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly1‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
Theorem	aks5lem2 42816*	Lemma for section 5 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. Construct the quotient for the AKS reduction. (Contributed by metakunt, 7-Jun-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   ⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   ⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   ⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ 𝐼 = (𝑠 ∈ (Base‘𝐴) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))    &   ⊢ 𝐴 = ((Poly1‘(ℤ/nℤ‘𝑁)) /s ((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿))    &   ⊢ 𝐿 = ((RSpan‘(Poly1‘(ℤ/nℤ‘𝑁)))‘{((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁))))})    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (𝐴 RingHom 𝐾) ∧ ∀𝑔 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑔]((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑔)))
Theorem	ply1asclzrhval 42817	Transfer results from algebraic scalars and ZR ring homomorphisms. (Contributed by metakunt, 17-Jun-2025.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐵 = (ℤRHom‘𝑊)    &   ⊢ 𝐶 = (ℤRHom‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴‘(𝐶‘𝑋)) = (𝐵‘𝑋))
Theorem	aks5lem3a 42818*	Lemma for AKS section 5. (Contributed by metakunt, 17-Jun-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   ⊢ 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ ∼ = {⟨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)))}    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   ⊢ 𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   ⊢ 𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ 𝐼 = (𝑠 ∈ (Base‘𝐵) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~QG 𝐿))    ⇒   ⊢ (𝜑 → (𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑀)))
Theorem	aks5lem4a 42819*	Lemma for AKS section 5, reduce hypotheses. (Contributed by metakunt, 17-Jun-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   ⊢ 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ ∼ = {⟨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)))}    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~QG 𝐿))    ⇒   ⊢ (𝜑 → (𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑀)))
Theorem	aks5lem5a 42820*	Lemma for AKS, section 5, connect to Theorem 6.1. (Contributed by metakunt, 17-Jun-2025.)
⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   ⊢ 𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ ∼ = {⟨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)))}    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
Theorem	aks5lem6 42821*	Connect results of section 5 and Theorem 6.1 AKS. (Contributed by metakunt, 25-Jun-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 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    ⇒   ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
Theorem	indstrd 42822*	Strong induction, deduction version. (Contributed by Steven Nguyen, 13-Jul-2025.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜒)) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	grpods 42823*	Relate sums of elements of orders and roots of unity. (Contributed by metakunt, 14-Jul-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝑚 ∈ (1...𝑁) ∣ 𝑚 ∥ 𝑁} (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝐵 ∣ (𝑁 ↑ 𝑥) = (0g‘𝐺)}))
Theorem	unitscyglem1 42824*	Lemma for unitscyg . (Contributed by metakunt, 13-Jul-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝐵 ∣ (((od‘𝐺)‘𝐴) ↑ 𝑥) = (0g‘𝐺)}) = ((od‘𝐺)‘𝐴))
Theorem	unitscyglem2 42825*	Lemma for unitscyg . (Contributed by metakunt, 13-Jul-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∥ (♯‘𝐵))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ((od‘𝐺)‘𝐴) = 𝐷)    &   ⊢ (𝜑 → ∀𝑐 ∈ ℕ (𝑐 < 𝐷 → ((𝑐 ∥ (♯‘𝐵) ∧ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑐} ≠ ∅) → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑐}) = (ϕ‘𝑐))))    ⇒   ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝐷}) = (ϕ‘𝐷))
Theorem	unitscyglem3 42826*	Lemma for unitscyg . (Contributed by metakunt, 14-Jul-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    ⇒   ⊢ (𝜑 → ∀𝑑 ∈ ℕ ((𝑑 ∥ (♯‘𝐵) ∧ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑑} ≠ ∅) → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑑}) = (ϕ‘𝑑)))
Theorem	unitscyglem4 42827*	Lemma for unitscyg . (Contributed by metakunt, 14-Jul-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∥ (♯‘𝐵))    ⇒   ⊢ (𝜑 → (♯‘{𝑦 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑦) = 𝐷}) = (ϕ‘𝐷))
Theorem	unitscyglem5 42828	Lemma for unitscyg . (Contributed by metakunt, 9-Aug-2025.)
⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → (Base‘𝑅) ∈ Fin)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∥ (♯‘(Base‘𝐺)))    ⇒   ⊢ (𝜑 → ((mulGrp‘𝑅) PrimRoots 𝐷) ≠ ∅)
Theorem	aks5lem7 42829*	Lemma for aks5. We clean up the hypotheses compared to aks5lem6 42821. (Contributed by metakunt, 9-Aug-2025.)
⊢ (𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁))    ⇒   ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
Theorem	aks5lem8 42830*	Lemma for aks5. Clean up the conclusion. (Contributed by metakunt, 9-Aug-2025.)
⊢ (𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   ⊢ 𝑃 = (chr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑃 ∥ 𝑁)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁))    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))
Axiom	ax-exfinfld 42831*	Existence axiom for finite fields, eventually we want to construct them. (Contributed by metakunt, 13-Jul-2025.)
⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
Theorem	exfinfldd 42832*	For any prime 𝑃 and any positive integer 𝑁 there exists a field 𝑘 such that 𝑘 contains 𝑃↑𝑁 elements. (Contributed by metakunt, 13-Jul-2025.)
⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))
Theorem	aks5 42833*	The AKS Primality test, given an integer 𝑁 greater than or equal to 3, find a coprime 𝑅 such that 𝑅 is big enough. Then, if a bunch of polynomial equalities in the residue ring hold then 𝑁 is a prime power. Currently depends on the axiom ax-exfinfld 42831, since we currently do not have the existence of finite fields in the database. (Contributed by metakunt, 16-Aug-2025.)
⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)    &   ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1)    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))
21.30  Mathbox for Steven Nguyen
21.30.1  Utility theorems
Theorem	jarrii 42834	Inference associated with jarri 108. A consequence of ax-mp 5 and ax-1 6. (Contributed by SN, 14-Oct-2025.)
⊢ 𝜓    &   ⊢ ((𝜑 → 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	intnanrt 42835	Introduction of conjunct inside of a contradiction. Would be used in elfvov1 7442. (Contributed by SN, 18-May-2025.)
⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
Theorem	ioin9i8 42836	Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    &   ⊢ (𝜒 → ¬ 𝜃)    &   ⊢ (𝜓 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	jaodd 42837	Double deduction form of jaoi 870. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) → 𝜃)))
Theorem	syl3an12 42838	A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
Theorem	exbiii 42839	Inference associated with exbii 1871. Weaker version of eximii 1860. (Contributed by SN, 14-Oct-2025.)
⊢ ∃𝑥𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ∃𝑥𝜓
Theorem	sbtd 42840*	A true statement is true upon substitution (deduction). A similar proof is possible for icht 48056. (Contributed by SN, 4-May-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → [𝑡 / 𝑥]𝜓)
Theorem	sbor2 42841	One direction of sbor 2343, using fewer axioms. Compare 19.33 1907. (Contributed by Steven Nguyen, 18-Aug-2023.)
⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓))
Theorem	sbalexi 42842*	Inference form of sbalex 2280, avoiding ax-10 2178 by using ax-gen 1818. (Contributed by SN, 12-Aug-2025.)
⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)    ⇒   ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)
Theorem	nfalh 42843	Version of nfal 2358 with an 'h' hypothesis, avoiding ax-12 2215. (Contributed by SN, 11-Feb-2026.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥∀𝑦𝜑
Theorem	nfe2 42844	An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∃𝑦∃𝑥𝜑
Theorem	nfale2 42845	An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∀𝑦∃𝑥𝜑
Theorem	19.9dev 42846*	19.9d 2241 in the case of an existential quantifier, avoiding the ax-10 2178 from nfex 2359 that would be used for the hypothesis of 19.9d 2241, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓))
Theorem	3rspcedvd 42847*	Triple application of rspcedvd 3586. (Contributed by Steven Nguyen, 27-Feb-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓)
Theorem	sn-axrep5v 42848*	A condensed form of axrep5 5240. (Contributed by SN, 21-Sep-2023.)
⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	sn-axprlem3 42849*	axprlem3 5387 using only Tarski's FOL axiom schemes and ax-rep 5232. (Contributed by SN, 22-Sep-2023.)
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
Theorem	sn-exelALT 42850*	Alternate proof of exel 5406, avoiding ax-pr 5395 but requiring ax-5 1933, ax-9 2155, and ax-pow 5327. This is similar to how elALT2 5331 uses ax-pow 5327 instead of ax-pr 5395 compared to el 5410. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦
Theorem	ssabdv 42851*	Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓})
Theorem	sn-iotalem 42852*	An unused lemma showing that many equivalences involving df-iota 6481 are potentially provable without ax-10 2178, ax-11 2194, ax-12 2215. (Contributed by SN, 6-Nov-2024.)
⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
Theorem	sn-iotalemcor 42853*	Corollary of sn-iotalem 42852. Compare sb8iota 6492. (Contributed by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})
Theorem	abbi1sn 42854*	Originally part of uniabio 6495. Convert a theorem about df-iota 6481 to one about dfiota2 6482, without ax-10 2178, ax-11 2194, ax-12 2215. Although, eu6 2604 uses ax-10 2178 and ax-12 2215. (Contributed by SN, 23-Nov-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
Theorem	brif2 42855	Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
⊢ (𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))
Theorem	brif12 42856	Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
⊢ (if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))
Theorem	pssexg 42857	The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	pssn0 42858	A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅)
Theorem	psspwb 42859	Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
Theorem	xppss12 42860	Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
Theorem	elpwbi 42861	Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)
Theorem	imaopab 42862*	The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
Theorem	eqresfnbd 42863	Property of being the restriction of a function. Note that this is closer to funssres 6569 than fnssres 6648. (Contributed by SN, 11-Mar-2025.)
⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))
Theorem	fmpocos 42864*	Composition of two functions. Variation of fmpoco 8078 with more context in the substitution hypothesis for 𝑇. (Contributed by SN, 14-Mar-2025.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
Theorem	ovmpogad 42865*	Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7554. (Contributed by SN, 14-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ofun 42866	A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
⊢ (𝜑 → 𝐴 Fn 𝑀)    &   ⊢ (𝜑 → 𝐵 Fn 𝑀)    &   ⊢ (𝜑 → 𝐶 Fn 𝑁)    &   ⊢ (𝜑 → 𝐷 Fn 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑀 ∩ 𝑁) = ∅)    ⇒   ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)))
Theorem	dfqs3 42867*	Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}
Theorem	qseq12d 42868	Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
Theorem	qsalrel 42869*	The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)    &   ⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})
Theorem	supinf 42870*	The supremum is the infimum of the upper bounds. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → < Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))
Theorem	mapcod 42871	Compose two mappings. (Contributed by SN, 11-Mar-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶))
Theorem	fisdomnn 42872	A finite set is dominated by the set of natural numbers. (Contributed by SN, 6-Jul-2025.)
⊢ (𝐴 ∈ Fin → 𝐴 ≺ ℕ)
Theorem	ltex 42873	The less-than relation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ < ∈ V
Theorem	leex 42874	The less-than-or-equal-to relation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ ≤ ∈ V
Theorem	subex 42875	The subtraction operation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ − ∈ V
Theorem	absex 42876	The absolute value function is a set. (Contributed by SN, 5-Jun-2025.)
⊢ abs ∈ V
Theorem	cjex 42877	The conjugate function is a set. (Contributed by SN, 5-Jun-2025.)
⊢ ∗ ∈ V
Theorem	fzosumm1 42878*	Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))
Theorem	ccatcan2d 42879	Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ Word 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ Word 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))
21.30.2  Arithmetic theorems Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results. For example, ax-1rid 11158 is used in mulrid 11194 related theorems, so one could trade off the extra axioms in mulrid 11194 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 11145; in the other direction, real number closure laws can be avoided by using ax-resscn 11145 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number). The natural numbers are especially amenable to axiom reductions, as the set ℕ is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This conveniently allows for adding natural numbers by rearranging parentheses, as shown below: (4 + 3) = 7 ((3 + 1) + (2 + 1)) = (6 + 1) ((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) = ((((((1 + 1) + 1) + 1) + 1) + 1) + 1) This only requires ax-addass 11153, ax-1cn 11146, and ax-addcl 11148. (And in practice, the expression isn't fully expanded into ones.) Multiplication by 1 requires either mullidi 11202 or (ax-1rid 11158 and 1re 11196) as seen in 1t1e1 12393 and 1t1e1ALT 12282. Multiplying with greater natural numbers uses ax-distr 11155. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = ;10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 11152 (see readdrid 43031 and readdlid 43024).
Theorem	c0exALT 42880	Alternate proof of c0ex 11188 using more set theory axioms but fewer complex number axioms (add ax-10 2178, ax-11 2194, ax-13 2406, ax-nul 5261, and remove ax-1cn 11146, ax-icn 11147, ax-addcl 11148, and ax-mulcl 11150). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ V
Theorem	0cnALT3 42881	Alternate proof of 0cn 11186 using ax-resscn 11145, ax-addrcl 11149, ax-rnegex 11159, ax-cnre 11161 instead of ax-icn 11147, ax-addcl 11148, ax-mulcl 11150, ax-i2m1 11156. Version of 0cnALT 11433 using ax-1cn 11146 instead of ax-icn 11147. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	elre0re 42882	Specialized version of 0red 11199 without using ax-1cn 11146 and ax-cnre 11161. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)
Theorem	lttrii 42883	'Less than' is transitive. (Contributed by SN, 26-Aug-2025.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ 𝐴 < 𝐶
Theorem	remulcan2d 42884	mulcan2d 11836 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
Theorem	readdridaddlidd 42885	Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 11372, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)
Theorem	1p3e4 42886	1 + 3 = 4. (Contributed by SN, 19-Nov-2025.)
⊢ (1 + 3) = 4
Theorem	5ne0 42887	The number 5 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 5 ≠ 0
Theorem	6ne0 42888	The number 6 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 6 ≠ 0
Theorem	7ne0 42889	The number 7 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 7 ≠ 0
Theorem	8ne0 42890	The number 8 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 8 ≠ 0
Theorem	9ne0 42891	The number 9 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 9 ≠ 0
Theorem	sn-1ne2 42892	A proof of 1ne2 12442 without using ax-mulcom 11152, ax-mulass 11154, ax-pre-mulgt0 11165. Based on mul02lem2 11375. (Contributed by SN, 13-Dec-2023.)
⊢ 1 ≠ 2
Theorem	nnn1suc 42893*	A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)
Theorem	readdrcl2d 42894	Reverse closure for addition: the second addend is real if the first addend is real and the sum is real. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	mvrrsubd 42895	Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11612. EDITORIAL: Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) = 𝐵)
Theorem	laddrotrd 42896	Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11613 with a commuted consequent, and of mvrladdd 11615 with a commuted hypothesis. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: ply1dg3rt0irred 33791. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐴) = 𝐵)
Theorem	raddswap12d 42897	Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11614 with a commuted consequent, and of mvlraddd 11612 with a commuted hypothesis. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐴 − 𝐶))
Theorem	lsubrotld 42898	Rotate the variables left in an equation with subtraction on the left, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐶) = 𝐴)
Theorem	rsubrotld 42899	Rotate the variables left in an equation with subtraction on the right, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 4-Jul-2025.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐶 + 𝐴))
Theorem	lsubswap23d 42900	Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 23-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) = 𝐵)