aks6d1c2lem4 - Mathbox for metakunt

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Theorem aks6d1c2lem4 43177

Description: Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof. (Contributed by metakunt, 1-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c2.1	⊢ ∼ = {⟨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)))}
aks6d1c2.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c2.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c2.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c2.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c2.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c2.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c2.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c2.9	⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ₀)
aks6d1c2.10	⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c2.11	⊢ (𝜑 → 𝐴 ∈ ℕ₀)
aks6d1c2.12	⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c2.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c2.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c2.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c2.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c2.17	⊢ 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c2.18	⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
aks6d1c2.19	⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
aks6d1c2.20	⊢ (𝜑 → 𝐼 ∈ 𝐶)
aks6d1c2.21	⊢ (𝜑 → 𝐽 ∈ 𝐶)
aks6d1c2.22	⊢ (𝜑 → 𝐼 < 𝐽)
aks6d1c2.23	⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
aks6d1c2.24	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c2.25	⊢ 𝑆 = ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))
aks6d1c2.26	⊢ (𝜑 → 𝑈 ∈ ℕ)
aks6d1c2.27	⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))

**Assertion**
Ref	Expression
aks6d1c2lem4	⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))

Distinct variable groups: ∼ ,𝑎 𝐴,𝑎 𝐴,𝑔,𝑖 𝐴,ℎ 𝐴,𝑘,𝑙 𝑥,𝐴 𝐵,𝑎 𝐵,𝑔,𝑖 𝐵,𝑘,𝑙 𝑥,𝐵 𝐸,𝑎 𝑔,𝐸,𝑖 𝑘,𝐸,𝑙 𝑥,𝐸 𝑒,𝐺,𝑓,𝑦 ℎ,𝐺 𝐼,𝑎 𝑔,𝐼,𝑖 𝑘,𝐼,𝑙 𝑥,𝐼 𝐽,𝑎 𝑔,𝐽,𝑖 𝑘,𝐽,𝑙 𝑥,𝐽 𝐾,𝑎 𝑒,𝐾,𝑓,𝑦 𝑔,𝐾,𝑖 ℎ,𝐾 𝑥,𝐾 ℎ,𝑀 𝑦,𝑀 𝑁,𝑎 𝑒,𝑁,𝑓,𝑦 𝑘,𝑁,𝑙 𝑥,𝑁 𝑃,𝑒,𝑓,𝑦 𝑃,𝑘,𝑙 𝑥,𝑃 𝑅,𝑒,𝑓,𝑦 𝑥,𝑅 𝜑,𝑎 𝜑,𝑔,𝑖 𝜑,ℎ 𝜑,𝑘,𝑙 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑦, 𝑒, 𝑓) 𝐴(𝑦, 𝑒, 𝑓) 𝐵(𝑦, 𝑒, 𝑓, ℎ) 𝐶(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝑃(𝑔, ℎ, 𝑖, 𝑎) ∼ (𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑙) 𝑅(𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝑆(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝑈(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝐸(𝑦, 𝑒, 𝑓, ℎ) ↑ (𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝐹(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝐺(𝑥, 𝑔, 𝑖, 𝑘, 𝑎, 𝑙) 𝐻(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝐼(𝑦, 𝑒, 𝑓, ℎ) 𝐽(𝑦, 𝑒, 𝑓, ℎ) 𝐾(𝑘, 𝑙) 𝐿(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙) 𝑀(𝑥, 𝑒, 𝑓, 𝑔, 𝑖, 𝑘, 𝑎, 𝑙) 𝑁(𝑔, ℎ, 𝑖) 𝑋(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙)

Proof of Theorem aks6d1c2lem4 Dummy variables 𝑜 𝑝 𝑞 𝑟 𝑠 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6900	. . . . . . . 8 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ V)
2		cnvexg 7936	. . . . . . . 8 ⊢ (((eval₁‘𝐾)‘𝑆) ∈ V → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
4	3	imaexd 7928	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V)
5		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑠𝜑
6		fvexd 6900	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀) ∈ V)
7		aks6d1c2.17	. . . . . . . . . 10 ⊢ 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀))
8	6, 7	fmptd 7114	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(ℕ₀ ↑_m (0...𝐴))⟶V)
9	8	ffnd 6710	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (ℕ₀ ↑_m (0...𝐴)))
10	9	fnfund 6640	. . . . . . 7 ⊢ (𝜑 → Fun 𝐻)
11		aks6d1c2.25	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑆 = ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))
13	12	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) = ((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
14	13	fveq1d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))‘(𝐻‘𝑠)))
15		eqid 2761	. . . . . . . . . . . . 13 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
16		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
17		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
19		aks6d1c2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Field)
20	19	fldcrngd 20995	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
21	20	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ CRing)
22	7	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
23		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ℎ = 𝑠)
24	23	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (𝐺‘ℎ) = (𝐺‘𝑠))
25	24	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ((eval₁‘𝐾)‘(𝐺‘ℎ)) = ((eval₁‘𝐾)‘(𝐺‘𝑠)))
26	25	fveq1d 6887	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀))
27		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)))
28		fvexd 6900	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ V)
29	22, 26, 27, 28	fvmptd 7001	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) = (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀))
30		aks6d1c2.16	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
31		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
32	31	crngmgp 20467	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
33	20, 32	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
34		aks6d1c2.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑅 ∈ ℕ)
35	34	nnnn0d 12667	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
36		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
37	33, 35, 36	isprimroot 43143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ₀ ((𝑣(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣))))
38	37	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ₀ ((𝑣(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣))))
39	30, 38	mpd 16	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ₀ ((𝑣(._g‘(mulGrp‘𝐾))𝑀) = (0_g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣)))
40	39	simp1d 1160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
41	31, 17	mgpbas 20365	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
42	40, 41	eleqtrrdi 2872	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
43	42	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑀 ∈ (Base‘𝐾))
44		aks6d1c2.1	. . . . . . . . . . . . . . . . 17 ⊢ ∼ = {⟨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)))}
45		aks6d1c2.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑃 = (chr‘𝐾)
46	19	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Field)
47		aks6d1c2.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℙ)
48	47	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∈ ℙ)
49	34	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑅 ∈ ℕ)
50		aks6d1c2.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
51	50	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑁 ∈ ℕ)
52		aks6d1c2.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
53	52	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∥ 𝑁)
54		aks6d1c2.8	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
55	54	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑁 gcd 𝑅) = 1)
56		elmapi 8869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) → 𝑠:(0...𝐴)⟶ℕ₀)
57	56	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑠:(0...𝐴)⟶ℕ₀)
58		aks6d1c2.10	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
59		aks6d1c2.11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
60	59	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐴 ∈ ℕ₀)
61		0nn0 12621	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ₀
62	61	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 0 ∈ ℕ₀)
63		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = ((𝑃↑0) · ((𝑁 / 𝑃)↑0))
64		aks6d1c2.14	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
65	64	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
66		aks6d1c2.15	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
67	66	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
68	44, 45, 46, 48, 49, 51, 53, 55, 57, 58, 60, 62, 62, 63, 65, 67	aks6d1c1rh 43175	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∼ (𝐺‘𝑠))
69	44, 68	aks6d1c1p1rcl 43158	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∈ ℕ ∧ (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾))))
70	69	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾)))
71	15, 16, 17, 18, 21, 43, 70	fveval1fvcl 22651	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ (Base‘𝐾))
72	29, 71	eqeltrd 2861	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (Base‘𝐾))
73		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
74		aks6d1c2.23	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
75	16	ply1crng 22516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
76	20, 75	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
77		eqid 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
78	77	crngmgp 20467	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Poly₁‘𝐾) ∈ CRing → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
79	76, 78	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
80	79	cmnmndd 20018	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
81		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 = (𝑟𝐸𝑜))
82		aks6d1c2.12	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
83	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))))
84		simprl 783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑘 = 𝑟)
85	84	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → (𝑃↑𝑘) = (𝑃↑𝑟))
86		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑙 = 𝑜)
87	86	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜))
88	85, 87	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
89		fz0ssnn0 13756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐵) ⊆ ℕ₀
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ₀)
91	90	sselda 3931	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
92	91	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
93	89	sseli 3927	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ₀)
94	93	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑜 ∈ ℕ₀)
95		ovexd 7455	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V)
96	83, 88, 92, 94, 95	ovmpod 7572	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
97		prmnn 16849	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
98	47, 97	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
99	98	nnnn0d 12667	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
100	99	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
101	100	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
102	101, 92	nn0expcld 14390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℕ₀)
103	99	nn0zd 12718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℤ)
104	98	nnne0d 12388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
105	50	nnnn0d 12667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
106	105	nn0zd 12718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
107		dvdsval2 16425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
108	103, 104, 106, 107	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
109	52, 108	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
110	50	nnred 12350	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℝ)
111	98	nnrpd 13162	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
112	105	nn0ge0d 12670	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 0 ≤ 𝑁)
113	110, 111, 112	divge0d 13204	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (𝑁 / 𝑃))
114	109, 113	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
115		elnn0z 12706	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 / 𝑃) ∈ ℕ₀ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
116	114, 115	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ₀)
117	116	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
118	117	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
119	118, 94	nn0expcld 14390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℕ₀)
120	102, 119	nn0mulcld 12672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ₀)
121	96, 120	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ∈ ℕ₀)
122	121	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ∈ ℕ₀)
123	81, 122	eqeltrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 ∈ ℕ₀)
124		aks6d1c2.21	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ 𝐶)
125		aks6d1c2.19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
126	125	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵))))
127	124, 126	eleqtrd 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
128	50, 47, 52, 82	aks6d1c2p1 43168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
129	128	ffnd 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
130	90, 90	jca 521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((0...𝐵) ⊆ ℕ₀ ∧ (0...𝐵) ⊆ ℕ₀))
131		aks6d1c2.18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
132	131	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
133		aks6d1c2.13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
134		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
135	50, 47, 52, 34, 54, 82, 133, 134	hashscontpowcl 43170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
136	135	nn0red 12668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
137	135	nn0ge0d 12670	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
138	136, 137	resqrtcld 15585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
139	138	flcld 13938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
140	136, 137	sqrtge0d 15588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
141		0zd 12705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ∈ ℤ)
142		flge 13945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
143	138, 141, 142	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
144	140, 143	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
145	139, 144	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
146		elnn0z 12706	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
147	145, 146	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
148	132, 147	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
149		elnn0z 12706	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
150	148, 149	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
151		0z 12704	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℤ
152		eluz1 12969	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0 ∈ ℤ → (𝐵 ∈ (ℤ_≥‘0) ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵)))
153	151, 152	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ∈ (ℤ_≥‘0) ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
154	150, 153	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘0))
155		fzn0 13671	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0...𝐵) ≠ ∅ ↔ 𝐵 ∈ (ℤ_≥‘0))
156	154, 155	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ≠ ∅)
157	156, 156	jca 521	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅))
158		xpnz 6150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
159	157, 158	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
160		ssxpb 6166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((0...𝐵) × (0...𝐵)) ≠ ∅ → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀) ↔ ((0...𝐵) ⊆ ℕ₀ ∧ (0...𝐵) ⊆ ℕ₀)))
161	159, 160	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀) ↔ ((0...𝐵) ⊆ ℕ₀ ∧ (0...𝐵) ⊆ ℕ₀)))
162	130, 161	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀))
163		ovelimab 7599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 Fn (ℕ₀ × ℕ₀) ∧ ((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀)) → (𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜)))
164	129, 162, 163	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜)))
165	127, 164	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
166	123, 165	r19.29vva 3223	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
167	20	crngringd 20473	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Ring)
168		aks6d1c2.24	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑋 = (var₁‘𝐾)
169	168, 16, 18	vr1cl 22535	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
170	167, 169	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
171	77, 18	mgpbas 20365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
172	171	eqcomi 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾))
173	172	eleq2i 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))) ↔ 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
174	170, 173	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
175	73, 74, 80, 166, 174	mulgnn0cld 19305	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
176	175, 171	eleqtrrdi 2872	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
177	176	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
178	170	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
179	15, 168, 17, 16, 18, 21, 72	evl1vard 22655	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
180	179	simprd 501	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠))
181	178, 180	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
182	166	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐽 ∈ ℕ₀)
183	15, 16, 17, 18, 21, 72, 181, 74, 36, 182	evl1expd 22663	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠))))
184	183	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
185	29	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (𝐽(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
186	19	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐾 ∈ Field)
187	47	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑃 ∈ ℙ)
188	34	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑅 ∈ ℕ)
189	50	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑁 ∈ ℕ)
190	52	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑃 ∥ 𝑁)
191	54	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑁 gcd 𝑅) = 1)
192		aks6d1c2.9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ₀)
193	192	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐹:(0...𝐴)⟶ℕ₀)
194	59	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐴 ∈ ℕ₀)
195	64	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
196	66	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
197	30	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
198		aks6d1c2.20	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ∈ 𝐶)
199	198	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ 𝐶)
200	124	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐽 ∈ 𝐶)
201		aks6d1c2.22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 < 𝐽)
202	201	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 < 𝐽)
203		aks6d1c2.26	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ ℕ)
204	203	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑈 ∈ ℕ)
205		aks6d1c2.27	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))
206	205	ad7antr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐽 = (𝐼 + (𝑈 · 𝑅)))
207	27	ad6antr 749	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)))
208		simp-6r 800	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑟 ∈ (0...𝐵))
209		simp-5r 798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑜 ∈ (0...𝐵))
210		simp-4r 796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐽 = (𝑟𝐸𝑜))
211		simpllr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑝 ∈ (0...𝐵))
212		simplr 781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑞 ∈ (0...𝐵))
213		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 = (𝑝𝐸𝑞))
214		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 = (𝑝𝐸𝑞))
215	82	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))))
216		simprl 783	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑘 = 𝑝)
217	216	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → (𝑃↑𝑘) = (𝑃↑𝑝))
218		simprr 785	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑙 = 𝑞)
219	218	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞))
220	217, 219	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
221	90	sselda 3931	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑝 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
222	221	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
223	222	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑝 ∈ ℕ₀)
224	89	sseli 3927	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ (0...𝐵) → 𝑞 ∈ ℕ₀)
225	224	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑞 ∈ ℕ₀)
226	225	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑞 ∈ ℕ₀)
227		ovexd 7455	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V)
228	215, 220, 223, 226, 227	ovmpod 7572	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
229	214, 228	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
230	99	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑃 ∈ ℕ₀)
231	230, 223	nn0expcld 14390	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑃↑𝑝) ∈ ℕ₀)
232	116	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
233	232	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑁 / 𝑃) ∈ ℕ₀)
234	233, 226	nn0expcld 14390	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑁 / 𝑃)↑𝑞) ∈ ℕ₀)
235	231, 234	nn0mulcld 12672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ₀)
236	229, 235	eqeltrd 2861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ ℕ₀)
237	198, 126	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
238		ovelimab 7599	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐸 Fn (ℕ₀ × ℕ₀) ∧ ((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀)) → (𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞)))
239	129, 162, 238	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞)))
240	237, 239	mpbid 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞))
241	236, 240	r19.29vva 3223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
242	241	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐼 ∈ ℕ₀)
243	242	ad6antr 749	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ ℕ₀)
244	44, 45, 186, 187, 188, 189, 190, 191, 193, 58, 194, 82, 133, 195, 196, 197, 7, 131, 125, 199, 200, 202, 74, 168, 11, 204, 206, 207, 208, 209, 210, 211, 212, 213, 243	aks6d1c2lem3 43176	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝐽(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
245	240	ad4antr 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞))
246	244, 245	r19.29vva 3223	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝐽(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
247	165	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
248	246, 247	r19.29vva 3223	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
249	29	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) = (𝐻‘𝑠))
250	249	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
251	185, 248, 250	3eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
252	15, 16, 17, 18, 21, 72, 181, 74, 36, 242	evl1expd 22663	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠))))
253	252	simprd 501	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
254	253	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
255	184, 251, 254	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
256	177, 255	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))))
257	73, 74, 80, 241, 174	mulgnn0cld 19305	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
258	171	eleq2i 2853	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ↔ (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
259	257, 258	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
260	259	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
261		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
262	260, 261	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))))
263		eqid 2761	. . . . . . . . . . . . 13 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
264		eqid 2761	. . . . . . . . . . . . 13 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
265	15, 16, 17, 18, 21, 72, 256, 262, 263, 264	evl1subd 22660	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))‘(𝐻‘𝑠)) = ((((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))(-_g‘𝐾)(((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))))
266	265	simprd 501	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))‘(𝐻‘𝑠)) = ((((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))(-_g‘𝐾)(((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))))
267	21	crnggrpd 20474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Grp)
268	15, 16, 17, 18, 21, 72, 260	fveval1fvcl 22651	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) ∈ (Base‘𝐾))
269		eqid 2761	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
270	17, 269, 264	grpsubid 19234	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Grp ∧ (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) ∈ (Base‘𝐾)) → ((((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))(-_g‘𝐾)(((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))) = (0_g‘𝐾))
271	267, 268, 270	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))(-_g‘𝐾)(((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠))) = (0_g‘𝐾))
272	266, 271	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))‘(𝐻‘𝑠)) = (0_g‘𝐾))
273	14, 272	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾))
274		fvexd 6900	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ V)
275		elsng 4598	. . . . . . . . . 10 ⊢ ((((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ V → ((((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)} ↔ (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾)))
276	274, 275	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)} ↔ (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾)))
277	273, 276	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)})
278	76	crnggrpd 20474	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Grp)
279	18, 263	grpsubcl 19230	. . . . . . . . . . . . . . . . 17 ⊢ (((Poly₁‘𝐾) ∈ Grp ∧ (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾))) → ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)) ∈ (Base‘(Poly₁‘𝐾)))
280	278, 176, 259, 279	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)) ∈ (Base‘(Poly₁‘𝐾)))
281	11, 280	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (Base‘(Poly₁‘𝐾)))
282		eqid 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↑_s (Base‘𝐾)) = (𝐾 ↑_s (Base‘𝐾))
283	15, 16, 282, 17	evl1rhm 22650	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ CRing → (eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))))
284	20, 283	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))))
285		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝐾 ↑_s (Base‘𝐾))) = (Base‘(𝐾 ↑_s (Base‘𝐾)))
286	18, 285	rhmf 20715	. . . . . . . . . . . . . . . . . 18 ⊢ ((eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))) → (eval₁‘𝐾):(Base‘(Poly₁‘𝐾))⟶(Base‘(𝐾 ↑_s (Base‘𝐾))))
287	284, 286	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (eval₁‘𝐾):(Base‘(Poly₁‘𝐾))⟶(Base‘(𝐾 ↑_s (Base‘𝐾))))
288	287	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾))))
289	288	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∈ (Base‘(Poly₁‘𝐾)) → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾)))))
290	281, 289	mpd 16	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾))))
291		fvexd 6900	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
292	282, 17	pwsbas 17658	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
293	19, 291, 292	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
294	290, 293	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)))
295	291, 291	elmapd 8860	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)) ↔ ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾)))
296	294, 295	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾))
297		ffn 6709	. . . . . . . . . . . 12 ⊢ (((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾) → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
298	296, 297	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
299	298	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
300		fnfun 6639	. . . . . . . . . 10 ⊢ (((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾) → Fun ((eval₁‘𝐾)‘𝑆))
301	299, 300	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → Fun ((eval₁‘𝐾)‘𝑆))
302		fndm 6642	. . . . . . . . . . . 12 ⊢ (((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾) → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
303	298, 302	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
304	303	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
305	72, 304	eleqtrrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ dom ((eval₁‘𝐾)‘𝑆))
306		fvimacnv 7052	. . . . . . . . 9 ⊢ ((Fun ((eval₁‘𝐾)‘𝑆) ∧ (𝐻‘𝑠) ∈ dom ((eval₁‘𝐾)‘𝑆)) → ((((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)} ↔ (𝐻‘𝑠) ∈ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
307	301, 305, 306	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)} ↔ (𝐻‘𝑠) ∈ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
308	277, 307	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
309	5, 10, 308	funimassd 6951	. . . . . 6 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ⊆ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
310	4, 309	ssexd 5286	. . . . 5 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
311	11	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)))
312	311	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
313		eqid 2761	. . . . . . . . . . 11 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
314		isfld 20993	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
315	314	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Field → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
316	315	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
317		drngnzr 21002	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
318	316, 317	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Field → 𝐾 ∈ NzRing)
319	19, 318	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
320	313, 16, 168, 77, 74	deg1pw 26439	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐼 ∈ ℕ₀) → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
321	319, 241, 320	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
322	321	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)))
323	313, 16, 168, 77, 74	deg1pw 26439	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐽 ∈ ℕ₀) → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
324	319, 166, 323	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
325	324	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
326	201, 322, 325	3brtr3d 5136	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) < ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
327	16, 313, 167, 18, 263, 176, 259, 326	deg1sub 26426	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
328	312, 327	eqtrd 2796	. . . . . . . . 9 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
329	328, 324	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = 𝐽)
330	329, 166	eqeltrd 2861	. . . . . . 7 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀)
331		eqid 2761	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
332		fldidom 21029	. . . . . . . . 9 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
333	19, 332	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ IDomn)
334	313, 16, 331, 18	deg1nn0clb 26408	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀))
335	167, 281, 334	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀))
336	330, 335	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≠ (0_g‘(Poly₁‘𝐾)))
337	16, 18, 313, 15, 269, 331, 333, 281, 336	fta1g 26488	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ ((deg₁‘𝐾)‘𝑆))
338		hashbnd 14480	. . . . . . 7 ⊢ (((^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V ∧ ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀ ∧ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ ((deg₁‘𝐾)‘𝑆)) → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin)
339	4, 330, 337, 338	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin)
340		hashcl 14500	. . . . . 6 ⊢ ((^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℕ₀)
341	339, 340	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℕ₀)
342		hashss 14553	. . . . . 6 ⊢ (((^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V ∧ (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ⊆ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
343	4, 309, 342	syl2anc 596	. . . . 5 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
344		hashbnd 14480	. . . . 5 ⊢ (((𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V ∧ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℕ₀ ∧ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin)
345	310, 341, 343, 344	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin)
346		hashcl 14500	. . . 4 ⊢ ((𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℕ₀)
347	345, 346	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℕ₀)
348	347	nn0red 12668	. 2 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ)
349	341	nn0red 12668	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℝ)
350	105, 148	nn0expcld 14390	. . 3 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℕ₀)
351	350	nn0red 12668	. 2 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
352	166	nn0red 12668	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
353	324, 352	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ∈ ℝ)
354	327, 353	eqeltrd 2861	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ∈ ℝ)
355	312, 354	eqeltrd 2861	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℝ)
356	97	nnred 12350	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
357	47, 356	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℝ)
358	357	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
359	358	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
360	359, 92	reexpcld 14306	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℝ)
361	110, 357, 104	redivcld 12145	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℝ)
362	361	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
363	362	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
364	363, 94	reexpcld 14306	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℝ)
365	360, 364	remulcld 11339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℝ)
366	357, 148	reexpcld 14306	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃↑𝐵) ∈ ℝ)
367	366	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
368	361, 148	reexpcld 14306	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
369	368	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
370	367, 369	remulcld 11339	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
371	370	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
372	110, 148	reexpcld 14306	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
373	372	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
374	373	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
375	367	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
376	369	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
377		0red 11311	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
378		1red 11309	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
379		0le1 11839	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 1
380	379	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 1)
381		prmgt1 16873	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
382	47, 381	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 < 𝑃)
383	378, 357, 382	ltled 11458	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑃)
384	377, 378, 357, 380, 383	letrd 11467	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝑃)
385	384	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ 𝑃)
386	385	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ 𝑃)
387	359, 92, 386	expge0d 14307	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑃↑𝑟))
388	113	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
389	388	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
390	363, 94, 389	expge0d 14307	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ ((𝑁 / 𝑃)↑𝑜))
391	98	nnge1d 12386	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑃)
392	391	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ 𝑃)
393	392	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ 𝑃)
394		elfzuz3 13653	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑟))
395	394	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
396	395	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
397	359, 393, 396	leexp2ad 14398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ≤ (𝑃↑𝐵))
398	357	recnd 11337	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
399	398	mullidd 11327	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 · 𝑃) = 𝑃)
400	98	nnzd 12719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℤ)
401		dvdsle 16480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
402	400, 50, 401	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
403	52, 402	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ≤ 𝑁)
404	399, 403	eqbrtrd 5127	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 · 𝑃) ≤ 𝑁)
405	378, 110, 111	lemuldivd 13213	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1 · 𝑃) ≤ 𝑁 ↔ 1 ≤ (𝑁 / 𝑃)))
406	404, 405	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (𝑁 / 𝑃))
407	406	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
408	407	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
409		elfzuz3 13653	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑜))
410	409	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑜))
411	363, 408, 410	leexp2ad 14398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ≤ ((𝑁 / 𝑃)↑𝐵))
412	360, 375, 364, 376, 387, 390, 397, 411	lemul12ad 12259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
413	110	recnd 11337	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
414	413, 398, 104	divcan2d 12095	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
415	414	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
416	415	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
417	416	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
418	417	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) = ((𝑃 · (𝑁 / 𝑃))↑𝐵))
419	359	recnd 11337	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℂ)
420	363	recnd 11337	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℂ)
421	148	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ ℕ₀)
422	419, 420, 421	mulexpd 14304	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃 · (𝑁 / 𝑃))↑𝐵) = ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
423	418, 422	eqtr2d 2797	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) = (𝑁↑𝐵))
424	374	leidd 11882	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ≤ (𝑁↑𝐵))
425	423, 424	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ≤ (𝑁↑𝐵))
426	365, 371, 374, 412, 425	letrd 11467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ (𝑁↑𝐵))
427	96, 426	eqbrtrd 5127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
428	427	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
429	81, 428	eqbrtrd 5127	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 ≤ (𝑁↑𝐵))
430	429, 165	r19.29vva 3223	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ (𝑁↑𝐵))
431	324, 430	eqbrtrd 5127	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ≤ (𝑁↑𝐵))
432	327, 431	eqbrtrd 5127	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ≤ (𝑁↑𝐵))
433	312, 432	eqbrtrd 5127	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ≤ (𝑁↑𝐵))
434	349, 355, 351, 337, 433	letrd 11467	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ (𝑁↑𝐵))
435	348, 349, 351, 343, 434	letrd 11467	1 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 Vcvv 3451 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 {copab 5167 ↦ cmpt 5186 × cxp 5649 ^◡ccnv 5650 dom cdm 5651 “ cima 5654 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8847 Fincfn 8973 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 / cdiv 11973 ℕcn 12335 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 ⌊cfl 13930 ↑cexp 14204 ♯chash 14474 √csqrt 15400 ∥ cdvds 16422 gcd cgcd 16664 ℙcprime 16846 Basecbs 17387 +_gcplusg 17428 0_gc0g 17610 Σ_g cgsu 17611 ↑_s cpws 17617 Grpcgrp 19144 -_gcsg 19146 ._gcmg 19277 CMndccmn 19994 mulGrpcmgp 20360 Ringcrg 20459 CRingccrg 20460 RingHom crh 20699 RingIso crs 20700 NzRingcnzr 20762 IDomncidom 20945 DivRingcdr 20980 Fieldcfield 20981 ℤRHomczrh 21805 chrcchr 21807 ℤ/nℤczn 21808 algSccascl 22160 var₁cv1 22494 Poly₁cpl1 22495 eval₁ce1 22632 deg₁cdg1 26372 PrimRoots cprimroots 43141

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-dvds 16423 df-gcd 16665 df-prm 16847 df-phi 16943 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-imas 17680 df-qus 17681 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-nsg 19334 df-eqg 19335 df-ghm 19428 df-cntz 19531 df-od 19742 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-srg 20413 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-rim 20703 df-nzr 20763 df-subrng 20798 df-subrg 20822 df-rlreg 20946 df-domn 20947 df-idom 20948 df-drng 20982 df-field 20983 df-lmod 21137 df-lss 21207 df-lsp 21247 df-sra 21448 df-rgmod 21449 df-lidl 21486 df-rsp 21487 df-2idl 21543 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-chr 21811 df-zn 21812 df-assa 22161 df-asp 22162 df-ascl 22163 df-psr 22217 df-mvr 22218 df-mpl 22219 df-opsr 22221 df-evls 22383 df-evl 22384 df-psr1 22498 df-vr1 22499 df-ply1 22500 df-coe1 22501 df-evl1 22634 df-mdeg 26373 df-deg1 26374 df-mon1 26449 df-uc1p 26450 df-q1p 26451 df-r1p 26452 df-primroots 43142

This theorem is used by: aks6d1c2 43180

W3C validator