aks6d1c2lem4 - Mathbox for metakunt

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Theorem aks6d1c2lem4 42381

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 6849	. . . . . . . 8 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ V)
2		cnvexg 7866	. . . . . . . 8 ⊢ (((eval₁‘𝐾)‘𝑆) ∈ V → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
4	3	imaexd 7858	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V)
5		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑠𝜑
6		fvexd 6849	. . . . . . . . . 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 7059	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(ℕ₀ ↑_m (0...𝐴))⟶V)
9	8	ffnd 6663	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (ℕ₀ ↑_m (0...𝐴)))
10	9	fnfund 6593	. . . . . . 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 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) = ((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
14	13	fveq1d 6836	. . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
16		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
17		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
19		aks6d1c2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Field)
20	19	fldcrngd 20675	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
21	20	adantr 480	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ℎ = 𝑠)
24	23	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (𝐺‘ℎ) = (𝐺‘𝑠))
25	24	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ((eval₁‘𝐾)‘(𝐺‘ℎ)) = ((eval₁‘𝐾)‘(𝐺‘𝑠)))
26	25	fveq1d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀))
27		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)))
28		fvexd 6849	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ V)
29	22, 26, 27, 28	fvmptd 6948	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
32	31	crngmgp 20176	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
33	20, 32	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
34		aks6d1c2.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑅 ∈ ℕ)
35	34	nnnn0d 12462	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
36		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
37	33, 35, 36	isprimroot 42347	. . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
41	31, 17	mgpbas 20080	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
42	40, 41	eleqtrrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
43	42	adantr 480	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Field)
47		aks6d1c2.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℙ)
48	47	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∈ ℙ)
49	34	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑅 ∈ ℕ)
50		aks6d1c2.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
51	50	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑁 ∈ ℕ)
52		aks6d1c2.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
53	52	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∥ 𝑁)
54		aks6d1c2.8	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
55	54	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑁 gcd 𝑅) = 1)
56		elmapi 8786	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) → 𝑠:(0...𝐴)⟶ℕ₀)
57	56	adantl 481	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐴 ∈ ℕ₀)
61		0nn0 12416	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ₀
62	61	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 0 ∈ ℕ₀)
63		eqid 2736	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 42379	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∼ (𝐺‘𝑠))
69	44, 68	aks6d1c1p1rcl 42362	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∈ ℕ ∧ (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾))))
70	69	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾)))
71	15, 16, 17, 18, 21, 43, 70	fveval1fvcl 22277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ (Base‘𝐾))
72	29, 71	eqeltrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (Base‘𝐾))
73		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
74		aks6d1c2.23	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
75	16	ply1crng 22139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
76	20, 75	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
77		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
78	77	crngmgp 20176	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Poly₁‘𝐾) ∈ CRing → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
79	76, 78	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
80	79	cmnmndd 19733	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
81		simpr 484	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑘 = 𝑟)
85	84	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → (𝑃↑𝑘) = (𝑃↑𝑟))
86		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑙 = 𝑜)
87	86	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜))
88	85, 87	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
89		fz0ssnn0 13538	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐵) ⊆ ℕ₀
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ₀)
91	90	sselda 3933	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
92	91	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
93	89	sseli 3929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ₀)
94	93	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑜 ∈ ℕ₀)
95		ovexd 7393	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V)
96	83, 88, 92, 94, 95	ovmpod 7510	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
97		prmnn 16601	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
98	47, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
99	98	nnnn0d 12462	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
100	99	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
101	100	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
102	101, 92	nn0expcld 14169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℕ₀)
103	99	nn0zd 12513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℤ)
104	98	nnne0d 12195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
105	50	nnnn0d 12462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
106	105	nn0zd 12513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
107		dvdsval2 16182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
108	103, 104, 106, 107	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
109	52, 108	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
110	50	nnred 12160	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℝ)
111	98	nnrpd 12947	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
112	105	nn0ge0d 12465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 0 ≤ 𝑁)
113	110, 111, 112	divge0d 12989	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (𝑁 / 𝑃))
114	109, 113	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
115		elnn0z 12501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 / 𝑃) ∈ ℕ₀ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
116	114, 115	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ₀)
117	116	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
118	117	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
119	118, 94	nn0expcld 14169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℕ₀)
120	102, 119	nn0mulcld 12467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ₀)
121	96, 120	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ∈ ℕ₀)
122	121	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ∈ ℕ₀)
123	81, 122	eqeltrd 2836	. . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
128	50, 47, 52, 82	aks6d1c2p1 42372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
129	128	ffnd 6663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
130	90, 90	jca 511	. . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
135	50, 47, 52, 34, 54, 82, 133, 134	hashscontpowcl 42374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
136	135	nn0red 12463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
137	135	nn0ge0d 12465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
138	136, 137	resqrtcld 15341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
139	138	flcld 13718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
140	136, 137	sqrtge0d 15344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
141		0zd 12500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ∈ ℤ)
142		flge 13725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
143	138, 141, 142	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
144	140, 143	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
145	139, 144	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
146		elnn0z 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
147	145, 146	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
148	132, 147	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
149		elnn0z 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
150	148, 149	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
151		0z 12499	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℤ
152		eluz1 12755	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘0))
155		fzn0 13454	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0...𝐵) ≠ ∅ ↔ 𝐵 ∈ (ℤ_≥‘0))
156	154, 155	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ≠ ∅)
157	156, 156	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅))
158		xpnz 6117	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
159	157, 158	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
160		ssxpb 6132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((0...𝐵) × (0...𝐵)) ≠ ∅ → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀) ↔ ((0...𝐵) ⊆ ℕ₀ ∧ (0...𝐵) ⊆ ℕ₀)))
161	159, 160	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀) ↔ ((0...𝐵) ⊆ ℕ₀ ∧ (0...𝐵) ⊆ ℕ₀)))
162	130, 161	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀))
163		ovelimab 7536	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜)))
165	127, 164	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
166	123, 165	r19.29vva 3196	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
167	20	crngringd 20181	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Ring)
168		aks6d1c2.24	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑋 = (var₁‘𝐾)
169	168, 16, 18	vr1cl 22158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
170	167, 169	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
171	77, 18	mgpbas 20080	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
172	171	eqcomi 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾))
173	172	eleq2i 2828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))) ↔ 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
174	170, 173	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
175	73, 74, 80, 166, 174	mulgnn0cld 19025	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
176	175, 171	eleqtrrdi 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
177	176	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
178	170	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
179	15, 168, 17, 16, 18, 21, 72	evl1vard 22281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
180	179	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠))
181	178, 180	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
182	166	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐽 ∈ ℕ₀)
183	15, 16, 17, 18, 21, 72, 181, 74, 36, 182	evl1expd 22289	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
185	29	oveq2d 7374	. . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑘 = 𝑝)
217	216	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → (𝑃↑𝑘) = (𝑃↑𝑝))
218		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑙 = 𝑞)
219	218	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞))
220	217, 219	oveq12d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑝 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
222	221	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
223	222	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑝 ∈ ℕ₀)
224	89	sseli 3929	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ (0...𝐵) → 𝑞 ∈ ℕ₀)
225	224	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑞 ∈ ℕ₀)
226	225	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑞 ∈ ℕ₀)
227		ovexd 7393	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V)
228	215, 220, 223, 226, 227	ovmpod 7510	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
229	214, 228	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
230	99	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑃 ∈ ℕ₀)
231	230, 223	nn0expcld 14169	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑃↑𝑝) ∈ ℕ₀)
232	116	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
233	232	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑁 / 𝑃) ∈ ℕ₀)
234	233, 226	nn0expcld 14169	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑁 / 𝑃)↑𝑞) ∈ ℕ₀)
235	231, 234	nn0mulcld 12467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ₀)
236	229, 235	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ ℕ₀)
237	198, 126	eleqtrd 2838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
238		ovelimab 7536	. . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞)))
240	237, 239	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞))
241	236, 240	r19.29vva 3196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
242	241	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐼 ∈ ℕ₀)
243	242	ad6antr 736	. . . . . . . . . . . . . . . . . . 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 42380	. . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . 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 3196	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
248	246, 247	r19.29vva 3196	. . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) = (𝐻‘𝑠))
250	249	oveq2d 7374	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 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 22289	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
254	253	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
255	184, 251, 254	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
256	177, 255	jca 511	. . . . . . . . . . . . 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 19025	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
258	171	eleq2i 2828	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ↔ (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
259	257, 258	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
260	259	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
261		eqidd 2737	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
262	260, 261	jca 511	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
264		eqid 2736	. . . . . . . . . . . . 13 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
265	15, 16, 17, 18, 21, 72, 256, 262, 263, 264	evl1subd 22286	. . . . . . . . . . . 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 495	. . . . . . . . . . 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 20182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Grp)
268	15, 16, 17, 18, 21, 72, 260	fveval1fvcl 22277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) ∈ (Base‘𝐾))
269		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
270	17, 269, 264	grpsubid 18954	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾))
274		fvexd 6849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ V)
275		elsng 4594	. . . . . . . . . 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 17	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)})
278	76	crnggrpd 20182	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Grp)
279	18, 263	grpsubcl 18950	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)) ∈ (Base‘(Poly₁‘𝐾)))
281	11, 280	eqeltrid 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (Base‘(Poly₁‘𝐾)))
282		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↑_s (Base‘𝐾)) = (𝐾 ↑_s (Base‘𝐾))
283	15, 16, 282, 17	evl1rhm 22276	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ CRing → (eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))))
284	20, 283	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))))
285		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝐾 ↑_s (Base‘𝐾))) = (Base‘(𝐾 ↑_s (Base‘𝐾)))
286	18, 285	rhmf 20420	. . . . . . . . . . . . . . . . . 18 ⊢ ((eval₁‘𝐾) ∈ ((Poly₁‘𝐾) RingHom (𝐾 ↑_s (Base‘𝐾))) → (eval₁‘𝐾):(Base‘(Poly₁‘𝐾))⟶(Base‘(𝐾 ↑_s (Base‘𝐾))))
287	284, 286	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (eval₁‘𝐾):(Base‘(Poly₁‘𝐾))⟶(Base‘(𝐾 ↑_s (Base‘𝐾))))
288	287	ffvelcdmda 7029	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾))))
289	288	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∈ (Base‘(Poly₁‘𝐾)) → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾)))))
290	281, 289	mpd 15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾))))
291		fvexd 6849	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
292	282, 17	pwsbas 17407	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
293	19, 291, 292	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
294	290, 293	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)))
295	291, 291	elmapd 8777	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)) ↔ ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾)))
296	294, 295	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾))
297		ffn 6662	. . . . . . . . . . . 12 ⊢ (((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾) → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
298	296, 297	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
299	298	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
300		fnfun 6592	. . . . . . . . . 10 ⊢ (((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾) → Fun ((eval₁‘𝐾)‘𝑆))
301	299, 300	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → Fun ((eval₁‘𝐾)‘𝑆))
302		fndm 6595	. . . . . . . . . . . 12 ⊢ (((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾) → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
303	298, 302	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
304	303	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
305	72, 304	eleqtrrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ dom ((eval₁‘𝐾)‘𝑆))
306		fvimacnv 6998	. . . . . . . . 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 584	. . . . . . . 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 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
309	5, 10, 308	funimassd 6900	. . . . . 6 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ⊆ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
310	4, 309	ssexd 5269	. . . . 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 6838	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
313		eqid 2736	. . . . . . . . . . 11 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
314		isfld 20673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
315	314	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Field → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
316	315	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
317		drngnzr 20681	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
318	316, 317	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ Field → 𝐾 ∈ NzRing)
319	19, 318	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
320	313, 16, 168, 77, 74	deg1pw 26082	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐼 ∈ ℕ₀) → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
321	319, 241, 320	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
322	321	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)))
323	313, 16, 168, 77, 74	deg1pw 26082	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐽 ∈ ℕ₀) → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
324	319, 166, 323	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
325	324	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
326	201, 322, 325	3brtr3d 5129	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) < ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
327	16, 313, 167, 18, 263, 176, 259, 326	deg1sub 26069	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
328	312, 327	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
329	328, 324	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = 𝐽)
330	329, 166	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀)
331		eqid 2736	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
332		fldidom 20704	. . . . . . . . 9 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
333	19, 332	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ IDomn)
334	313, 16, 331, 18	deg1nn0clb 26051	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀))
335	167, 281, 334	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀))
336	330, 335	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≠ (0_g‘(Poly₁‘𝐾)))
337	16, 18, 313, 15, 269, 331, 333, 281, 336	fta1g 26131	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ ((deg₁‘𝐾)‘𝑆))
338		hashbnd 14259	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin)
340		hashcl 14279	. . . . . 6 ⊢ ((^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℕ₀)
341	339, 340	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℕ₀)
342		hashss 14332	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
344		hashbnd 14259	. . . . 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 1373	. . . 4 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin)
346		hashcl 14279	. . . 4 ⊢ ((𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℕ₀)
347	345, 346	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℕ₀)
348	347	nn0red 12463	. 2 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ)
349	341	nn0red 12463	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℝ)
350	105, 148	nn0expcld 14169	. . 3 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℕ₀)
351	350	nn0red 12463	. 2 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
352	166	nn0red 12463	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
353	324, 352	eqeltrd 2836	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ∈ ℝ)
354	327, 353	eqeltrd 2836	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ∈ ℝ)
355	312, 354	eqeltrd 2836	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℝ)
356	97	nnred 12160	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
357	47, 356	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℝ)
358	357	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
359	358	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
360	359, 92	reexpcld 14086	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℝ)
361	110, 357, 104	redivcld 11969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℝ)
362	361	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
363	362	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
364	363, 94	reexpcld 14086	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℝ)
365	360, 364	remulcld 11162	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℝ)
366	357, 148	reexpcld 14086	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃↑𝐵) ∈ ℝ)
367	366	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
368	361, 148	reexpcld 14086	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
369	368	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
370	367, 369	remulcld 11162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
371	370	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
372	110, 148	reexpcld 14086	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
373	372	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
374	373	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
375	367	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
376	369	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
377		0red 11135	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
378		1red 11133	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
379		0le1 11660	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 1
380	379	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 1)
381		prmgt1 16624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
382	47, 381	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 < 𝑃)
383	378, 357, 382	ltled 11281	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑃)
384	377, 378, 357, 380, 383	letrd 11290	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝑃)
385	384	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ 𝑃)
386	385	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ 𝑃)
387	359, 92, 386	expge0d 14087	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑃↑𝑟))
388	113	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
389	388	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
390	363, 94, 389	expge0d 14087	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ ((𝑁 / 𝑃)↑𝑜))
391	98	nnge1d 12193	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑃)
392	391	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ 𝑃)
393	392	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ 𝑃)
394		elfzuz3 13437	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑟))
395	394	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
396	395	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
397	359, 393, 396	leexp2ad 14177	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ≤ (𝑃↑𝐵))
398	357	recnd 11160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
399	398	mullidd 11150	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 · 𝑃) = 𝑃)
400	98	nnzd 12514	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℤ)
401		dvdsle 16237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
402	400, 50, 401	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
403	52, 402	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ≤ 𝑁)
404	399, 403	eqbrtrd 5120	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 · 𝑃) ≤ 𝑁)
405	378, 110, 111	lemuldivd 12998	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1 · 𝑃) ≤ 𝑁 ↔ 1 ≤ (𝑁 / 𝑃)))
406	404, 405	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (𝑁 / 𝑃))
407	406	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
408	407	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
409		elfzuz3 13437	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑜))
410	409	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑜))
411	363, 408, 410	leexp2ad 14177	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ≤ ((𝑁 / 𝑃)↑𝐵))
412	360, 375, 364, 376, 387, 390, 397, 411	lemul12ad 12084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
413	110	recnd 11160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
414	413, 398, 104	divcan2d 11919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
415	414	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
416	415	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
417	416	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
418	417	oveq1d 7373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) = ((𝑃 · (𝑁 / 𝑃))↑𝐵))
419	359	recnd 11160	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℂ)
420	363	recnd 11160	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℂ)
421	148	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ ℕ₀)
422	419, 420, 421	mulexpd 14084	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃 · (𝑁 / 𝑃))↑𝐵) = ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
423	418, 422	eqtr2d 2772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) = (𝑁↑𝐵))
424	374	leidd 11703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ≤ (𝑁↑𝐵))
425	423, 424	eqbrtrd 5120	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ≤ (𝑁↑𝐵))
426	365, 371, 374, 412, 425	letrd 11290	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ (𝑁↑𝐵))
427	96, 426	eqbrtrd 5120	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
428	427	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
429	81, 428	eqbrtrd 5120	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 ≤ (𝑁↑𝐵))
430	429, 165	r19.29vva 3196	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ (𝑁↑𝐵))
431	324, 430	eqbrtrd 5120	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ≤ (𝑁↑𝐵))
432	327, 431	eqbrtrd 5120	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ≤ (𝑁↑𝐵))
433	312, 432	eqbrtrd 5120	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ≤ (𝑁↑𝐵))
434	349, 355, 351, 337, 433	letrd 11290	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ (𝑁↑𝐵))
435	348, 349, 351, 343, 434	letrd 11290	1 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3440 ⊆ wss 3901 ∅c0 4285 {csn 4580 class class class wbr 5098 {copab 5160 ↦ cmpt 5179 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 “ cima 5627 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ↑_m cmap 8763 Fincfn 8883 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 / cdiv 11794 ℕcn 12145 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 ⌊cfl 13710 ↑cexp 13984 ♯chash 14253 √csqrt 15156 ∥ cdvds 16179 gcd cgcd 16421 ℙcprime 16598 Basecbs 17136 +_gcplusg 17177 0_gc0g 17359 Σ_g cgsu 17360 ↑_s cpws 17366 Grpcgrp 18863 -_gcsg 18865 ._gcmg 18997 CMndccmn 19709 mulGrpcmgp 20075 Ringcrg 20168 CRingccrg 20169 RingHom crh 20405 RingIso crs 20406 NzRingcnzr 20445 IDomncidom 20626 DivRingcdr 20662 Fieldcfield 20663 ℤRHomczrh 21454 chrcchr 21456 ℤ/nℤczn 21457 algSccascl 21807 var₁cv1 22116 Poly₁cpl1 22117 eval₁ce1 22258 deg₁cdg1 26015 PrimRoots cprimroots 42345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-prm 16599 df-phi 16693 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-imas 17429 df-qus 17430 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-nsg 19054 df-eqg 19055 df-ghm 19142 df-cntz 19246 df-od 19457 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-srg 20122 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-rhm 20408 df-rim 20409 df-nzr 20446 df-subrng 20479 df-subrg 20503 df-rlreg 20627 df-domn 20628 df-idom 20629 df-drng 20664 df-field 20665 df-lmod 20813 df-lss 20883 df-lsp 20923 df-sra 21125 df-rgmod 21126 df-lidl 21163 df-rsp 21164 df-2idl 21205 df-cnfld 21310 df-zring 21402 df-zrh 21458 df-chr 21460 df-zn 21461 df-assa 21808 df-asp 21809 df-ascl 21810 df-psr 21865 df-mvr 21866 df-mpl 21867 df-opsr 21869 df-evls 22029 df-evl 22030 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-evl1 22260 df-mdeg 26016 df-deg1 26017 df-mon1 26092 df-uc1p 26093 df-q1p 26094 df-r1p 26095 df-primroots 42346

This theorem is referenced by: aks6d1c2 42384

W3C validator