aks6d1c2lem4 - Mathbox for metakunt

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Theorem aks6d1c2lem4 42140

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 6891	. . . . . . . 8 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ V)
2		cnvexg 7920	. . . . . . . 8 ⊢ (((eval₁‘𝐾)‘𝑆) ∈ V → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
4	3	imaexd 7912	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V)
5		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑠𝜑
6		fvexd 6891	. . . . . . . . . 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 7104	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(ℕ₀ ↑_m (0...𝐴))⟶V)
9	8	ffnd 6707	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (ℕ₀ ↑_m (0...𝐴)))
10	9	fnfund 6639	. . . . . . 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 6880	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) = ((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
14	13	fveq1d 6878	. . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
16		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
17		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
19		aks6d1c2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Field)
20	19	fldcrngd 20702	. . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (𝐺‘ℎ) = (𝐺‘𝑠))
25	24	fveq2d 6880	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ((eval₁‘𝐾)‘(𝐺‘ℎ)) = ((eval₁‘𝐾)‘(𝐺‘𝑠)))
26	25	fveq1d 6878	. . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ V)
29	22, 26, 27, 28	fvmptd 6993	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
32	31	crngmgp 20201	. . . . . . . . . . . . . . . . . . . . . 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 12562	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
36		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
37	33, 35, 36	isprimroot 42106	. . . . . . . . . . . . . . . . . . . 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 20105	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
42	40, 41	eleqtrrdi 2845	. . . . . . . . . . . . . . . 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 8863	. . . . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ₀
62	61	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 0 ∈ ℕ₀)
63		eqid 2735	. . . . . . . . . . . . . . . . . 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 42138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∼ (𝐺‘𝑠))
69	44, 68	aks6d1c1p1rcl 42121	. . . . . . . . . . . . . . . 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 22271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ (Base‘𝐾))
72	29, 71	eqeltrd 2834	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (Base‘𝐾))
73		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
74		aks6d1c2.23	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
75	16	ply1crng 22134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
76	20, 75	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
77		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
78	77	crngmgp 20201	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Poly₁‘𝐾) ∈ CRing → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
79	76, 78	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
80	79	cmnmndd 19785	. . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜))
88	85, 87	oveq12d 7423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
89		fz0ssnn0 13639	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐵) ⊆ ℕ₀
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ₀)
91	90	sselda 3958	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
92	91	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
93	89	sseli 3954	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ₀)
94	93	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑜 ∈ ℕ₀)
95		ovexd 7440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V)
96	83, 88, 92, 94, 95	ovmpod 7559	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
97		prmnn 16693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
98	47, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
99	98	nnnn0d 12562	. . . . . . . . . . . . . . . . . . . . . . . . 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 14264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℕ₀)
103	99	nn0zd 12614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℤ)
104	98	nnne0d 12290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
105	50	nnnn0d 12562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
106	105	nn0zd 12614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
107		dvdsval2 16275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12255	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℝ)
111	98	nnrpd 13049	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
112	105	nn0ge0d 12565	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 0 ≤ 𝑁)
113	110, 111, 112	divge0d 13091	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (𝑁 / 𝑃))
114	109, 113	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
115		elnn0z 12601	. . . . . . . . . . . . . . . . . . . . . . . . . 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 14264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℕ₀)
120	102, 119	nn0mulcld 12567	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ₀)
121	96, 120	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
128	50, 47, 52, 82	aks6d1c2p1 42131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
129	128	ffnd 6707	. . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
135	50, 47, 52, 34, 54, 82, 133, 134	hashscontpowcl 42133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
136	135	nn0red 12563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
137	135	nn0ge0d 12565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
138	136, 137	resqrtcld 15436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
139	138	flcld 13815	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
140	136, 137	sqrtge0d 15439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
141		0zd 12600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ∈ ℤ)
142		flge 13822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
149		elnn0z 12601	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
150	148, 149	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
151		0z 12599	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℤ
152		eluz1 12856	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13555	. . . . . . . . . . . . . . . . . . . . . . . . 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 6148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
159	157, 158	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
160		ssxpb 6163	. . . . . . . . . . . . . . . . . . . . . 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 7585	. . . . . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
167	20	crngringd 20206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Ring)
168		aks6d1c2.24	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑋 = (var₁‘𝐾)
169	168, 16, 18	vr1cl 22153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
170	167, 169	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
171	77, 18	mgpbas 20105	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
172	171	eqcomi 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾))
173	172	eleq2i 2826	. . . . . . . . . . . . . . . . . 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 19078	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
176	175, 171	eleqtrrdi 2845	. . . . . . . . . . . . . . 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 22275	. . . . . . . . . . . . . . . . . . 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 22283	. . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞))
220	217, 219	oveq12d 7423	. . . . . . . . . . . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V)
228	215, 220, 223, 226, 227	ovmpod 7559	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
229	214, 228	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . 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 14264	. . . . . . . . . . . . . . . . . . . . . . . 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 14264	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑁 / 𝑃)↑𝑞) ∈ ℕ₀)
235	231, 234	nn0mulcld 12567	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ₀)
236	229, 235	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ ℕ₀)
237	198, 126	eleqtrd 2836	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
238		ovelimab 7585	. . . . . . . . . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . . . . . . . . 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 42139	. . . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) = (𝐻‘𝑠))
250	249	oveq2d 7421	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 22283	. . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
255	184, 251, 254	3eqtrd 2774	. . . . . . . . . . . . . 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 19078	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
258	171	eleq2i 2826	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
264		eqid 2735	. . . . . . . . . . . . 13 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
265	15, 16, 17, 18, 21, 72, 256, 262, 263, 264	evl1subd 22280	. . . . . . . . . . . 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 20207	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Grp)
268	15, 16, 17, 18, 21, 72, 260	fveval1fvcl 22271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) ∈ (Base‘𝐾))
269		eqid 2735	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
270	17, 269, 264	grpsubid 19007	. . . . . . . . . . . 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 2770	. . . . . . . . . 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 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾))
274		fvexd 6891	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ V)
275		elsng 4615	. . . . . . . . . 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 20207	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Grp)
279	18, 263	grpsubcl 19003	. . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (Base‘(Poly₁‘𝐾)))
282		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↑_s (Base‘𝐾)) = (𝐾 ↑_s (Base‘𝐾))
283	15, 16, 282, 17	evl1rhm 22270	. . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝐾 ↑_s (Base‘𝐾))) = (Base‘(𝐾 ↑_s (Base‘𝐾)))
286	18, 285	rhmf 20445	. . . . . . . . . . . . . . . . . 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 7074	. . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
292	282, 17	pwsbas 17501	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)))
295	291, 291	elmapd 8854	. . . . . . . . . . . . 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 6706	. . . . . . . . . . . 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 6638	. . . . . . . . . 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 6641	. . . . . . . . . . . 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 2837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ dom ((eval₁‘𝐾)‘𝑆))
306		fvimacnv 7043	. . . . . . . . 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 6945	. . . . . 6 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ⊆ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
310	4, 309	ssexd 5294	. . . . 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 6880	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
313		eqid 2735	. . . . . . . . . . 11 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
314		isfld 20700	. . . . . . . . . . . . . . . . . 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 20708	. . . . . . . . . . . . . . . 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 26078	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)))
323	313, 16, 168, 77, 74	deg1pw 26078	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
326	201, 322, 325	3brtr3d 5150	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐼 ↑ 𝑋)) < ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
327	16, 313, 167, 18, 263, 176, 259, 326	deg1sub 26065	. . . . . . . . . 10 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
328	312, 327	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)))
329	328, 324	eqtrd 2770	. . . . . . . 8 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) = 𝐽)
330	329, 166	eqeltrd 2834	. . . . . . 7 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℕ₀)
331		eqid 2735	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
332		fldidom 20731	. . . . . . . . 9 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
333	19, 332	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ IDomn)
334	313, 16, 331, 18	deg1nn0clb 26047	. . . . . . . . . 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 26127	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ ((deg₁‘𝐾)‘𝑆))
338		hashbnd 14354	. . . . . . 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 14374	. . . . . 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 14427	. . . . . 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 14354	. . . . 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 14374	. . . 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 12563	. 2 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ)
349	341	nn0red 12563	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℝ)
350	105, 148	nn0expcld 14264	. . 3 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℕ₀)
351	350	nn0red 12563	. 2 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
352	166	nn0red 12563	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
353	324, 352	eqeltrd 2834	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ∈ ℝ)
354	327, 353	eqeltrd 2834	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ∈ ℝ)
355	312, 354	eqeltrd 2834	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ∈ ℝ)
356	97	nnred 12255	. . . . . . . . . . . . . . . 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 14181	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℝ)
361	110, 357, 104	redivcld 12069	. . . . . . . . . . . . . . 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 14181	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℝ)
365	360, 364	remulcld 11265	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℝ)
366	357, 148	reexpcld 14181	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃↑𝐵) ∈ ℝ)
367	366	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
368	361, 148	reexpcld 14181	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
369	368	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
370	367, 369	remulcld 11265	. . . . . . . . . . . 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 14181	. . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
378		1red 11236	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
379		0le1 11760	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 1
380	379	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 1)
381		prmgt1 16716	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
382	47, 381	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 < 𝑃)
383	378, 357, 382	ltled 11383	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑃)
384	377, 378, 357, 380, 383	letrd 11392	. . . . . . . . . . . . . . 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 14182	. . . . . . . . . . . 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 14182	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ ((𝑁 / 𝑃)↑𝑜))
391	98	nnge1d 12288	. . . . . . . . . . . . . . 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 13538	. . . . . . . . . . . . . . 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 14272	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ≤ (𝑃↑𝐵))
398	357	recnd 11263	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
399	398	mullidd 11253	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 · 𝑃) = 𝑃)
400	98	nnzd 12615	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℤ)
401		dvdsle 16329	. . . . . . . . . . . . . . . . . . 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 5141	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 · 𝑃) ≤ 𝑁)
405	378, 110, 111	lemuldivd 13100	. . . . . . . . . . . . . . . 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 13538	. . . . . . . . . . . . . 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 14272	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ≤ ((𝑁 / 𝑃)↑𝐵))
412	360, 375, 364, 376, 387, 390, 397, 411	lemul12ad 12184	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
413	110	recnd 11263	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
414	413, 398, 104	divcan2d 12019	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
415	414	eqcomd 2741	. . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) = ((𝑃 · (𝑁 / 𝑃))↑𝐵))
419	359	recnd 11263	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℂ)
420	363	recnd 11263	. . . . . . . . . . . . . 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 14179	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃 · (𝑁 / 𝑃))↑𝐵) = ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
423	418, 422	eqtr2d 2771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) = (𝑁↑𝐵))
424	374	leidd 11803	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ≤ (𝑁↑𝐵))
425	423, 424	eqbrtrd 5141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ≤ (𝑁↑𝐵))
426	365, 371, 374, 412, 425	letrd 11392	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ (𝑁↑𝐵))
427	96, 426	eqbrtrd 5141	. . . . . . . . 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 5141	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 ≤ (𝑁↑𝐵))
430	429, 165	r19.29vva 3201	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ (𝑁↑𝐵))
431	324, 430	eqbrtrd 5141	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐽 ↑ 𝑋)) ≤ (𝑁↑𝐵))
432	327, 431	eqbrtrd 5141	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ≤ (𝑁↑𝐵))
433	312, 432	eqbrtrd 5141	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘𝑆) ≤ (𝑁↑𝐵))
434	349, 355, 351, 337, 433	letrd 11392	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ (𝑁↑𝐵))
435	348, 349, 351, 343, 434	letrd 11392	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 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ⊆ wss 3926 ∅c0 4308 {csn 4601 class class class wbr 5119 {copab 5181 ↦ cmpt 5201 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 “ cima 5657 Fun wfun 6525 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 ↑_m cmap 8840 Fincfn 8959 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 / cdiv 11894 ℕcn 12240 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13524 ⌊cfl 13807 ↑cexp 14079 ♯chash 14348 √csqrt 15252 ∥ cdvds 16272 gcd cgcd 16513 ℙcprime 16690 Basecbs 17228 +_gcplusg 17271 0_gc0g 17453 Σ_g cgsu 17454 ↑_s cpws 17460 Grpcgrp 18916 -_gcsg 18918 ._gcmg 19050 CMndccmn 19761 mulGrpcmgp 20100 Ringcrg 20193 CRingccrg 20194 RingHom crh 20429 RingIso crs 20430 NzRingcnzr 20472 IDomncidom 20653 DivRingcdr 20689 Fieldcfield 20690 ℤRHomczrh 21460 chrcchr 21462 ℤ/nℤczn 21463 algSccascl 21812 var₁cv1 22111 Poly₁cpl1 22112 eval₁ce1 22252 deg₁cdg1 26011 PrimRoots cprimroots 42104

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-ofr 7672 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-dvds 16273 df-gcd 16514 df-prm 16691 df-phi 16785 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-imas 17522 df-qus 17523 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-nsg 19107 df-eqg 19108 df-ghm 19196 df-cntz 19300 df-od 19509 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-srg 20147 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-rim 20433 df-nzr 20473 df-subrng 20506 df-subrg 20530 df-rlreg 20654 df-domn 20655 df-idom 20656 df-drng 20691 df-field 20692 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-2idl 21211 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-chr 21466 df-zn 21467 df-assa 21813 df-asp 21814 df-ascl 21815 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-evls 22032 df-evl 22033 df-psr1 22115 df-vr1 22116 df-ply1 22117 df-coe1 22118 df-evl1 22254 df-mdeg 26012 df-deg1 26013 df-mon1 26088 df-uc1p 26089 df-q1p 26090 df-r1p 26091 df-primroots 42105

This theorem is referenced by: aks6d1c2 42143

W3C validator