aks6d1c2lem4 - Mathbox for metakunt

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Theorem aks6d1c2lem4 41630

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 6917	. . . . . . . 8 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ V)
2		cnvexg 7938	. . . . . . . 8 ⊢ (((eval₁‘𝐾)‘𝑆) ∈ V → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → ^◡((eval₁‘𝐾)‘𝑆) ∈ V)
4	3	imaexd 7930	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ V)
5		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑠𝜑
6		fvexd 6917	. . . . . . . . . 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 7129	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(ℕ₀ ↑_m (0...𝐴))⟶V)
9	8	ffnd 6728	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (ℕ₀ ↑_m (0...𝐴)))
10	9	fnfund 6660	. . . . . . 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 6906	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) = ((eval₁‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
14	13	fveq1d 6904	. . . . . . . . . 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 2728	. . . . . . . . . . . . 13 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
16		eqid 2728	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
17		eqid 2728	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2728	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
19		aks6d1c2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Field)
20	19	fldcrngd 20644	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
21	20	adantr 479	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ℎ = 𝑠)
24	23	fveq2d 6906	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (𝐺‘ℎ) = (𝐺‘𝑠))
25	24	fveq2d 6906	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → ((eval₁‘𝐾)‘(𝐺‘ℎ)) = ((eval₁‘𝐾)‘(𝐺‘𝑠)))
26	25	fveq1d 6904	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) ∧ ℎ = 𝑠) → (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀))
27		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)))
28		fvexd 6917	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ V)
29	22, 26, 27, 28	fvmptd 7017	. . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
32	31	crngmgp 20188	. . . . . . . . . . . . . . . . . . . . . 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 12570	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
36		eqid 2728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
37	33, 35, 36	isprimroot 41596	. . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
41	31, 17	mgpbas 20087	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
42	40, 41	eleqtrrdi 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
43	42	adantr 479	. . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Field)
47		aks6d1c2.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℙ)
48	47	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∈ ℙ)
49	34	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑅 ∈ ℕ)
50		aks6d1c2.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
51	50	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑁 ∈ ℕ)
52		aks6d1c2.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
53	52	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑃 ∥ 𝑁)
54		aks6d1c2.8	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
55	54	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑁 gcd 𝑅) = 1)
56		elmapi 8874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) → 𝑠:(0...𝐴)⟶ℕ₀)
57	56	adantl 480	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐴 ∈ ℕ₀)
61		0nn0 12525	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ₀
62	61	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 0 ∈ ℕ₀)
63		eqid 2728	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 41628	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∼ (𝐺‘𝑠))
69	44, 68	aks6d1c1p1rcl 41611	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((𝑃↑0) · ((𝑁 / 𝑃)↑0)) ∈ ℕ ∧ (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾))))
70	69	simprd 494	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐺‘𝑠) ∈ (Base‘(Poly₁‘𝐾)))
71	15, 16, 17, 18, 21, 43, 70	fveval1fvcl 22259	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) ∈ (Base‘𝐾))
72	29, 71	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (Base‘𝐾))
73		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
74		aks6d1c2.23	. . . . . . . . . . . . . . . . 17 ⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
75	16	ply1crng 22124	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
76	20, 75	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
77		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
78	77	crngmgp 20188	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Poly₁‘𝐾) ∈ CRing → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
79	76, 78	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
80	79	cmnmndd 19766	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
81		simpr 483	. . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑘 = 𝑟)
85	84	oveq2d 7442	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → (𝑃↑𝑘) = (𝑃↑𝑟))
86		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑙 = 𝑜)
87	86	oveq2d 7442	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜))
88	85, 87	oveq12d 7444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
89		fz0ssnn0 13636	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐵) ⊆ ℕ₀
90	89	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ₀)
91	90	sselda 3982	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
92	91	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑟 ∈ ℕ₀)
93	89	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ₀)
94	93	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑜 ∈ ℕ₀)
95		ovexd 7461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V)
96	83, 88, 92, 94, 95	ovmpod 7579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
97		prmnn 16652	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
98	47, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
99	98	nnnn0d 12570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
100	99	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
101	100	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℕ₀)
102	101, 92	nn0expcld 14248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℕ₀)
103	99	nn0zd 12622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℤ)
104	98	nnne0d 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
105	50	nnnn0d 12570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
106	105	nn0zd 12622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
107		dvdsval2 16241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
108	103, 104, 106, 107	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
109	52, 108	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
110	50	nnred 12265	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℝ)
111	98	nnrpd 13054	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
112	105	nn0ge0d 12573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 0 ≤ 𝑁)
113	110, 111, 112	divge0d 13096	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (𝑁 / 𝑃))
114	109, 113	jca 510	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
115		elnn0z 12609	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 / 𝑃) ∈ ℕ₀ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 ≤ (𝑁 / 𝑃)))
116	114, 115	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ₀)
117	116	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
118	117	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
119	118, 94	nn0expcld 14248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℕ₀)
120	102, 119	nn0mulcld 12575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ₀)
121	96, 120	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ∈ ℕ₀)
122	121	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ∈ ℕ₀)
123	81, 122	eqeltrd 2829	. . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
128	50, 47, 52, 82	aks6d1c2p1 41621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
129	128	ffnd 6728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
130	90, 90	jca 510	. . . . . . . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
135	50, 47, 52, 34, 54, 82, 133, 134	hashscontpowcl 41623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
136	135	nn0red 12571	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
137	135	nn0ge0d 12573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
138	136, 137	resqrtcld 15404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
139	138	flcld 13803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
140	136, 137	sqrtge0d 15407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
141		0zd 12608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 0 ∈ ℤ)
142		flge 13810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
143	138, 141, 142	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
144	140, 143	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
145	139, 144	jca 510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
146		elnn0z 12609	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
147	145, 146	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
148	132, 147	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
149		elnn0z 12609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
150	148, 149	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
151		0z 12607	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℤ
152		eluz1 12864	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘0))
155		fzn0 13555	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0...𝐵) ≠ ∅ ↔ 𝐵 ∈ (ℤ_≥‘0))
156	154, 155	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0...𝐵) ≠ ∅)
157	156, 156	jca 510	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅))
158		xpnz 6168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
159	157, 158	sylib 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
160		ssxpb 6183	. . . . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ⊆ (ℕ₀ × ℕ₀))
163		ovelimab 7605	. . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜)))
165	127, 164	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
166	123, 165	r19.29vva 3211	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
167	20	crngringd 20193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Ring)
168		aks6d1c2.24	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑋 = (var₁‘𝐾)
169	168, 16, 18	vr1cl 22143	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Ring → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
170	167, 169	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
171	77, 18	mgpbas 20087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
172	171	eqcomi 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾))
173	172	eleq2i 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))) ↔ 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
174	170, 173	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
175	73, 74, 80, 166, 174	mulgnn0cld 19057	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
176	175, 171	eleqtrrdi 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
177	176	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐽 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
178	170	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
179	15, 168, 17, 16, 18, 21, 72	evl1vard 22263	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
180	179	simprd 494	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠))
181	178, 180	jca 510	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘(𝐻‘𝑠)) = (𝐻‘𝑠)))
182	166	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐽 ∈ ℕ₀)
183	15, 16, 17, 18, 21, 72, 181, 74, 36, 182	evl1expd 22271	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐽(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
185	29	oveq2d 7442	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 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 ∈ 𝐶)
200	124	ad7antr 736	. . . . . . . . . . . . . . . . . . 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 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 < 𝐽)
203		aks6d1c2.26	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑈 ∈ ℕ)
204	203	ad7antr 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑘 = 𝑝)
217	216	oveq2d 7442	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → (𝑃↑𝑘) = (𝑃↑𝑝))
218		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑙 = 𝑞)
219	218	oveq2d 7442	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞))
220	217, 219	oveq12d 7444	. . . . . . . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑝 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
222	221	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑝 ∈ ℕ₀)
223	222	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑝 ∈ ℕ₀)
224	89	sseli 3978	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ (0...𝐵) → 𝑞 ∈ ℕ₀)
225	224	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → 𝑞 ∈ ℕ₀)
226	225	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑞 ∈ ℕ₀)
227		ovexd 7461	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V)
228	215, 220, 223, 226, 227	ovmpod 7579	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
229	214, 228	eqtrd 2768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
230	99	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝑃 ∈ ℕ₀)
231	230, 223	nn0expcld 14248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑃↑𝑝) ∈ ℕ₀)
232	116	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℕ₀)
233	232	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → (𝑁 / 𝑃) ∈ ℕ₀)
234	233, 226	nn0expcld 14248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑁 / 𝑃)↑𝑞) ∈ ℕ₀)
235	231, 234	nn0mulcld 12575	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ₀)
236	229, 235	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑝 ∈ (0...𝐵)) ∧ 𝑞 ∈ (0...𝐵)) ∧ 𝐼 = (𝑝𝐸𝑞)) → 𝐼 ∈ ℕ₀)
237	198, 126	eleqtrd 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
238		ovelimab 7605	. . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐼 ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞)))
240	237, 239	mpbid 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∃𝑝 ∈ (0...𝐵)∃𝑞 ∈ (0...𝐵)𝐼 = (𝑝𝐸𝑞))
241	236, 240	r19.29vva 3211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
242	241	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐼 ∈ ℕ₀)
243	242	ad6antr 734	. . . . . . . . . . . . . . . . . . 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 41629	. . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 3211	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ∃𝑟 ∈ (0...𝐵)∃𝑜 ∈ (0...𝐵)𝐽 = (𝑟𝐸𝑜))
248	246, 247	r19.29vva 3211	. . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐺‘𝑠))‘𝑀) = (𝐻‘𝑠))
250	249	oveq2d 7442	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 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 22271	. . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)))
254	253	eqcomd 2734	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼(._g‘(mulGrp‘𝐾))(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
255	184, 251, 254	3eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐽 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
256	177, 255	jca 510	. . . . . . . . . . . . 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 19057	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
258	171	eleq2i 2821	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ↔ (𝐼 ↑ 𝑋) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
259	257, 258	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
260	259	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐼 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)))
261		eqidd 2729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) = (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)))
262	260, 261	jca 510	. . . . . . . . . . . . 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 2728	. . . . . . . . . . . . 13 ⊢ (-_g‘(Poly₁‘𝐾)) = (-_g‘(Poly₁‘𝐾))
264		eqid 2728	. . . . . . . . . . . . 13 ⊢ (-_g‘𝐾) = (-_g‘𝐾)
265	15, 16, 17, 18, 21, 72, 256, 262, 263, 264	evl1subd 22268	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 20194	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → 𝐾 ∈ Grp)
268	15, 16, 17, 18, 21, 72, 260	fveval1fvcl 22259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘(𝐼 ↑ 𝑋))‘(𝐻‘𝑠)) ∈ (Base‘𝐾))
269		eqid 2728	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
270	17, 269, 264	grpsubid 18987	. . . . . . . . . . . 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 582	. . . . . . . . . . 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 2768	. . . . . . . . . 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 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) = (0_g‘𝐾))
274		fvexd 6917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ V)
275		elsng 4646	. . . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (((eval₁‘𝐾)‘𝑆)‘(𝐻‘𝑠)) ∈ {(0_g‘𝐾)})
278	76	crnggrpd 20194	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Grp)
279	18, 263	grpsubcl 18983	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋)) ∈ (Base‘(Poly₁‘𝐾)))
281	11, 280	eqeltrid 2833	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (Base‘(Poly₁‘𝐾)))
282		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↑_s (Base‘𝐾)) = (𝐾 ↑_s (Base‘𝐾))
283	15, 16, 282, 17	evl1rhm 22258	. . . . . . . . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝐾 ↑_s (Base‘𝐾))) = (Base‘(𝐾 ↑_s (Base‘𝐾)))
286	18, 285	rhmf 20431	. . . . . . . . . . . . . . . . . 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 7099	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → ((eval₁‘𝐾)‘𝑆) ∈ (Base‘(𝐾 ↑_s (Base‘𝐾))))
289	288	ex 411	. . . . . . . . . . . . . . 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 6917	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
292	282, 17	pwsbas 17476	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
293	19, 291, 292	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝐾) ↑_m (Base‘𝐾)) = (Base‘(𝐾 ↑_s (Base‘𝐾))))
294	290, 293	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)))
295	291, 291	elmapd 8865	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((eval₁‘𝐾)‘𝑆) ∈ ((Base‘𝐾) ↑_m (Base‘𝐾)) ↔ ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾)))
296	294, 295	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval₁‘𝐾)‘𝑆):(Base‘𝐾)⟶(Base‘𝐾))
297		ffn 6727	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → ((eval₁‘𝐾)‘𝑆) Fn (Base‘𝐾))
300		fnfun 6659	. . . . . . . . . 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 6662	. . . . . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → dom ((eval₁‘𝐾)‘𝑆) = (Base‘𝐾))
305	72, 304	eleqtrrd 2832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ dom ((eval₁‘𝐾)‘𝑆))
306		fvimacnv 7067	. . . . . . . . 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 582	. . . . . . . 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 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (ℕ₀ ↑_m (0...𝐴))) → (𝐻‘𝑠) ∈ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
309	5, 10, 308	funimassd 6970	. . . . . 6 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ⊆ (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}))
310	4, 309	ssexd 5328	. . . . 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 6906	. . . . . . . . . 10 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) = (( deg₁ ‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))))
313		eqid 2728	. . . . . . . . . . 11 ⊢ ( deg₁ ‘𝐾) = ( deg₁ ‘𝐾)
314		isfld 20642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
315	314	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Field → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
316	315	simpld 493	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
317		drngnzr 20651	. . . . . . . . . . . . . . . 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 26076	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐼 ∈ ℕ₀) → (( deg₁ ‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
321	319, 241, 320	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘(𝐼 ↑ 𝑋)) = 𝐼)
322	321	eqcomd 2734	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = (( deg₁ ‘𝐾)‘(𝐼 ↑ 𝑋)))
323	313, 16, 168, 77, 74	deg1pw 26076	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ NzRing ∧ 𝐽 ∈ ℕ₀) → (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
324	319, 166, 323	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)) = 𝐽)
325	324	eqcomd 2734	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 = (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)))
326	201, 322, 325	3brtr3d 5183	. . . . . . . . . . 11 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘(𝐼 ↑ 𝑋)) < (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)))
327	16, 313, 167, 18, 263, 176, 259, 326	deg1sub 26064	. . . . . . . . . 10 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) = (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)))
328	312, 327	eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) = (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)))
329	328, 324	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) = 𝐽)
330	329, 166	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) ∈ ℕ₀)
331		eqid 2728	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
332		fldidom 21265	. . . . . . . . 9 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
333	19, 332	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ IDomn)
334	313, 16, 331, 18	deg1nn0clb 26046	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Ring ∧ 𝑆 ∈ (Base‘(Poly₁‘𝐾))) → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ (( deg₁ ‘𝐾)‘𝑆) ∈ ℕ₀))
335	167, 281, 334	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ≠ (0_g‘(Poly₁‘𝐾)) ↔ (( deg₁ ‘𝐾)‘𝑆) ∈ ℕ₀))
336	330, 335	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≠ (0_g‘(Poly₁‘𝐾)))
337	16, 18, 313, 15, 269, 331, 333, 281, 336	fta1g 26124	. . . . . . 7 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ (( deg₁ ‘𝐾)‘𝑆))
338		hashbnd 14335	. . . . . . 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 1368	. . . . . 6 ⊢ (𝜑 → (^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)}) ∈ Fin)
340		hashcl 14355	. . . . . 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 14408	. . . . . 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 582	. . . . 5 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})))
344		hashbnd 14335	. . . . 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 1368	. . . 4 ⊢ (𝜑 → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ Fin)
346		hashcl 14355	. . . 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 12571	. 2 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ)
349	341	nn0red 12571	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ∈ ℝ)
350	105, 148	nn0expcld 14248	. . 3 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℕ₀)
351	350	nn0red 12571	. 2 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
352	166	nn0red 12571	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
353	324, 352	eqeltrd 2829	. . . . 5 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)) ∈ ℝ)
354	327, 353	eqeltrd 2829	. . . 4 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ∈ ℝ)
355	312, 354	eqeltrd 2829	. . 3 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) ∈ ℝ)
356	97	nnred 12265	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
357	47, 356	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℝ)
358	357	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
359	358	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℝ)
360	359, 92	reexpcld 14167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ∈ ℝ)
361	110, 357, 104	redivcld 12080	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℝ)
362	361	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
363	362	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℝ)
364	363, 94	reexpcld 14167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ∈ ℝ)
365	360, 364	remulcld 11282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℝ)
366	357, 148	reexpcld 14167	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃↑𝐵) ∈ ℝ)
367	366	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
368	361, 148	reexpcld 14167	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
369	368	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
370	367, 369	remulcld 11282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
371	370	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ∈ ℝ)
372	110, 148	reexpcld 14167	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
373	372	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
374	373	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ∈ ℝ)
375	367	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝐵) ∈ ℝ)
376	369	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝐵) ∈ ℝ)
377		0red 11255	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℝ)
378		1red 11253	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
379		0le1 11775	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 1
380	379	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 1)
381		prmgt1 16675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
382	47, 381	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 < 𝑃)
383	378, 357, 382	ltled 11400	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑃)
384	377, 378, 357, 380, 383	letrd 11409	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝑃)
385	384	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ 𝑃)
386	385	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ 𝑃)
387	359, 92, 386	expge0d 14168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑃↑𝑟))
388	113	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
389	388	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ (𝑁 / 𝑃))
390	363, 94, 389	expge0d 14168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 0 ≤ ((𝑁 / 𝑃)↑𝑜))
391	98	nnge1d 12298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑃)
392	391	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ 𝑃)
393	392	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ 𝑃)
394		elfzuz3 13538	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑟))
395	394	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
396	395	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑟))
397	359, 393, 396	leexp2ad 14256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑃↑𝑟) ≤ (𝑃↑𝐵))
398	357	recnd 11280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℂ)
399	398	mullidd 11270	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 · 𝑃) = 𝑃)
400	98	nnzd 12623	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℤ)
401		dvdsle 16294	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
402	400, 50, 401	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 ∥ 𝑁 → 𝑃 ≤ 𝑁))
403	52, 402	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ≤ 𝑁)
404	399, 403	eqbrtrd 5174	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 · 𝑃) ≤ 𝑁)
405	378, 110, 111	lemuldivd 13105	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1 · 𝑃) ≤ 𝑁 ↔ 1 ≤ (𝑁 / 𝑃)))
406	404, 405	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (𝑁 / 𝑃))
407	406	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
408	407	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 1 ≤ (𝑁 / 𝑃))
409		elfzuz3 13538	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ (0...𝐵) → 𝐵 ∈ (ℤ_≥‘𝑜))
410	409	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ (ℤ_≥‘𝑜))
411	363, 408, 410	leexp2ad 14256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑁 / 𝑃)↑𝑜) ≤ ((𝑁 / 𝑃)↑𝐵))
412	360, 375, 364, 376, 387, 390, 397, 411	lemul12ad 12194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
413	110	recnd 11280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℂ)
414	413, 398, 104	divcan2d 12030	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
415	414	eqcomd 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
416	415	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
417	416	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
418	417	oveq1d 7441	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) = ((𝑃 · (𝑁 / 𝑃))↑𝐵))
419	359	recnd 11280	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝑃 ∈ ℂ)
420	363	recnd 11280	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁 / 𝑃) ∈ ℂ)
421	148	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → 𝐵 ∈ ℕ₀)
422	419, 420, 421	mulexpd 14165	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃 · (𝑁 / 𝑃))↑𝐵) = ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)))
423	418, 422	eqtr2d 2769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) = (𝑁↑𝐵))
424	374	leidd 11818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑁↑𝐵) ≤ (𝑁↑𝐵))
425	423, 424	eqbrtrd 5174	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝐵) · ((𝑁 / 𝑃)↑𝐵)) ≤ (𝑁↑𝐵))
426	365, 371, 374, 412, 425	letrd 11409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ≤ (𝑁↑𝐵))
427	96, 426	eqbrtrd 5174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
428	427	adantr 479	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → (𝑟𝐸𝑜) ≤ (𝑁↑𝐵))
429	81, 428	eqbrtrd 5174	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ (0...𝐵)) ∧ 𝑜 ∈ (0...𝐵)) ∧ 𝐽 = (𝑟𝐸𝑜)) → 𝐽 ≤ (𝑁↑𝐵))
430	429, 165	r19.29vva 3211	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ (𝑁↑𝐵))
431	324, 430	eqbrtrd 5174	. . . . 5 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘(𝐽 ↑ 𝑋)) ≤ (𝑁↑𝐵))
432	327, 431	eqbrtrd 5174	. . . 4 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘((𝐽 ↑ 𝑋)(-_g‘(Poly₁‘𝐾))(𝐼 ↑ 𝑋))) ≤ (𝑁↑𝐵))
433	312, 432	eqbrtrd 5174	. . 3 ⊢ (𝜑 → (( deg₁ ‘𝐾)‘𝑆) ≤ (𝑁↑𝐵))
434	349, 355, 351, 337, 433	letrd 11409	. 2 ⊢ (𝜑 → (♯‘(^◡((eval₁‘𝐾)‘𝑆) “ {(0_g‘𝐾)})) ≤ (𝑁↑𝐵))
435	348, 349, 351, 343, 434	letrd 11409	1 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 Vcvv 3473 ⊆ wss 3949 ∅c0 4326 {csn 4632 class class class wbr 5152 {copab 5214 ↦ cmpt 5235 × cxp 5680 ^◡ccnv 5681 dom cdm 5682 “ cima 5685 Fun wfun 6547 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ↑_m cmap 8851 Fincfn 8970 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 / cdiv 11909 ℕcn 12250 ℕ₀cn0 12510 ℤcz 12596 ℤ_≥cuz 12860 ...cfz 13524 ⌊cfl 13795 ↑cexp 14066 ♯chash 14329 √csqrt 15220 ∥ cdvds 16238 gcd cgcd 16476 ℙcprime 16649 Basecbs 17187 +_gcplusg 17240 0_gc0g 17428 Σ_g cgsu 17429 ↑_s cpws 17435 Grpcgrp 18897 -_gcsg 18899 ._gcmg 19030 CMndccmn 19742 mulGrpcmgp 20081 Ringcrg 20180 CRingccrg 20181 RingHom crh 20415 RingIso crs 20416 NzRingcnzr 20458 DivRingcdr 20631 Fieldcfield 20632 IDomncidom 21235 ℤRHomczrh 21432 chrcchr 21434 ℤ/nℤczn 21435 algSccascl 21793 var₁cv1 22102 Poly₁cpl1 22103 eval₁ce1 22240 deg₁ cdg1 26007 PrimRoots cprimroots 41594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-gcd 16477 df-prm 16650 df-phi 16742 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-imas 17497 df-qus 17498 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-nsg 19086 df-eqg 19087 df-ghm 19175 df-cntz 19275 df-od 19490 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-rhm 20418 df-rim 20419 df-nzr 20459 df-subrng 20490 df-subrg 20515 df-drng 20633 df-field 20634 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-lidl 21111 df-rsp 21112 df-2idl 21151 df-rlreg 21237 df-domn 21238 df-idom 21239 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-chr 21438 df-zn 21439 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-evls 22025 df-evl 22026 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-coe1 22109 df-evl1 22242 df-mdeg 26008 df-deg1 26009 df-mon1 26086 df-uc1p 26087 df-q1p 26088 df-r1p 26089 df-primroots 41595

This theorem is referenced by: aks6d1c2 41633

W3C validator