Theorem aks6d1c6lem4 42998

Description: Claim 6 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf Add hypothesis on coprimality, lift function to the integers so that group operations may be applied. Inline definition. (Contributed by metakunt, 14-May-2025.)

Hypotheses

Ref	Expression
aks6d1c6lem4.1	⊢ ∼ = {⟨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)))}
aks6d1c6lem4.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c6lem4.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c6lem4.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c6lem4.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c6lem4.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c6lem4.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c6lem4.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6lem4.9	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks6d1c6lem4.10	⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c6lem4.11	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
aksaks6dlem4.12	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6lem4.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6lem4.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6lem4.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6lem4.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6lem4.17	⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c6lem4.18	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
aks6d1c6lem4.19	⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
aks6d1c6lem4.20	⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
aks6d1c6lem4.21	⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
aks6d1c6lem4.22	⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}

Assertion

Ref	Expression
aks6d1c6lem4	⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

Distinct variable groups:   ∼ ,𝑎   𝑃,𝑘,𝑙,𝑠   𝜑,ℎ   𝑁,𝑠   𝜑,𝑘,𝑙   ℎ,𝐾   𝑦,𝑘,𝑙,𝜑   𝑔,𝐾,𝑥   𝑒,𝐾,𝑓   𝑚,𝐾,𝑛   𝑘,𝑁,𝑙,𝑥   𝑥,𝑅   𝑃,𝑗   𝑒,𝑁,𝑓   𝑆,𝑠,𝑡   𝑃,𝑒,𝑓   𝑗,𝑁   𝑅,𝑒,𝑓,𝑦   𝑗,𝐾   𝑦,𝑀   𝑁,𝑎   ℎ,𝑀,𝑗   𝑥,𝑃   𝑆,ℎ,𝑗   𝜑,𝑗   𝑈,𝑗   𝑆,𝑎   𝑆,𝑔,𝑖,𝑥,𝑦   𝜑,𝑔,𝑖,𝑥   𝜑,𝑠,𝑡   𝜑,𝑎   𝑃,𝑏   𝑁,𝑏   𝐾,𝑎   𝑖,𝐾,𝑡,𝑦,𝑥   𝐷,𝑠   ℎ,𝐺   𝑡,𝐺   𝑔,𝐺,𝑖,𝑦   𝐻,𝑠,𝑡   ℎ,𝐻,𝑗   𝑥,𝐸   𝑒,𝐸,𝑓,𝑦   𝑗,𝐸   𝑔,𝐻,𝑖,𝑥,𝑦   𝐴,𝑎   𝑒,𝐺,𝑓   𝐴,𝑏   𝐻,𝑎   𝐴,𝑔,𝑖,𝑥   𝐴,ℎ,𝑗   𝐴,𝑠,𝑡

Allowed substitution hints:   𝜑(𝑒, 𝑓, 𝑚, 𝑛, 𝑏)   𝐴(𝑦, 𝑒, 𝑓, 𝑘, 𝑚, 𝑛, 𝑙)   𝐷(𝑥, 𝑦, 𝑡, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑎, 𝑏, 𝑙)   𝑃(𝑦, 𝑡, 𝑔, ℎ, 𝑖, 𝑚, 𝑛, 𝑎)   ∼ (𝑥, 𝑦, 𝑡, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑠, 𝑏, 𝑙)   𝑅(𝑡, 𝑔, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝑆(𝑒, 𝑓, 𝑘, 𝑚, 𝑛, 𝑏, 𝑙)   𝑈(𝑥, 𝑦, 𝑡, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝐸(𝑡, 𝑔, ℎ, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝐺(𝑥, 𝑗, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝐻(𝑒, 𝑓, 𝑘, 𝑚, 𝑛, 𝑏, 𝑙)   𝐽(𝑥, 𝑦, 𝑡, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝐾(𝑘, 𝑠, 𝑏, 𝑙)   𝐿(𝑥, 𝑦, 𝑡, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝑀(𝑥, 𝑡, 𝑒, 𝑓, 𝑔, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠, 𝑎, 𝑏, 𝑙)   𝑁(𝑦, 𝑡, 𝑔, ℎ, 𝑖, 𝑚, 𝑛)

Proof of Theorem aks6d1c6lem4

Dummy variables 𝑣 𝑤 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c6lem4.1	. 2 ⊢ ∼ = {⟨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)))}
2		aks6d1c6lem4.2	. 2 ⊢ 𝑃 = (chr‘𝐾)
3		aks6d1c6lem4.3	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
4		aks6d1c6lem4.4	. 2 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		aks6d1c6lem4.5	. 2 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6		aks6d1c6lem4.6	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		aks6d1c6lem4.7	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
8		aks6d1c6lem4.8	. 2 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
9		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝑃) → 𝐴 < 𝑃)
10		prmnn 16754	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
11	4, 10	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
12	11	nnred 12263	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13		aks6d1c6lem4.11	. . . . . . . . 9 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
14	5	phicld 16853	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
15	14	nnred 12263	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
16	14	nnnn0d 12580	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ0)
17	16	nn0ge0d 12583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
18	15, 17	resqrtcld 15493	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
19		2re 12330	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
21		2pos 12360	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
23	6	nnred 12263	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
24	6	nngt0d 12300	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
25		1red 11224	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
26		1lt2 12428	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
27	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
28	25, 27	ltned 11361	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
29	28	necomd 3015	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
30	20, 22, 23, 24, 29	relogbcld 42799	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ)
31	18, 30	remulcld 11254	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ)
32	31	flcld 13849	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ)
33	15, 17	sqrtge0d 15496	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
34	20	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℂ)
35	22	gt0ne0d 11793	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 0)
36		logb1 26985	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0)
37	34, 35, 29, 36	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 logb 1) = 0)
38	37	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 = (2 logb 1))
39		2z 12641	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℤ)
41	20	leidd 11795	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ 2)
42		0lt1 11751	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
44	6	nnge1d 12299	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑁)
45	40, 41, 25, 43, 23, 24, 44	logblebd 42802	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 logb 1) ≤ (2 logb 𝑁))
46	38, 45	eqbrtrd 5135	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 logb 𝑁))
47	18, 30, 33, 46	mulge0d 11806	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
48		0zd 12618	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
49		flge 13856	. . . . . . . . . . . . 13 ⊢ ((((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
50	31, 48, 49	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
51	47, 50	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))
52	32, 51	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
53		elnn0z 12619	. . . . . . . . . 10 ⊢ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0 ↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
54	52, 53	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0)
55	13, 54	eqeltrid 2869	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
56	55	nn0red 12581	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	12, 56	lenltd 11371	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑃))
58	57	biimpar 483	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃 ≤ 𝐴)
59		oveq1 7426	. . . . . . . . 9 ⊢ (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁))
60	59	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑏 = 𝑃 → ((𝑏 gcd 𝑁) = 1 ↔ (𝑃 gcd 𝑁) = 1))
61		aks6d1c6lem4.9	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
62	61	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
63		1zzd 12640	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ∈ ℤ)
64	13, 32	eqeltrid 2869	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
65	64	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝐴 ∈ ℤ)
66	11	nnzd 12632	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	66	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ ℤ)
68	11	nnge1d 12299	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ 𝑃)
69	68	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ≤ 𝑃)
70		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ≤ 𝐴)
71	63, 65, 67, 69, 70	elfzd 13559	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ (1...𝐴))
72	60, 62, 71	rspcdva 3584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → (𝑃 gcd 𝑁) = 1)
73	72	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1))
74	73	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1))
75	58, 74	mpd 16	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) = 1)
76	6	nnzd 12632	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
77		coprm 16792	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
78	4, 76, 77	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
79	78	con1bid 358	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝑃 gcd 𝑁) = 1 ↔ 𝑃 ∥ 𝑁))
80	79	bicomd 226	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ ¬ (𝑃 gcd 𝑁) = 1))
81	80	biimpd 232	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ 𝑁 → ¬ (𝑃 gcd 𝑁) = 1))
82	7, 81	mpd 16	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 gcd 𝑁) = 1)
83	82	neqned 2967	. . . . . 6 ⊢ (𝜑 → (𝑃 gcd 𝑁) ≠ 1)
84	83	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1)
85	84	neneqd 2965	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → ¬ (𝑃 gcd 𝑁) = 1)
86	75, 85	pm2.21dd 198	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝐴 < 𝑃)
87	9, 86	pm2.61dan 825	. 2 ⊢ (𝜑 → 𝐴 < 𝑃)
88		aks6d1c6lem4.10	. 2 ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
89		aksaks6dlem4.12	. 2 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
90		aks6d1c6lem4.13	. 2 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
91		aks6d1c6lem4.14	. 2 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
92		aks6d1c6lem4.15	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
93		aks6d1c6lem4.16	. 2 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
94		aks6d1c6lem4.17	. 2 ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
95		aks6d1c6lem4.18	. 2 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
96		aks6d1c6lem4.19	. 2 ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
97		eqid 2765	. 2 ⊢ (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))
98		aks6d1c6lem4.21	. . 3 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
99		imaco 6254	. . . . . 6 ⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))
100	99	eqcomi 2774	. . . . 5 ⊢ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
101		resima 6016	. . . . . . . 8 ⊢ (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
102	101	eqcomi 2774	. . . . . . 7 ⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0))
103	102	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)))
104	66	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
105		xp1st 8024	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (1st ‘𝑣) ∈ ℕ0)
106	105	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑣) ∈ ℕ0)
107	104, 106	zexpcld 14141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑣)) ∈ ℤ)
108	11	nnne0d 12301	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ≠ 0)
109		dvdsval2 16335	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
110	66, 108, 76, 109	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
111	7, 110	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
112	111	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
113		xp2nd 8025	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑣) ∈ ℕ0)
114	113	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑣) ∈ ℕ0)
115	112, 114	zexpcld 14141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) ∈ ℤ)
116	107, 115	zmulcld 12722	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) ∈ ℤ)
117		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
118		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑙 ∈ V
119	117, 118	op1std 8002	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑣) = 𝑘)
120	119	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑣)) = (𝑃↑𝑘))
121	117, 118	op2ndd 8003	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑣) = 𝑙)
122	121	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
123	120, 122	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
124	123	mpompt 7533	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
125	89, 124	eqtr4i 2791	. . . . . . . . . . 11 ⊢ 𝐸 = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))
126	125	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))))
127		aks6d1c6lem4.20	. . . . . . . . . . 11 ⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
128	127	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
129		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑗 = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
130	116, 126, 128, 129	fmptco 7129	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∘ 𝐸) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
131	130	reseq1d 5979	. . . . . . . 8 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) = ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)))
132		ssidd 3961	. . . . . . . . . 10 ⊢ (𝜑 → (ℕ0 × ℕ0) ⊆ (ℕ0 × ℕ0))
133	132	resmptd 6044	. . . . . . . . 9 ⊢ (𝜑 → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
134	126, 116	fvmpt2d 7007	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑣) = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))
135	134	oveq1d 7434	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
136	135	mpteq2dva 5206	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
137	136	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
138		ovexd 7454	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
139		eqid 2765	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
140	138, 139	fmptd 7113	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V)
141		ffn 6709	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 × ℕ0))
142	140, 141	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 × ℕ0))
143		ovexd 7454	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
144	143, 97	fmptd 7113	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V)
145		ffn 6709	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 × ℕ0))
146	144, 145	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 × ℕ0))
147		eqidd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
148		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → 𝑣 = 𝑐)
149	148	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸‘𝑣) = (𝐸‘𝑐))
150	149	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
151		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑐 ∈ (ℕ0 × ℕ0))
152		ovexd 7454	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153	147, 150, 151, 152	fvmptd 7001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
154		eqid 2765	. . . . . . . . . . . . 13 ⊢ ((mulGrp‘𝐾) ↾s 𝑈) = ((mulGrp‘𝐾) ↾s 𝑈)
155		aks6d1c6lem4.22	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}
156	155	ssrab3 4037	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ⊆ (Base‘(mulGrp‘𝐾))
157	156	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
158	157	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
159	3	fldcrngd 20892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ CRing)
160		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
161	160	crngmgp 20367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
162	159, 161	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
163	162, 5, 155	primrootsunit 42923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
164	163	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
165	93, 164	eleqtrd 2867	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
166	163	simprd 501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
167		ablcmn 19901	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
168	166, 167	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
169	5	nnnn0d 12580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
170		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
171	168, 169, 170	isprimroot 42918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤))))
172	171	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤))))
173	165, 172	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑤 ∈ ℕ0 ((𝑤(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑤)))
174	173	simp1d 1160	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
175		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
176	154, 175	ressbas2 17320	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ⊆ (Base‘(mulGrp‘𝐾)) → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
177	157, 176	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
178	174, 177	eleqtrrd 2868	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
179	178	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ 𝑈)
180	6, 4, 7, 89	aks6d1c2p1 42943	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
181	180	ffvelcdmda 7083	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑐) ∈ ℕ)
182	154, 158, 179, 181	ressmulgnnd 19188	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
183		eqidd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
184		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → 𝑗 = 𝑐)
185	184	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸‘𝑗) = (𝐸‘𝑐))
186	185	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
187		ovexd 7454	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
188	183, 186, 151, 187	fvmptd 7001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
189	188	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐))
190	153, 182, 189	3eqtrd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐))
191	142, 146, 190	eqfnfvd 7032	. . . . . . . . . 10 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
192	137, 191	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
193	133, 192	eqtrd 2800	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
194	131, 193	eqtrd 2800	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
195	194	imaeq1d 6063	. . . . . 6 ⊢ (𝜑 → (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
196	103, 195	eqtrd 2800	. . . . 5 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
197	100, 196	eqtrid 2812	. . . 4 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
198	197	fveq2d 6889	. . 3 ⊢ (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
199	98, 198	breqtrd 5139	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
200	1, 2, 3, 4, 5, 6, 7, 8, 87, 88, 55, 89, 90, 91, 92, 93, 94, 95, 96, 97, 199	aks6d1c6lem3 42997	1 ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111  {copab 5175   ↦ cmpt 5194   × cxp 5661   ↾ cres 5665   “ cima 5666   ∘ ccom 5667   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991   ↑_m cmap 8830  ℂcc 11113  ℝcr 11114  0cc0 11115  1c1 11116   + caddc 11118   · cmul 11120   < clt 11258   ≤ cle 11259   − cmin 11456   / cdiv 11886  ℕcn 12248  2c2 12310  ℕ₀cn0 12519  ℤcz 12606  ...cfz 13551  ⌊cfl 13841  ↑cexp 14115  Ccbc 14356  ♯chash 14384  √csqrt 15308  Σcsu 15761   ∥ cdvds 16332   gcd cgcd 16574  ℙcprime 16751  ϕcphi 16845  Basecbs 17291   ↾_s cress 17312  +_gcplusg 17332  0_gc0g 17514   Σ_g cgsu 17515  ._gcmg 19177  CMndccmn 19894  Abelcabl 19895  mulGrpcmgp 20260  CRingccrg 20360   RingIso crs 20598  Fieldcfield 20878  ℤRHomczrh 21699  chrcchr 21701  ℤ/nℤczn 21702  algSccascl 22052  var₁cv1 22386  Poly₁cpl1 22387  eval₁ce1 22524   log_b clogb 26980   PrimRoots cprimroots 42916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193  ax-addf 11194  ax-mulf 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-ofr 7685  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-dju 9903  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-xnn0 12593  df-z 12607  df-dec 12728  df-uz 12879  df-q 12989  df-rp 13033  df-xneg 13153  df-xadd 13154  df-xmul 13155  df-ioo 13392  df-ioc 13393  df-ico 13394  df-icc 13395  df-fz 13552  df-fzo 13700  df-fl 13843  df-mod 13921  df-seq 14056  df-exp 14116  df-fac 14328  df-bc 14357  df-hash 14385  df-shft 15128  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-limsup 15546  df-clim 15563  df-rlim 15564  df-sum 15762  df-ef 16143  df-sin 16145  df-cos 16146  df-pi 16148  df-dvds 16333  df-gcd 16575  df-prm 16752  df-phi 16847  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-starv 17347  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-unif 17355  df-hom 17356  df-cco 17357  df-rest 17497  df-topn 17498  df-0g 17516  df-gsum 17517  df-topgen 17518  df-pt 17519  df-prds 17522  df-pws 17524  df-xrs 17578  df-qtop 17583  df-imas 17584  df-qus 17585  df-xps 17586  df-mre 17660  df-mrc 17661  df-acs 17663  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-mhm 18878  df-submnd 18879  df-grp 19047  df-minusg 19048  df-sbg 19049  df-mulg 19178  df-subg 19233  df-nsg 19234  df-eqg 19235  df-ghm 19328  df-cntz 19431  df-od 19642  df-cmn 19896  df-abl 19897  df-mgp 20261  df-rng 20275  df-ur 20308  df-srg 20313  df-ring 20361  df-cring 20362  df-oppr 20465  df-dvdsr 20485  df-unit 20486  df-invr 20516  df-dvr 20529  df-rhm 20600  df-rim 20601  df-nzr 20660  df-subrng 20695  df-subrg 20719  df-rlreg 20843  df-domn 20844  df-idom 20845  df-drng 20879  df-field 20880  df-lmod 21033  df-lss 21103  df-lsp 21143  df-sra 21344  df-rgmod 21345  df-lidl 21382  df-rsp 21383  df-2idl 21439  df-psmet 21564  df-xmet 21565  df-met 21566  df-bl 21567  df-mopn 21568  df-fbas 21569  df-fg 21570  df-cnfld 21573  df-zring 21647  df-zrh 21703  df-chr 21705  df-zn 21706  df-assa 22053  df-asp 22054  df-ascl 22055  df-psr 22109  df-mvr 22110  df-mpl 22111  df-opsr 22113  df-evls 22275  df-evl 22276  df-psr1 22390  df-vr1 22391  df-ply1 22392  df-coe1 22393  df-evl1 22526  df-top 23101  df-topon 23118  df-topsp 23140  df-bases 23153  df-cld 23226  df-ntr 23227  df-cls 23228  df-nei 23305  df-lp 23343  df-perf 23344  df-cn 23434  df-cnp 23435  df-haus 23522  df-tx 23770  df-hmeo 23963  df-fil 24054  df-fm 24146  df-flim 24147  df-flf 24148  df-xms 24528  df-ms 24529  df-tms 24530  df-cncf 25088  df-limc 26076  df-dv 26077  df-mdeg 26263  df-deg1 26264  df-mon1 26339  df-uc1p 26340  df-q1p 26341  df-r1p 26342  df-log 26772  df-logb 26981  df-primroots 42917

This theorem is used by:  aks6d1c6lem5  43002