Theorem aks6d1c6lem4 42825

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 489	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝑃) → 𝐴 < 𝑃)
10		prmnn 16728	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
11	4, 10	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
12	11	nnred 12244	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13		aks6d1c6lem4.11	. . . . . . . . 9 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
14	5	phicld 16827	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
15	14	nnred 12244	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
16	14	nnnn0d 12561	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ0)
17	16	nn0ge0d 12564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
18	15, 17	resqrtcld 15465	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
19		2re 12311	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
21		2pos 12341	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
23	6	nnred 12244	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
24	6	nngt0d 12281	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
25		1red 11205	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
26		1lt2 12409	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
27	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
28	25, 27	ltned 11342	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
29	28	necomd 3019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
30	20, 22, 23, 24, 29	relogbcld 42626	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ)
31	18, 30	remulcld 11235	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ)
32	31	flcld 13827	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ)
33	15, 17	sqrtge0d 15468	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
34	20	recnd 11233	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℂ)
35	22	gt0ne0d 11774	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 0)
36		logb1 26896	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0)
37	34, 35, 29, 36	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 logb 1) = 0)
38	37	eqcomd 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 = (2 logb 1))
39		2z 12622	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℤ)
41	20	leidd 11776	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ 2)
42		0lt1 11732	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
44	6	nnge1d 12280	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑁)
45	40, 41, 25, 43, 23, 24, 44	logblebd 42629	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 logb 1) ≤ (2 logb 𝑁))
46	38, 45	eqbrtrd 5134	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 logb 𝑁))
47	18, 30, 33, 46	mulge0d 11787	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
48		0zd 12599	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
49		flge 13834	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 520	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
53		elnn0z 12600	. . . . . . . . . 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 2873	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
56	55	nn0red 12562	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	12, 56	lenltd 11352	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑃))
58	57	biimpar 482	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃 ≤ 𝐴)
59		oveq1 7415	. . . . . . . . 9 ⊢ (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁))
60	59	eqeq1d 2771	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
63		1zzd 12621	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ∈ ℤ)
64	13, 32	eqeltrid 2873	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
65	64	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝐴 ∈ ℤ)
66	11	nnzd 12613	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	66	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ ℤ)
68	11	nnge1d 12280	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ 𝑃)
69	68	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ≤ 𝑃)
70		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ≤ 𝐴)
71	63, 65, 67, 69, 70	elfzd 13539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ (1...𝐴))
72	60, 62, 71	rspcdva 3591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → (𝑃 gcd 𝑁) = 1)
73	72	ex 417	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1))
74	73	adantr 485	. . . . 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 12613	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
77		coprm 16766	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
78	4, 76, 77	syl2anc 595	. . . . . . . . . . 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 2971	. . . . . 6 ⊢ (𝜑 → (𝑃 gcd 𝑁) ≠ 1)
84	83	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1)
85	84	neneqd 2969	. . . 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 824	. 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 2769	. 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 6250	. . . . . 6 ⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))
100	99	eqcomi 2778	. . . . 5 ⊢ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
101		resima 6012	. . . . . . . 8 ⊢ (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
102	101	eqcomi 2778	. . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
105		xp1st 8014	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (1st ‘𝑣) ∈ ℕ0)
106	105	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑣) ∈ ℕ0)
107	104, 106	zexpcld 14119	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑣)) ∈ ℤ)
108	11	nnne0d 12282	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ≠ 0)
109		dvdsval2 16309	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
110	66, 108, 76, 109	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
111	7, 110	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
112	111	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
113		xp2nd 8015	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑣) ∈ ℕ0)
114	113	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑣) ∈ ℕ0)
115	112, 114	zexpcld 14119	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) ∈ ℤ)
116	107, 115	zmulcld 12702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) ∈ ℤ)
117		vex 3467	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
118		vex 3467	. . . . . . . . . . . . . . . 16 ⊢ 𝑙 ∈ V
119	117, 118	op1std 7992	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑣) = 𝑘)
120	119	oveq2d 7424	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑣)) = (𝑃↑𝑘))
121	117, 118	op2ndd 7993	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑣) = 𝑙)
122	121	oveq2d 7424	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
123	120, 122	oveq12d 7426	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
124	123	mpompt 7522	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
125	89, 124	eqtr4i 2795	. . . . . . . . . . 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 7415	. . . . . . . . . 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 7123	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∘ 𝐸) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
131	130	reseq1d 5975	. . . . . . . 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 3968	. . . . . . . . . 10 ⊢ (𝜑 → (ℕ0 × ℕ0) ⊆ (ℕ0 × ℕ0))
133	132	resmptd 6040	. . . . . . . . 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 7001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑣) = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))
135	134	oveq1d 7423	. . . . . . . . . . . 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 5205	. . . . . . . . . . 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 2775	. . . . . . . . . 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 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
139		eqid 2769	. . . . . . . . . . . . 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 7107	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V)
141		ffn 6703	. . . . . . . . . . . 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 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
144	143, 97	fmptd 7107	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V)
145		ffn 6703	. . . . . . . . . . . 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 2770	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → 𝑣 = 𝑐)
149	148	fveq2d 6883	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸‘𝑣) = (𝐸‘𝑐))
150	149	oveq1d 7423	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑐 ∈ (ℕ0 × ℕ0))
152		ovexd 7443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153	147, 150, 151, 152	fvmptd 6995	. . . . . . . . . . . 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 2769	. . . . . . . . . . . . 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 4044	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ⊆ (Base‘(mulGrp‘𝐾))
157	156	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
158	157	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
159	3	fldcrngd 20822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ CRing)
160		eqid 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
161	160	crngmgp 20319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
162	159, 161	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
163	162, 5, 155	primrootsunit 42750	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
164	163	simpld 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
165	93, 164	eleqtrd 2871	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
166	163	simprd 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
167		ablcmn 19853	. . . . . . . . . . . . . . . . . . . 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 12561	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
170		eqid 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
171	168, 169, 170	isprimroot 42745	. . . . . . . . . . . . . . . . . 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 1158	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
175		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
176	154, 175	ressbas2 17294	. . . . . . . . . . . . . . . 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 2872	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
179	178	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ 𝑈)
180	6, 4, 7, 89	aks6d1c2p1 42770	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
181	180	ffvelcdmda 7077	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑐) ∈ ℕ)
182	154, 158, 179, 181	ressmulgnnd 19140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
183		eqidd 2770	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → 𝑗 = 𝑐)
185	184	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸‘𝑗) = (𝐸‘𝑐))
186	185	oveq1d 7423	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
187		ovexd 7443	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
188	183, 186, 151, 187	fvmptd 6995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
189	188	eqcomd 2775	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 7026	. . . . . . . . . 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 2804	. . . . . . . . 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 2804	. . . . . . . 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 2804	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
195	194	imaeq1d 6059	. . . . . 6 ⊢ (𝜑 → (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
196	103, 195	eqtrd 2804	. . . . 5 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
197	100, 196	eqtrid 2816	. . . 4 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
198	197	fveq2d 6883	. . 3 ⊢ (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
199	98, 198	breqtrd 5138	. 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 42824	1 ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ⟨cop 4597   class class class wbr 5110  {copab 5174   ↦ cmpt 5193   × cxp 5657   ↾ cres 5661   “ cima 5662   ∘ ccom 5663   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7980  2^nd c2nd 7981   ↑_m cmap 8820  ℂcc 11094  ℝcr 11095  0cc0 11096  1c1 11097   + caddc 11099   · cmul 11101   < clt 11239   ≤ cle 11240   − cmin 11437   / cdiv 11867  ℕcn 12229  2c2 12291  ℕ₀cn0 12500  ℤcz 12587  ...cfz 13531  ⌊cfl 13819  ↑cexp 14093  Ccbc 14334  ♯chash 14362  √csqrt 15280  Σcsu 15733   ∥ cdvds 16306   gcd cgcd 16548  ℙcprime 16725  ϕcphi 16819  Basecbs 17265   ↾_s cress 17286  +_gcplusg 17306  0_gc0g 17488   Σ_g cgsu 17489  ._gcmg 19129  CMndccmn 19846  Abelcabl 19847  mulGrpcmgp 20212  CRingccrg 20312   RingIso crs 20548  Fieldcfield 20810  ℤRHomczrh 21614  chrcchr 21616  ℤ/nℤczn 21617  algSccascl 21967  var₁cv1 22301  Poly₁cpl1 22302  eval₁ce1 22439   log_b clogb 26891   PrimRoots cprimroots 42743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174  ax-addf 11175  ax-mulf 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-ofr 7673  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-dju 9883  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-xnn0 12574  df-z 12588  df-dec 12708  df-uz 12859  df-q 12969  df-rp 13013  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13372  df-ioc 13373  df-ico 13374  df-icc 13375  df-fz 13532  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-fac 14306  df-bc 14335  df-hash 14363  df-shft 15100  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-limsup 15518  df-clim 15535  df-rlim 15536  df-sum 15734  df-ef 16117  df-sin 16119  df-cos 16120  df-pi 16122  df-dvds 16307  df-gcd 16549  df-prm 16726  df-phi 16821  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-starv 17321  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-unif 17329  df-hom 17330  df-cco 17331  df-rest 17471  df-topn 17472  df-0g 17490  df-gsum 17491  df-topgen 17492  df-pt 17493  df-prds 17496  df-pws 17498  df-xrs 17552  df-qtop 17557  df-imas 17558  df-qus 17559  df-xps 17560  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-mhm 18837  df-submnd 18838  df-grp 18999  df-minusg 19000  df-sbg 19001  df-mulg 19130  df-subg 19185  df-nsg 19186  df-eqg 19187  df-ghm 19280  df-cntz 19383  df-od 19594  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-srg 20265  df-ring 20313  df-cring 20314  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-dvr 20479  df-rhm 20550  df-rim 20551  df-nzr 20592  df-subrng 20627  df-subrg 20651  df-rlreg 20775  df-domn 20776  df-idom 20777  df-drng 20811  df-field 20812  df-lmod 20957  df-lss 21027  df-lsp 21067  df-sra 21268  df-rgmod 21269  df-lidl 21306  df-rsp 21307  df-2idl 21356  df-psmet 21479  df-xmet 21480  df-met 21481  df-bl 21482  df-mopn 21483  df-fbas 21484  df-fg 21485  df-cnfld 21488  df-zring 21562  df-zrh 21618  df-chr 21620  df-zn 21621  df-assa 21968  df-asp 21969  df-ascl 21970  df-psr 22024  df-mvr 22025  df-mpl 22026  df-opsr 22028  df-evls 22190  df-evl 22191  df-psr1 22305  df-vr1 22306  df-ply1 22307  df-coe1 22308  df-evl1 22441  df-top 23016  df-topon 23033  df-topsp 23055  df-bases 23068  df-cld 23141  df-ntr 23142  df-cls 23143  df-nei 23220  df-lp 23258  df-perf 23259  df-cn 23349  df-cnp 23350  df-haus 23437  df-tx 23684  df-hmeo 23877  df-fil 23968  df-fm 24060  df-flim 24061  df-flf 24062  df-xms 24442  df-ms 24443  df-tms 24444  df-cncf 25002  df-limc 25990  df-dv 25991  df-mdeg 26177  df-deg1 26178  df-mon1 26253  df-uc1p 26254  df-q1p 26255  df-r1p 26256  df-log 26683  df-logb 26892  df-primroots 42744

This theorem is referenced by:  aks6d1c6lem5  42829