Theorem aks6d1c6lem4 42427

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 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝑃) → 𝐴 < 𝑃)
10		prmnn 16601	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
11	4, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
12	11	nnred 12160	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13		aks6d1c6lem4.11	. . . . . . . . 9 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
14	5	phicld 16699	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
15	14	nnred 12160	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
16	14	nnnn0d 12462	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ0)
17	16	nn0ge0d 12465	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
18	15, 17	resqrtcld 15341	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
19		2re 12219	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
21		2pos 12248	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
23	6	nnred 12160	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
24	6	nngt0d 12194	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
25		1red 11133	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
26		1lt2 12311	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
27	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
28	25, 27	ltned 11269	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
29	28	necomd 2987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
30	20, 22, 23, 24, 29	relogbcld 42227	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ)
31	18, 30	remulcld 11162	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ)
32	31	flcld 13718	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ)
33	15, 17	sqrtge0d 15344	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
34	20	recnd 11160	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℂ)
35	22	gt0ne0d 11701	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 0)
36		logb1 26735	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0)
37	34, 35, 29, 36	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 logb 1) = 0)
38	37	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 = (2 logb 1))
39		2z 12523	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℤ)
41	20	leidd 11703	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ 2)
42		0lt1 11659	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
44	6	nnge1d 12193	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑁)
45	40, 41, 25, 43, 23, 24, 44	logblebd 42230	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 logb 1) ≤ (2 logb 𝑁))
46	38, 45	eqbrtrd 5120	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 logb 𝑁))
47	18, 30, 33, 46	mulge0d 11714	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
48		0zd 12500	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
49		flge 13725	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
51	47, 50	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))
52	32, 51	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
53		elnn0z 12501	. . . . . . . . . 10 ⊢ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0 ↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
54	52, 53	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0)
55	13, 54	eqeltrid 2840	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
56	55	nn0red 12463	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	12, 56	lenltd 11279	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑃))
58	57	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃 ≤ 𝐴)
59		oveq1 7365	. . . . . . . . 9 ⊢ (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁))
60	59	eqeq1d 2738	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
63		1zzd 12522	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ∈ ℤ)
64	13, 32	eqeltrid 2840	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
65	64	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝐴 ∈ ℤ)
66	11	nnzd 12514	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	66	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ ℤ)
68	11	nnge1d 12193	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ 𝑃)
69	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ≤ 𝑃)
70		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ≤ 𝐴)
71	63, 65, 67, 69, 70	elfzd 13431	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ (1...𝐴))
72	60, 62, 71	rspcdva 3577	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → (𝑃 gcd 𝑁) = 1)
73	72	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1))
74	73	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 ≤ 𝐴 → (𝑃 gcd 𝑁) = 1))
75	58, 74	mpd 15	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) = 1)
76	6	nnzd 12514	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
77		coprm 16638	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
78	4, 76, 77	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1))
79	78	con1bid 355	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝑃 gcd 𝑁) = 1 ↔ 𝑃 ∥ 𝑁))
80	79	bicomd 223	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ ¬ (𝑃 gcd 𝑁) = 1))
81	80	biimpd 229	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ 𝑁 → ¬ (𝑃 gcd 𝑁) = 1))
82	7, 81	mpd 15	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑃 gcd 𝑁) = 1)
83	82	neqned 2939	. . . . . 6 ⊢ (𝜑 → (𝑃 gcd 𝑁) ≠ 1)
84	83	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1)
85	84	neneqd 2937	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → ¬ (𝑃 gcd 𝑁) = 1)
86	75, 85	pm2.21dd 195	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝐴 < 𝑃)
87	9, 86	pm2.61dan 812	. 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 2736	. 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 6209	. . . . . 6 ⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))
100	99	eqcomi 2745	. . . . 5 ⊢ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
101		resima 5974	. . . . . . . 8 ⊢ (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
102	101	eqcomi 2745	. . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
105		xp1st 7965	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (1st ‘𝑣) ∈ ℕ0)
106	105	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑣) ∈ ℕ0)
107	104, 106	zexpcld 14010	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑣)) ∈ ℤ)
108	11	nnne0d 12195	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ≠ 0)
109		dvdsval2 16182	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
110	66, 108, 76, 109	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
111	7, 110	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
112	111	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
113		xp2nd 7966	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑣) ∈ ℕ0)
114	113	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑣) ∈ ℕ0)
115	112, 114	zexpcld 14010	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) ∈ ℤ)
116	107, 115	zmulcld 12602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) ∈ ℤ)
117		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
118		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑙 ∈ V
119	117, 118	op1std 7943	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑣) = 𝑘)
120	119	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑣)) = (𝑃↑𝑘))
121	117, 118	op2ndd 7944	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑣) = 𝑙)
122	121	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
123	120, 122	oveq12d 7376	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
124	123	mpompt 7472	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
125	89, 124	eqtr4i 2762	. . . . . . . . . . 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 7365	. . . . . . . . . 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 7074	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∘ 𝐸) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
131	130	reseq1d 5937	. . . . . . . 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 3957	. . . . . . . . . 10 ⊢ (𝜑 → (ℕ0 × ℕ0) ⊆ (ℕ0 × ℕ0))
133	132	resmptd 5999	. . . . . . . . 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 6954	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑣) = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))
135	134	oveq1d 7373	. . . . . . . . . . . 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 5191	. . . . . . . . . . 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 2742	. . . . . . . . . 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 7393	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
139		eqid 2736	. . . . . . . . . . . . 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 7059	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V)
141		ffn 6662	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 × ℕ0))
143		ovexd 7393	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
144	143, 97	fmptd 7059	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V)
145		ffn 6662	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 × ℕ0))
147		eqidd 2737	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → 𝑣 = 𝑐)
149	148	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸‘𝑣) = (𝐸‘𝑐))
150	149	oveq1d 7373	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑐 ∈ (ℕ0 × ℕ0))
152		ovexd 7393	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153	147, 150, 151, 152	fvmptd 6948	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 4034	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ⊆ (Base‘(mulGrp‘𝐾))
157	156	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
158	157	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
159	3	fldcrngd 20675	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ CRing)
160		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
161	160	crngmgp 20176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
162	159, 161	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
163	162, 5, 155	primrootsunit 42352	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
164	163	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
165	93, 164	eleqtrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
166	163	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
167		ablcmn 19716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
168	166, 167	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
169	5	nnnn0d 12462	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
170		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
171	168, 169, 170	isprimroot 42347	. . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
175		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
176	154, 175	ressbas2 17165	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ⊆ (Base‘(mulGrp‘𝐾)) → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
177	157, 176	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
178	174, 177	eleqtrrd 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
179	178	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ 𝑈)
180	6, 4, 7, 89	aks6d1c2p1 42372	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
181	180	ffvelcdmda 7029	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑐) ∈ ℕ)
182	154, 158, 179, 181	ressmulgnnd 19008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
183		eqidd 2737	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → 𝑗 = 𝑐)
185	184	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸‘𝑗) = (𝐸‘𝑐))
186	185	oveq1d 7373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
187		ovexd 7393	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
188	183, 186, 151, 187	fvmptd 6948	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
189	188	eqcomd 2742	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 6979	. . . . . . . . . 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 2771	. . . . . . . . 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 2771	. . . . . . . 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 2771	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
195	194	imaeq1d 6018	. . . . . 6 ⊢ (𝜑 → (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
196	103, 195	eqtrd 2771	. . . . 5 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
197	100, 196	eqtrid 2783	. . . 4 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
198	197	fveq2d 6838	. . 3 ⊢ (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
199	98, 198	breqtrd 5124	. 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 42426	1 ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901  ⟨cop 4586   class class class wbr 5098  {copab 5160   ↦ cmpt 5179   × cxp 5622   ↾ cres 5626   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8763  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   · cmul 11031   < clt 11166   ≤ cle 11167   − cmin 11364   / cdiv 11794  ℕcn 12145  2c2 12200  ℕ₀cn0 12401  ℤcz 12488  ...cfz 13423  ⌊cfl 13710  ↑cexp 13984  Ccbc 14225  ♯chash 14253  √csqrt 15156  Σcsu 15609   ∥ cdvds 16179   gcd cgcd 16421  ℙcprime 16598  ϕcphi 16691  Basecbs 17136   ↾_s cress 17157  +_gcplusg 17177  0_gc0g 17359   Σ_g cgsu 17360  ._gcmg 18997  CMndccmn 19709  Abelcabl 19710  mulGrpcmgp 20075  CRingccrg 20169   RingIso crs 20406  Fieldcfield 20663  ℤRHomczrh 21454  chrcchr 21456  ℤ/nℤczn 21457  algSccascl 21807  var₁cv1 22116  Poly₁cpl1 22117  eval₁ce1 22258   log_b clogb 26730   PrimRoots cprimroots 42345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105  ax-mulf 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ioc 13266  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-seq 13925  df-exp 13985  df-fac 14197  df-bc 14226  df-hash 14254  df-shft 14990  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-limsup 15394  df-clim 15411  df-rlim 15412  df-sum 15610  df-ef 15990  df-sin 15992  df-cos 15993  df-pi 15995  df-dvds 16180  df-gcd 16422  df-prm 16599  df-phi 16693  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-pws 17369  df-xrs 17423  df-qtop 17428  df-imas 17429  df-qus 17430  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-nsg 19054  df-eqg 19055  df-ghm 19142  df-cntz 19246  df-od 19457  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-dvr 20337  df-rhm 20408  df-rim 20409  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-rlreg 20627  df-domn 20628  df-idom 20629  df-drng 20664  df-field 20665  df-lmod 20813  df-lss 20883  df-lsp 20923  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-rsp 21164  df-2idl 21205  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-fbas 21306  df-fg 21307  df-cnfld 21310  df-zring 21402  df-zrh 21458  df-chr 21460  df-zn 21461  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evl1 22260  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-lp 23080  df-perf 23081  df-cn 23171  df-cnp 23172  df-haus 23259  df-tx 23506  df-hmeo 23699  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25823  df-dv 25824  df-mdeg 26016  df-deg1 26017  df-mon1 26092  df-uc1p 26093  df-q1p 26094  df-r1p 26095  df-log 26521  df-logb 26731  df-primroots 42346

This theorem is referenced by:  aks6d1c6lem5  42431