Theorem aks6d1c6lem4 42154

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   𝐴,𝑎   𝐴,𝑏   𝐴,𝑔,𝑖,𝑥   𝐴,ℎ,𝑗   𝐴,𝑠,𝑡   𝐷,𝑠   𝑗,𝐸   𝑒,𝐸,𝑓,𝑦   𝑥,𝐸,𝑦   𝑒,𝐺,𝑓,𝑦   𝑔,𝐺,𝑖,𝑦   ℎ,𝐺   𝑡,𝐺,𝑖,𝑦   𝐻,𝑎   𝑔,𝐻,𝑖,𝑥,𝑦   ℎ,𝐻,𝑗   𝐻,𝑠,𝑡   𝐾,𝑎   𝑗,𝐾   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖,𝑥   ℎ,𝐾   𝑚,𝐾,𝑛   𝑡,𝐾,𝑥   𝑗,𝑀,ℎ   𝑦,𝑀   𝑁,𝑎   𝑁,𝑏   𝑒,𝑁,𝑓   𝑗,𝑁   𝑘,𝑁,𝑙,𝑠   𝑥,𝑁,𝑘,𝑙   𝑃,𝑏   𝑃,𝑒,𝑓   𝑃,𝑗   𝑃,𝑘,𝑙,𝑠   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝑆,𝑎   𝑆,𝑔,𝑖,𝑥,𝑦   𝑆,ℎ,𝑗   𝑆,𝑠,𝑡   𝑈,𝑗   𝜑,𝑎   𝜑,𝑗   𝜑,𝑔,𝑖,𝑥,𝑦   𝜑,ℎ   𝜑,𝑘,𝑙,𝑠   𝑦,𝑘,𝑙   𝜑,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 𝑐 𝑣 𝑤 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 16707	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
11	4, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
12	11	nnred 12278	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
13		aks6d1c6lem4.11	. . . . . . . . 9 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
14	5	phicld 16805	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
15	14	nnred 12278	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
16	14	nnnn0d 12584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ0)
17	16	nn0ge0d 12587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
18	15, 17	resqrtcld 15452	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
19		2re 12337	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
21		2pos 12366	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
23	6	nnred 12278	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
24	6	nngt0d 12312	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
25		1red 11259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
26		1lt2 12434	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
27	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
28	25, 27	ltned 11394	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
29	28	necomd 2993	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
30	20, 22, 23, 24, 29	relogbcld 41954	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ)
31	18, 30	remulcld 11288	. . . . . . . . . . . 12 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ)
32	31	flcld 13834	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ)
33	15, 17	sqrtge0d 15455	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
34	20	recnd 11286	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℂ)
35	22	gt0ne0d 11824	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≠ 0)
36		logb1 26826	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0)
37	34, 35, 29, 36	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 logb 1) = 0)
38	37	eqcomd 2740	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 = (2 logb 1))
39		2z 12646	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℤ)
41	20	leidd 11826	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ 2)
42		0lt1 11782	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 1)
44	6	nnge1d 12311	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑁)
45	40, 41, 25, 43, 23, 24, 44	logblebd 41957	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 logb 1) ≤ (2 logb 𝑁))
46	38, 45	eqbrtrd 5169	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 logb 𝑁))
47	18, 30, 33, 46	mulge0d 11837	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
48		0zd 12622	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
49		flge 13841	. . . . . . . . . . . . 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 12623	. . . . . . . . . 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 2842	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
56	55	nn0red 12585	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	12, 56	lenltd 11404	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑃))
58	57	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃 ≤ 𝐴)
59		oveq1 7437	. . . . . . . . 9 ⊢ (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁))
60	59	eqeq1d 2736	. . . . . . . 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 12645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 1 ∈ ℤ)
64	13, 32	eqeltrid 2842	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℤ)
65	64	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝐴 ∈ ℤ)
66	11	nnzd 12637	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
67	66	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ ℤ)
68	11	nnge1d 12311	. . . . . . . . . 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 13551	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≤ 𝐴) → 𝑃 ∈ (1...𝐴))
72	60, 62, 71	rspcdva 3622	. . . . . . 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 12637	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
77		coprm 16744	. . . . . . . . . . . 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 2944	. . . . . 6 ⊢ (𝜑 → (𝑃 gcd 𝑁) ≠ 1)
84	83	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1)
85	84	neneqd 2942	. . . 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 813	. 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 2734	. 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 6272	. . . . . 6 ⊢ ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))
100	99	eqcomi 2743	. . . . 5 ⊢ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
101		resima 6034	. . . . . . . 8 ⊢ (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0))
102	101	eqcomi 2743	. . . . . . 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 8044	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (1st ‘𝑣) ∈ ℕ0)
106	105	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑣) ∈ ℕ0)
107	104, 106	zexpcld 14124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑣)) ∈ ℤ)
108	11	nnne0d 12313	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ≠ 0)
109		dvdsval2 16289	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
110	66, 108, 76, 109	syl3anc 1370	. . . . . . . . . . . . . 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 8045	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑣) ∈ ℕ0)
114	113	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑣) ∈ ℕ0)
115	112, 114	zexpcld 14124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) ∈ ℤ)
116	107, 115	zmulcld 12725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) ∈ ℤ)
117		vex 3481	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
118		vex 3481	. . . . . . . . . . . . . . . 16 ⊢ 𝑙 ∈ V
119	117, 118	op1std 8022	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑣) = 𝑘)
120	119	oveq2d 7446	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑣)) = (𝑃↑𝑘))
121	117, 118	op2ndd 8023	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑣) = 𝑙)
122	121	oveq2d 7446	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
123	120, 122	oveq12d 7448	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
124	123	mpompt 7546	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
125	89, 124	eqtr4i 2765	. . . . . . . . . . 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 7437	. . . . . . . . . 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 7148	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∘ 𝐸) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
131	130	reseq1d 5998	. . . . . . . 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 4018	. . . . . . . . . 10 ⊢ (𝜑 → (ℕ0 × ℕ0) ⊆ (ℕ0 × ℕ0))
133	132	resmptd 6059	. . . . . . . . 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 7028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑣) = ((𝑃↑(1st ‘𝑣)) · ((𝑁 / 𝑃)↑(2nd ‘𝑣))))
135	134	oveq1d 7445	. . . . . . . . . . . 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 5247	. . . . . . . . . . 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 2740	. . . . . . . . . 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 7465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
139		eqid 2734	. . . . . . . . . . . . 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 7133	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V)
141		ffn 6736	. . . . . . . . . . . 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 7465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
144	143, 97	fmptd 7133	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V)
145		ffn 6736	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸‘𝑣) = (𝐸‘𝑐))
150	149	oveq1d 7445	. . . . . . . . . . . . 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 7465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153	147, 150, 151, 152	fvmptd 7022	. . . . . . . . . . . 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 2734	. . . . . . . . . . . . 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 4091	. . . . . . . . . . . . . . 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 20758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ CRing)
160		eqid 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
161	160	crngmgp 20258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
162	159, 161	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
163	162, 5, 155	primrootsunit 42079	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
166	163	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
167		ablcmn 19819	. . . . . . . . . . . . . . . . . . . 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 12584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
170		eqid 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
171	168, 169, 170	isprimroot 42074	. . . . . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
175		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
176	154, 175	ressbas2 17282	. . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
179	178	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ 𝑈)
180	6, 4, 7, 89	aks6d1c2p1 42099	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
181	180	ffvelcdmda 7103	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑐) ∈ ℕ)
182	154, 158, 179, 181	ressmulgnnd 19108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
183		eqidd 2735	. . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸‘𝑗) = (𝐸‘𝑐))
186	185	oveq1d 7445	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
187		ovexd 7465	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
188	183, 186, 151, 187	fvmptd 7022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸‘𝑐)(.g‘(mulGrp‘𝐾))𝑀))
189	188	eqcomd 2740	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 7053	. . . . . . . . . 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 2774	. . . . . . . . 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 2774	. . . . . . . 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 2774	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
195	194	imaeq1d 6078	. . . . . 6 ⊢ (𝜑 → (((𝐽 ∘ 𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
196	103, 195	eqtrd 2774	. . . . 5 ⊢ (𝜑 → ((𝐽 ∘ 𝐸) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
197	100, 196	eqtrid 2786	. . . 4 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
198	197	fveq2d 6910	. . 3 ⊢ (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
199	98, 198	breqtrd 5173	. 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 42153	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 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3432  Vcvv 3477   ⊆ wss 3962  ⟨cop 4636   class class class wbr 5147  {copab 5209   ↦ cmpt 5230   × cxp 5686   ↾ cres 5690   “ cima 5691   ∘ ccom 5692   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8010  2^nd c2nd 8011   ↑_m cmap 8864  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292   ≤ cle 11293   − cmin 11489   / cdiv 11917  ℕcn 12263  2c2 12318  ℕ₀cn0 12523  ℤcz 12610  ...cfz 13543  ⌊cfl 13826  ↑cexp 14098  Ccbc 14337  ♯chash 14365  √csqrt 15268  Σcsu 15718   ∥ cdvds 16286   gcd cgcd 16527  ℙcprime 16704  ϕcphi 16797  Basecbs 17244   ↾_s cress 17273  +_gcplusg 17297  0_gc0g 17485   Σ_g cgsu 17486  ._gcmg 19097  CMndccmn 19812  Abelcabl 19813  mulGrpcmgp 20151  CRingccrg 20251   RingIso crs 20486  Fieldcfield 20746  ℤRHomczrh 21527  chrcchr 21529  ℤ/nℤczn 21530  algSccascl 21889  var₁cv1 22192  Poly₁cpl1 22193  eval₁ce1 22333   log_b clogb 26821   PrimRoots cprimroots 42072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-ef 16099  df-sin 16101  df-cos 16102  df-pi 16104  df-dvds 16287  df-gcd 16528  df-prm 16705  df-phi 16799  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-pws 17495  df-xrs 17548  df-qtop 17553  df-imas 17554  df-qus 17555  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-nsg 19154  df-eqg 19155  df-ghm 19243  df-cntz 19347  df-od 19560  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-rhm 20488  df-rim 20489  df-nzr 20529  df-subrng 20562  df-subrg 20586  df-rlreg 20710  df-domn 20711  df-idom 20712  df-drng 20747  df-field 20748  df-lmod 20876  df-lss 20947  df-lsp 20987  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-rsp 21236  df-2idl 21277  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-chr 21533  df-zn 21534  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-evls 22115  df-evl 22116  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-evl1 22335  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-mdeg 26108  df-deg1 26109  df-mon1 26184  df-uc1p 26185  df-q1p 26186  df-r1p 26187  df-log 26612  df-logb 26822  df-primroots 42073

This theorem is referenced by:  aks6d1c6lem5  42158