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Theorem aks6d1c6lem4 42174
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‘(Poly1𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘𝑓)‘𝑦)) = (((eval1𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
aks6d1c6lem4.2 𝑃 = (chr‘𝐾)
aks6d1c6lem4.3 (𝜑𝐾 ∈ Field)
aks6d1c6lem4.4 (𝜑𝑃 ∈ ℙ)
aks6d1c6lem4.5 (𝜑𝑅 ∈ ℕ)
aks6d1c6lem4.6 (𝜑𝑁 ∈ ℕ)
aks6d1c6lem4.7 (𝜑𝑃𝑁)
aks6d1c6lem4.8 (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6lem4.9 (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks6d1c6lem4.10 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)(.g‘(mulGrp‘(Poly1𝐾)))((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c6lem4.11 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
aksaks6dlem4.12 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6lem4.13 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6lem4.14 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6lem4.15 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6lem4.16 (𝜑𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6lem4.17 𝐻 = ( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀))
aks6d1c6lem4.18 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
aks6d1c6lem4.19 𝑆 = {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (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 “ (ℕ0m (0...𝐴)))))
Distinct variable groups:   ,𝑎   𝑃,𝑘,𝑙,𝑠   𝜑,   𝑁,𝑠   𝜑,𝑘,𝑙   ,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   ,𝑀,𝑗   𝑥,𝑃   𝑆,,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   ,𝐺   𝑡,𝐺   𝑔,𝐺,𝑖,𝑦   𝐻,𝑠,𝑡   ,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,,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,,𝑖,𝑗,𝑘,𝑚,𝑛,𝑎,𝑏,𝑙)   𝑃(𝑦,𝑡,𝑔,,𝑖,𝑚,𝑛,𝑎)   (𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑚,𝑛,𝑠,𝑏,𝑙)   𝑅(𝑡,𝑔,,𝑖,𝑗,𝑘,𝑚,𝑛,𝑠,𝑎,𝑏,𝑙)   𝑆(𝑒,𝑓,𝑘,𝑚,𝑛,𝑏,𝑙)   𝑈(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑘,𝑚,𝑛,𝑠,𝑎,𝑏,𝑙)   𝐸(𝑡,𝑔,,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,,𝑖,𝑗,𝑘,𝑚,𝑛,𝑠,𝑎,𝑏,𝑙)   𝐾(𝑘,𝑠,𝑏,𝑙)   𝐿(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,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,,𝑖,𝑚,𝑛)

Proof of Theorem aks6d1c6lem4
Dummy variables 𝑣 𝑤 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aks6d1c6lem4.1 . 2 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘𝑓)‘𝑦)) = (((eval1𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
2 aks6d1c6lem4.2 . 2 𝑃 = (chr‘𝐾)
3 aks6d1c6lem4.3 . 2 (𝜑𝐾 ∈ Field)
4 aks6d1c6lem4.4 . 2 (𝜑𝑃 ∈ ℙ)
5 aks6d1c6lem4.5 . 2 (𝜑𝑅 ∈ ℕ)
6 aks6d1c6lem4.6 . 2 (𝜑𝑁 ∈ ℕ)
7 aks6d1c6lem4.7 . 2 (𝜑𝑃𝑁)
8 aks6d1c6lem4.8 . 2 (𝜑 → (𝑁 gcd 𝑅) = 1)
9 simpr 484 . . 3 ((𝜑𝐴 < 𝑃) → 𝐴 < 𝑃)
10 prmnn 16711 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
114, 10syl 17 . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1211nnred 12281 . . . . . . 7 (𝜑𝑃 ∈ ℝ)
13 aks6d1c6lem4.11 . . . . . . . . 9 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
145phicld 16809 . . . . . . . . . . . . . . 15 (𝜑 → (ϕ‘𝑅) ∈ ℕ)
1514nnred 12281 . . . . . . . . . . . . . 14 (𝜑 → (ϕ‘𝑅) ∈ ℝ)
1614nnnn0d 12587 . . . . . . . . . . . . . . 15 (𝜑 → (ϕ‘𝑅) ∈ ℕ0)
1716nn0ge0d 12590 . . . . . . . . . . . . . 14 (𝜑 → 0 ≤ (ϕ‘𝑅))
1815, 17resqrtcld 15456 . . . . . . . . . . . . 13 (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
19 2re 12340 . . . . . . . . . . . . . . 15 2 ∈ ℝ
2019a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 2 ∈ ℝ)
21 2pos 12369 . . . . . . . . . . . . . . 15 0 < 2
2221a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 0 < 2)
236nnred 12281 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℝ)
246nngt0d 12315 . . . . . . . . . . . . . 14 (𝜑 → 0 < 𝑁)
25 1red 11262 . . . . . . . . . . . . . . . 16 (𝜑 → 1 ∈ ℝ)
26 1lt2 12437 . . . . . . . . . . . . . . . . 17 1 < 2
2726a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 1 < 2)
2825, 27ltned 11397 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≠ 2)
2928necomd 2996 . . . . . . . . . . . . . 14 (𝜑 → 2 ≠ 1)
3020, 22, 23, 24, 29relogbcld 41974 . . . . . . . . . . . . 13 (𝜑 → (2 logb 𝑁) ∈ ℝ)
3118, 30remulcld 11291 . . . . . . . . . . . 12 (𝜑 → ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ)
3231flcld 13838 . . . . . . . . . . 11 (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ)
3315, 17sqrtge0d 15459 . . . . . . . . . . . . 13 (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
3420recnd 11289 . . . . . . . . . . . . . . . 16 (𝜑 → 2 ∈ ℂ)
3522gt0ne0d 11827 . . . . . . . . . . . . . . . 16 (𝜑 → 2 ≠ 0)
36 logb1 26812 . . . . . . . . . . . . . . . 16 ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0)
3734, 35, 29, 36syl3anc 1373 . . . . . . . . . . . . . . 15 (𝜑 → (2 logb 1) = 0)
3837eqcomd 2743 . . . . . . . . . . . . . 14 (𝜑 → 0 = (2 logb 1))
39 2z 12649 . . . . . . . . . . . . . . . 16 2 ∈ ℤ
4039a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 2 ∈ ℤ)
4120leidd 11829 . . . . . . . . . . . . . . 15 (𝜑 → 2 ≤ 2)
42 0lt1 11785 . . . . . . . . . . . . . . . 16 0 < 1
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 0 < 1)
446nnge1d 12314 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ 𝑁)
4540, 41, 25, 43, 23, 24, 44logblebd 41977 . . . . . . . . . . . . . 14 (𝜑 → (2 logb 1) ≤ (2 logb 𝑁))
4638, 45eqbrtrd 5165 . . . . . . . . . . . . 13 (𝜑 → 0 ≤ (2 logb 𝑁))
4718, 30, 33, 46mulge0d 11840 . . . . . . . . . . . 12 (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
48 0zd 12625 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ ℤ)
49 flge 13845 . . . . . . . . . . . . 13 ((((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
5031, 48, 49syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 logb 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
5147, 50mpbid 232 . . . . . . . . . . 11 (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))))
5232, 51jca 511 . . . . . . . . . 10 (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
53 elnn0z 12626 . . . . . . . . . 10 ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0 ↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))))
5452, 53sylibr 234 . . . . . . . . 9 (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) ∈ ℕ0)
5513, 54eqeltrid 2845 . . . . . . . 8 (𝜑𝐴 ∈ ℕ0)
5655nn0red 12588 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
5712, 56lenltd 11407 . . . . . 6 (𝜑 → (𝑃𝐴 ↔ ¬ 𝐴 < 𝑃))
5857biimpar 477 . . . . 5 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝑃𝐴)
59 oveq1 7438 . . . . . . . . 9 (𝑏 = 𝑃 → (𝑏 gcd 𝑁) = (𝑃 gcd 𝑁))
6059eqeq1d 2739 . . . . . . . 8 (𝑏 = 𝑃 → ((𝑏 gcd 𝑁) = 1 ↔ (𝑃 gcd 𝑁) = 1))
61 aks6d1c6lem4.9 . . . . . . . . 9 (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
6261adantr 480 . . . . . . . 8 ((𝜑𝑃𝐴) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
63 1zzd 12648 . . . . . . . . 9 ((𝜑𝑃𝐴) → 1 ∈ ℤ)
6413, 32eqeltrid 2845 . . . . . . . . . 10 (𝜑𝐴 ∈ ℤ)
6564adantr 480 . . . . . . . . 9 ((𝜑𝑃𝐴) → 𝐴 ∈ ℤ)
6611nnzd 12640 . . . . . . . . . 10 (𝜑𝑃 ∈ ℤ)
6766adantr 480 . . . . . . . . 9 ((𝜑𝑃𝐴) → 𝑃 ∈ ℤ)
6811nnge1d 12314 . . . . . . . . . 10 (𝜑 → 1 ≤ 𝑃)
6968adantr 480 . . . . . . . . 9 ((𝜑𝑃𝐴) → 1 ≤ 𝑃)
70 simpr 484 . . . . . . . . 9 ((𝜑𝑃𝐴) → 𝑃𝐴)
7163, 65, 67, 69, 70elfzd 13555 . . . . . . . 8 ((𝜑𝑃𝐴) → 𝑃 ∈ (1...𝐴))
7260, 62, 71rspcdva 3623 . . . . . . 7 ((𝜑𝑃𝐴) → (𝑃 gcd 𝑁) = 1)
7372ex 412 . . . . . 6 (𝜑 → (𝑃𝐴 → (𝑃 gcd 𝑁) = 1))
7473adantr 480 . . . . 5 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃𝐴 → (𝑃 gcd 𝑁) = 1))
7558, 74mpd 15 . . . 4 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) = 1)
766nnzd 12640 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
77 coprm 16748 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑃𝑁 ↔ (𝑃 gcd 𝑁) = 1))
784, 76, 77syl2anc 584 . . . . . . . . . . 11 (𝜑 → (¬ 𝑃𝑁 ↔ (𝑃 gcd 𝑁) = 1))
7978con1bid 355 . . . . . . . . . 10 (𝜑 → (¬ (𝑃 gcd 𝑁) = 1 ↔ 𝑃𝑁))
8079bicomd 223 . . . . . . . . 9 (𝜑 → (𝑃𝑁 ↔ ¬ (𝑃 gcd 𝑁) = 1))
8180biimpd 229 . . . . . . . 8 (𝜑 → (𝑃𝑁 → ¬ (𝑃 gcd 𝑁) = 1))
827, 81mpd 15 . . . . . . 7 (𝜑 → ¬ (𝑃 gcd 𝑁) = 1)
8382neqned 2947 . . . . . 6 (𝜑 → (𝑃 gcd 𝑁) ≠ 1)
8483adantr 480 . . . . 5 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → (𝑃 gcd 𝑁) ≠ 1)
8584neneqd 2945 . . . 4 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → ¬ (𝑃 gcd 𝑁) = 1)
8675, 85pm2.21dd 195 . . 3 ((𝜑 ∧ ¬ 𝐴 < 𝑃) → 𝐴 < 𝑃)
879, 86pm2.61dan 813 . 2 (𝜑𝐴 < 𝑃)
88 aks6d1c6lem4.10 . 2 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)(.g‘(mulGrp‘(Poly1𝐾)))((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
89 aksaks6dlem4.12 . 2 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
90 aks6d1c6lem4.13 . 2 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
91 aks6d1c6lem4.14 . 2 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
92 aks6d1c6lem4.15 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
93 aks6d1c6lem4.16 . 2 (𝜑𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
94 aks6d1c6lem4.17 . 2 𝐻 = ( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀))
95 aks6d1c6lem4.18 . 2 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
96 aks6d1c6lem4.19 . 2 𝑆 = {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)}
97 eqid 2737 . 2 (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))
98 aks6d1c6lem4.21 . . 3 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
99 imaco 6271 . . . . . 6 ((𝐽𝐸) “ (ℕ0 × ℕ0)) = (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))
10099eqcomi 2746 . . . . 5 (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝐽𝐸) “ (ℕ0 × ℕ0))
101 resima 6033 . . . . . . . 8 (((𝐽𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝐽𝐸) “ (ℕ0 × ℕ0))
102101eqcomi 2746 . . . . . . 7 ((𝐽𝐸) “ (ℕ0 × ℕ0)) = (((𝐽𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0))
103102a1i 11 . . . . . 6 (𝜑 → ((𝐽𝐸) “ (ℕ0 × ℕ0)) = (((𝐽𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)))
10466adantr 480 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
105 xp1st 8046 . . . . . . . . . . . . 13 (𝑣 ∈ (ℕ0 × ℕ0) → (1st𝑣) ∈ ℕ0)
106105adantl 481 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → (1st𝑣) ∈ ℕ0)
107104, 106zexpcld 14128 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st𝑣)) ∈ ℤ)
10811nnne0d 12316 . . . . . . . . . . . . . . 15 (𝜑𝑃 ≠ 0)
109 dvdsval2 16293 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
11066, 108, 76, 109syl3anc 1373 . . . . . . . . . . . . . 14 (𝜑 → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
1117, 110mpbid 232 . . . . . . . . . . . . 13 (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
112111adantr 480 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
113 xp2nd 8047 . . . . . . . . . . . . 13 (𝑣 ∈ (ℕ0 × ℕ0) → (2nd𝑣) ∈ ℕ0)
114113adantl 481 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → (2nd𝑣) ∈ ℕ0)
115112, 114zexpcld 14128 . . . . . . . . . . 11 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd𝑣)) ∈ ℤ)
116107, 115zmulcld 12728 . . . . . . . . . 10 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣))) ∈ ℤ)
117 vex 3484 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
118 vex 3484 . . . . . . . . . . . . . . . 16 𝑙 ∈ V
119117, 118op1std 8024 . . . . . . . . . . . . . . 15 (𝑣 = ⟨𝑘, 𝑙⟩ → (1st𝑣) = 𝑘)
120119oveq2d 7447 . . . . . . . . . . . . . 14 (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st𝑣)) = (𝑃𝑘))
121117, 118op2ndd 8025 . . . . . . . . . . . . . . 15 (𝑣 = ⟨𝑘, 𝑙⟩ → (2nd𝑣) = 𝑙)
122121oveq2d 7447 . . . . . . . . . . . . . 14 (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd𝑣)) = ((𝑁 / 𝑃)↑𝑙))
123120, 122oveq12d 7449 . . . . . . . . . . . . 13 (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣))) = ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
124123mpompt 7547 . . . . . . . . . . . 12 (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
12589, 124eqtr4i 2768 . . . . . . . . . . 11 𝐸 = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣))))
126125a1i 11 . . . . . . . . . 10 (𝜑𝐸 = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))))
127 aks6d1c6lem4.20 . . . . . . . . . . 11 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
128127a1i 11 . . . . . . . . . 10 (𝜑𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
129 oveq1 7438 . . . . . . . . . 10 (𝑗 = ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣))) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
130116, 126, 128, 129fmptco 7149 . . . . . . . . 9 (𝜑 → (𝐽𝐸) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
131130reseq1d 5996 . . . . . . . 8 (𝜑 → ((𝐽𝐸) ↾ (ℕ0 × ℕ0)) = ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)))
132 ssidd 4007 . . . . . . . . . 10 (𝜑 → (ℕ0 × ℕ0) ⊆ (ℕ0 × ℕ0))
133132resmptd 6058 . . . . . . . . 9 (𝜑 → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
134126, 116fvmpt2d 7029 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → (𝐸𝑣) = ((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣))))
135134oveq1d 7446 . . . . . . . . . . . 12 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
136135mpteq2dva 5242 . . . . . . . . . . 11 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
137136eqcomd 2743 . . . . . . . . . 10 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
138 ovexd 7466 . . . . . . . . . . . . 13 ((𝜑𝑣 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
139 eqid 2737 . . . . . . . . . . . . 13 (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
140138, 139fmptd 7134 . . . . . . . . . . . 12 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V)
141 ffn 6736 . . . . . . . . . . . 12 ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)):(ℕ0 × ℕ0)⟶V → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 × ℕ0))
142140, 141syl 17 . . . . . . . . . . 11 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) Fn (ℕ0 × ℕ0))
143 ovexd 7466 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
144143, 97fmptd 7134 . . . . . . . . . . . 12 (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V)
145 ffn 6736 . . . . . . . . . . . 12 ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)):(ℕ0 × ℕ0)⟶V → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 × ℕ0))
146144, 145syl 17 . . . . . . . . . . 11 (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) Fn (ℕ0 × ℕ0))
147 eqidd 2738 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
148 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → 𝑣 = 𝑐)
149148fveq2d 6910 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → (𝐸𝑣) = (𝐸𝑐))
150149oveq1d 7446 . . . . . . . . . . . . 13 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑣 = 𝑐) → ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
151 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → 𝑐 ∈ (ℕ0 × ℕ0))
152 ovexd 7466 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153147, 150, 151, 152fvmptd 7023 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝐸𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
154 eqid 2737 . . . . . . . . . . . . 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))}
156155ssrab3 4082 . . . . . . . . . . . . . . 15 𝑈 ⊆ (Base‘(mulGrp‘𝐾))
157156a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
158157adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → 𝑈 ⊆ (Base‘(mulGrp‘𝐾)))
1593fldcrngd 20742 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐾 ∈ CRing)
160 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (mulGrp‘𝐾) = (mulGrp‘𝐾)
161160crngmgp 20238 . . . . . . . . . . . . . . . . . . . . 21 (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
162159, 161syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
163162, 5, 155primrootsunit 42099 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
164163simpld 494 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
16593, 164eleqtrd 2843 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
166163simprd 495 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
167 ablcmn 19805 . . . . . . . . . . . . . . . . . . . 20 (((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
168166, 167syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
1695nnnn0d 12587 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑅 ∈ ℕ0)
170 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
171168, 169, 170isprimroot 42094 . . . . . . . . . . . . . . . . . 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)) → 𝑅𝑤))))
172171biimpd 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)) → 𝑅𝑤))))
173165, 172mpd 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)) → 𝑅𝑤)))
174173simp1d 1143 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
175 eqid 2737 . . . . . . . . . . . . . . . . 17 (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
176154, 175ressbas2 17283 . . . . . . . . . . . . . . . 16 (𝑈 ⊆ (Base‘(mulGrp‘𝐾)) → 𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
177157, 176syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
178174, 177eleqtrrd 2844 . . . . . . . . . . . . . 14 (𝜑𝑀𝑈)
179178adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → 𝑀𝑈)
1806, 4, 7, 89aks6d1c2p1 42119 . . . . . . . . . . . . . 14 (𝜑𝐸:(ℕ0 × ℕ0)⟶ℕ)
181180ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → (𝐸𝑐) ∈ ℕ)
182154, 158, 179, 181ressmulgnnd 19096 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑐)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = ((𝐸𝑐)(.g‘(mulGrp‘𝐾))𝑀))
183 eqidd 2738 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
184 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → 𝑗 = 𝑐)
185184fveq2d 6910 . . . . . . . . . . . . . . 15 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → (𝐸𝑗) = (𝐸𝑐))
186185oveq1d 7446 . . . . . . . . . . . . . 14 (((𝜑𝑐 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑐) → ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸𝑐)(.g‘(mulGrp‘𝐾))𝑀))
187 ovexd 7466 . . . . . . . . . . . . . 14 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑐)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
188183, 186, 151, 187fvmptd 7023 . . . . . . . . . . . . 13 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐) = ((𝐸𝑐)(.g‘(mulGrp‘𝐾))𝑀))
189188eqcomd 2743 . . . . . . . . . . . 12 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑐)(.g‘(mulGrp‘𝐾))𝑀) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐))
190153, 182, 1893eqtrd 2781 . . . . . . . . . . 11 ((𝜑𝑐 ∈ (ℕ0 × ℕ0)) → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))‘𝑐) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))‘𝑐))
191142, 146, 190eqfnfvd 7054 . . . . . . . . . 10 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑣)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
192137, 191eqtrd 2777 . . . . . . . . 9 (𝜑 → (𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
193133, 192eqtrd 2777 . . . . . . . 8 (𝜑 → ((𝑣 ∈ (ℕ0 × ℕ0) ↦ (((𝑃↑(1st𝑣)) · ((𝑁 / 𝑃)↑(2nd𝑣)))(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
194131, 193eqtrd 2777 . . . . . . 7 (𝜑 → ((𝐽𝐸) ↾ (ℕ0 × ℕ0)) = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
195194imaeq1d 6077 . . . . . 6 (𝜑 → (((𝐽𝐸) ↾ (ℕ0 × ℕ0)) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
196103, 195eqtrd 2777 . . . . 5 (𝜑 → ((𝐽𝐸) “ (ℕ0 × ℕ0)) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
197100, 196eqtrid 2789 . . . 4 (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) = ((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0)))
198197fveq2d 6910 . . 3 (𝜑 → (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
19998, 198breqtrd 5169 . 2 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘((𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) “ (ℕ0 × ℕ0))))
2001, 2, 3, 4, 5, 6, 7, 8, 87, 88, 55, 89, 90, 91, 92, 93, 94, 95, 96, 97, 199aks6d1c6lem3 42173 1 (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0m (0...𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  cop 4632   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  cres 5687  cima 5688  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  m cmap 8866  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  cz 12613  ...cfz 13547  cfl 13830  cexp 14102  Ccbc 14341  chash 14369  csqrt 15272  Σcsu 15722  cdvds 16290   gcd cgcd 16531  cprime 16708  ϕcphi 16801  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484   Σg cgsu 17485  .gcmg 19085  CMndccmn 19798  Abelcabl 19799  mulGrpcmgp 20137  CRingccrg 20231   RingIso crs 20470  Fieldcfield 20730  ℤRHomczrh 21510  chrcchr 21512  ℤ/nczn 21513  algSccascl 21872  var1cv1 22177  Poly1cpl1 22178  eval1ce1 22318   logb clogb 26807   PrimRoots cprimroots 42092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-pi 16108  df-dvds 16291  df-gcd 16532  df-prm 16709  df-phi 16803  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-pws 17494  df-xrs 17547  df-qtop 17552  df-imas 17553  df-qus 17554  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-rhm 20472  df-rim 20473  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-chr 21516  df-zn 21517  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evl1 22320  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-mdeg 26094  df-deg1 26095  df-mon1 26170  df-uc1p 26171  df-q1p 26172  df-r1p 26173  df-log 26598  df-logb 26808  df-primroots 42093
This theorem is referenced by:  aks6d1c6lem5  42178
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