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Theorem aks6d1c6lem2 42153
Description: Every primitive root is root of G(u)-G(v). (Contributed by metakunt, 8-May-2025.)
Hypotheses
Ref Expression
aks6d1c6.1 = {⟨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)))}
aks6d1c6.2 𝑃 = (chr‘𝐾)
aks6d1c6.3 (𝜑𝐾 ∈ Field)
aks6d1c6.4 (𝜑𝑃 ∈ ℙ)
aks6d1c6.5 (𝜑𝑅 ∈ ℕ)
aks6d1c6.6 (𝜑𝑁 ∈ ℕ)
aks6d1c6.7 (𝜑𝑃𝑁)
aks6d1c6.8 (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6.9 (𝜑𝐴 < 𝑃)
aks6d1c6.10 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)(.g‘(mulGrp‘(Poly1𝐾)))((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c6.11 (𝜑𝐴 ∈ ℕ0)
aks6d1c6.12 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6.13 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6.14 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6.15 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6.16 (𝜑𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6.17 𝐻 = ( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀))
aks6d1c6.18 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
aks6d1c6.19 𝑆 = {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)}
aks6d1c6lem2.1 (𝜑𝑈𝑆)
aks6d1c6lem2.2 (𝜑𝑉𝑆)
aks6d1c6lem2.3 (𝜑 → ((𝐻𝑆)‘𝑈) = ((𝐻𝑆)‘𝑉))
aks6d1c6lem2.4 (𝜑𝑈𝑉)
aks6d1c6lem2.5 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))
aks6d1c6lem2.6 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
Assertion
Ref Expression
aks6d1c6lem2 (𝜑𝐷 ≤ (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
Distinct variable groups:   ,𝑎   𝐴,𝑎   𝐴,𝑔,𝑖   𝐴,   𝐴,𝑠   𝑥,𝐴   𝑒,𝐸,𝑓,𝑦   𝑗,𝐸   𝑒,𝐺,𝑓,𝑦   ,𝐺   𝐾,𝑎   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖   ,𝐾   𝑗,𝐾   𝑥,𝐾   ,𝑀   𝑗,𝑀   𝑦,𝑀   𝑁,𝑎   𝑒,𝑁,𝑓   𝑘,𝑁,𝑙,𝑠   𝑥,𝑁   𝑃,𝑒,𝑓   𝑃,𝑘,𝑙,𝑠   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝑆,   𝑈,𝑒,𝑓,𝑦   𝑈,𝑔,𝑖   𝑈,   𝑒,𝑉,𝑓,𝑦   𝑔,𝑉,𝑖   ,𝑉   𝜑,𝑎   𝜑,𝑔,𝑖   𝜑,   𝜑,𝑗   𝜑,𝑠   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑡,𝑒,𝑓,𝑘,𝑙)   𝐴(𝑦,𝑡,𝑒,𝑓,𝑗,𝑘,𝑙)   𝐷(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑃(𝑦,𝑡,𝑔,,𝑖,𝑗,𝑎)   (𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑙)   𝑅(𝑡,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑆(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑈(𝑥,𝑡,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐸(𝑥,𝑡,𝑔,,𝑖,𝑘,𝑠,𝑎,𝑙)   𝐺(𝑥,𝑡,𝑔,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐻(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐽(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐾(𝑡,𝑘,𝑠,𝑙)   𝐿(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑀(𝑥,𝑡,𝑒,𝑓,𝑔,𝑖,𝑘,𝑠,𝑎,𝑙)   𝑁(𝑦,𝑡,𝑔,,𝑖,𝑗)   𝑉(𝑥,𝑡,𝑗,𝑘,𝑠,𝑎,𝑙)

Proof of Theorem aks6d1c6lem2
Dummy variables 𝑤 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aks6d1c6.18 . . 3 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
2 aks6d1c6.13 . . . . . 6 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
3 fvexd 6922 . . . . . 6 (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
42, 3eqeltrid 2843 . . . . 5 (𝜑𝐿 ∈ V)
54imaexd 7939 . . . 4 (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
6 hashxrcl 14393 . . . 4 ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
75, 6syl 17 . . 3 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
81, 7eqeltrid 2843 . 2 (𝜑𝐷 ∈ ℝ*)
9 aks6d1c6lem2.5 . . . . . 6 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀))
109a1i 11 . . . . 5 (𝜑𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
11 nn0ex 12530 . . . . . . . 8 0 ∈ V
1211a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
1312, 12xpexd 7770 . . . . . 6 (𝜑 → (ℕ0 × ℕ0) ∈ V)
1413mptexd 7244 . . . . 5 (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
1510, 14eqeltrd 2839 . . . 4 (𝜑𝐽 ∈ V)
1615imaexd 7939 . . 3 (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ∈ V)
17 hashxrcl 14393 . . 3 ((𝐽 “ (ℕ0 × ℕ0)) ∈ V → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
1816, 17syl 17 . 2 (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
19 fvexd 6922 . . . . 5 (𝜑 → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ V)
20 cnvexg 7947 . . . . 5 (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ V → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ V)
2119, 20syl 17 . . . 4 (𝜑((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ V)
2221imaexd 7939 . . 3 (𝜑 → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}) ∈ V)
23 hashxrcl 14393 . . 3 ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}) ∈ V → (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})) ∈ ℝ*)
2422, 23syl 17 . 2 (𝜑 → (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})) ∈ ℝ*)
251a1i 11 . . 3 (𝜑𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
26 aks6d1c6lem2.6 . . 3 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
2725, 26eqbrtrd 5170 . 2 (𝜑𝐷 ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
2822elexd 3502 . . 3 (𝜑 → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}) ∈ V)
29 nfv 1912 . . . 4 𝑤𝜑
30 ovexd 7466 . . . . . 6 ((𝜑𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
3130, 9fmptd 7134 . . . . 5 (𝜑𝐽:(ℕ0 × ℕ0)⟶V)
32 ffun 6740 . . . . 5 (𝐽:(ℕ0 × ℕ0)⟶V → Fun 𝐽)
3331, 32syl 17 . . . 4 (𝜑 → Fun 𝐽)
349a1i 11 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
35 simpr 484 . . . . . . . 8 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → 𝑗 = 𝑤)
3635fveq2d 6911 . . . . . . 7 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → (𝐸𝑗) = (𝐸𝑤))
3736oveq1d 7446 . . . . . 6 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))
38 simpr 484 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑤 ∈ (ℕ0 × ℕ0))
39 ovexd 7466 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
4034, 37, 38, 39fvmptd 7023 . . . . 5 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽𝑤) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))
41 eqid 2735 . . . . . . . . . 10 (eval1𝐾) = (eval1𝐾)
42 eqid 2735 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
43 eqid 2735 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
44 eqid 2735 . . . . . . . . . 10 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
45 aks6d1c6.3 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Field)
4645fldcrngd 20759 . . . . . . . . . . 11 (𝜑𝐾 ∈ CRing)
4746adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ CRing)
48 eqid 2735 . . . . . . . . . . . 12 (mulGrp‘𝐾) = (mulGrp‘𝐾)
4948, 43mgpbas 20158 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
50 eqid 2735 . . . . . . . . . . 11 (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾))
5146crngringd 20264 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Ring)
5248ringmgp 20257 . . . . . . . . . . . . 13 (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
5351, 52syl 17 . . . . . . . . . . . 12 (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
5453adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (mulGrp‘𝐾) ∈ Mnd)
55 aks6d1c6.6 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℕ)
56 aks6d1c6.4 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℙ)
57 aks6d1c6.7 . . . . . . . . . . . . . 14 (𝜑𝑃𝑁)
58 aks6d1c6.12 . . . . . . . . . . . . . 14 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
5955, 56, 57, 58aks6d1c2p1 42100 . . . . . . . . . . . . 13 (𝜑𝐸:(ℕ0 × ℕ0)⟶ℕ)
6059ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) ∈ ℕ)
6160nnnn0d 12585 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) ∈ ℕ0)
62 aks6d1c6.16 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
6348crngmgp 20259 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
6446, 63syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
65 aks6d1c6.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ ℕ)
6665nnnn0d 12585 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ ℕ0)
6764, 66, 50isprimroot 42075 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑜 ∈ ℕ0 ((𝑜(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅𝑜))))
6862, 67mpbid 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑜 ∈ ℕ0 ((𝑜(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅𝑜)))
6968simp1d 1141 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (Base‘(mulGrp‘𝐾)))
7069, 49eleqtrrdi 2850 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (Base‘𝐾))
7170adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ (Base‘𝐾))
7249, 50, 54, 61, 71mulgnn0cld 19126 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾))
73 aks6d1c6.2 . . . . . . . . . . . . . 14 𝑃 = (chr‘𝐾)
74 aks6d1c6.11 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ ℕ0)
75 aks6d1c6.9 . . . . . . . . . . . . . 14 (𝜑𝐴 < 𝑃)
76 eqid 2735 . . . . . . . . . . . . . 14 (var1𝐾) = (var1𝐾)
77 eqid 2735 . . . . . . . . . . . . . 14 (.g‘(mulGrp‘(Poly1𝐾))) = (.g‘(mulGrp‘(Poly1𝐾)))
78 aks6d1c6.10 . . . . . . . . . . . . . 14 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ ((mulGrp‘(Poly1𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)(.g‘(mulGrp‘(Poly1𝐾)))((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
7945, 56, 73, 74, 75, 76, 77, 78aks6d1c5lem0 42117 . . . . . . . . . . . . 13 (𝜑𝐺:(ℕ0m (0...𝐴))⟶(Base‘(Poly1𝐾)))
80 aks6d1c6lem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑆)
81 aks6d1c6.19 . . . . . . . . . . . . . . . 16 𝑆 = {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)}
8281eleq2i 2831 . . . . . . . . . . . . . . 15 (𝑈𝑆𝑈 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)})
8380, 82sylib 218 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)})
84 elrabi 3690 . . . . . . . . . . . . . . 15 (𝑈 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0m (0...𝐴)))
8584a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0m (0...𝐴))))
8683, 85mpd 15 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ (ℕ0m (0...𝐴)))
8779, 86ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → (𝐺𝑈) ∈ (Base‘(Poly1𝐾)))
8887adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺𝑈) ∈ (Base‘(Poly1𝐾)))
89 eqidd 2736 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
9088, 89jca 511 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐺𝑈) ∈ (Base‘(Poly1𝐾)) ∧ (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
91 aks6d1c6lem2.2 . . . . . . . . . . . . . . 15 (𝜑𝑉𝑆)
9281eleq2i 2831 . . . . . . . . . . . . . . 15 (𝑉𝑆𝑉 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)})
9391, 92sylib 218 . . . . . . . . . . . . . 14 (𝜑𝑉 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)})
94 elrabi 3690 . . . . . . . . . . . . . . 15 (𝑉 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0m (0...𝐴)))
9594a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (𝑉 ∈ {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0m (0...𝐴))))
9693, 95mpd 15 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ (ℕ0m (0...𝐴)))
9779, 96ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → (𝐺𝑉) ∈ (Base‘(Poly1𝐾)))
9897adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺𝑉) ∈ (Base‘(Poly1𝐾)))
99 eqidd 2736 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
10098, 99jca 511 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐺𝑉) ∈ (Base‘(Poly1𝐾)) ∧ (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
101 eqid 2735 . . . . . . . . . 10 (-g‘(Poly1𝐾)) = (-g‘(Poly1𝐾))
102 eqid 2735 . . . . . . . . . 10 (-g𝐾) = (-g𝐾)
10341, 42, 43, 44, 47, 72, 90, 100, 101, 102evl1subd 22362 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)) ∈ (Base‘(Poly1𝐾)) ∧ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g𝐾)(((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))))
104103simprd 495 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g𝐾)(((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
105 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑈))‘𝑦) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
106105oveq2d 7447 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑦)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)))
107 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))
108107fveq2d 6911 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
109106, 108eqeq12d 2751 . . . . . . . . . . . . 13 (𝑦 = 𝑀 → (((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
110 vex 3482 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 ∈ V
111 vex 3482 . . . . . . . . . . . . . . . . . . . . . . 23 𝑙 ∈ V
112110, 111op1std 8023 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = ⟨𝑘, 𝑙⟩ → (1st𝑠) = 𝑘)
113112oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st𝑠)) = (𝑃𝑘))
114110, 111op2ndd 8024 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = ⟨𝑘, 𝑙⟩ → (2nd𝑠) = 𝑙)
115114oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd𝑠)) = ((𝑁 / 𝑃)↑𝑙))
116113, 115oveq12d 7449 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠))) = ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
117116mpompt 7547 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
11858eqcomi 2744 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙))) = 𝐸
119117, 118eqtri 2763 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠)))) = 𝐸
120119eqcomi 2744 . . . . . . . . . . . . . . . . 17 𝐸 = (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠))))
121120a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐸 = (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠)))))
122 simpr 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → 𝑠 = 𝑤)
123122fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (1st𝑠) = (1st𝑤))
124123oveq2d 7447 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (𝑃↑(1st𝑠)) = (𝑃↑(1st𝑤)))
125122fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (2nd𝑠) = (2nd𝑤))
126125oveq2d 7447 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑁 / 𝑃)↑(2nd𝑠)) = ((𝑁 / 𝑃)↑(2nd𝑤)))
127124, 126oveq12d 7449 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠))) = ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))))
128 ovexd 7466 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) ∈ V)
129121, 127, 38, 128fvmptd 7023 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) = ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))))
130 aks6d1c6.1 . . . . . . . . . . . . . . . 16 = {⟨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)))}
13145adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Field)
13256adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℙ)
13365adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℕ)
13455adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℕ)
13557adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃𝑁)
136 aks6d1c6.8 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 gcd 𝑅) = 1)
137136adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝑁 gcd 𝑅) = 1)
138 ovexd 7466 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...𝐴) ∈ V)
13912, 138elmapd 8879 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑈 ∈ (ℕ0m (0...𝐴)) ↔ 𝑈:(0...𝐴)⟶ℕ0))
14086, 139mpbid 232 . . . . . . . . . . . . . . . . 17 (𝜑𝑈:(0...𝐴)⟶ℕ0)
141140adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑈:(0...𝐴)⟶ℕ0)
14274adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐴 ∈ ℕ0)
143 xp1st 8045 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (ℕ0 × ℕ0) → (1st𝑤) ∈ ℕ0)
144143adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (1st𝑤) ∈ ℕ0)
145 xp2nd 8046 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (ℕ0 × ℕ0) → (2nd𝑤) ∈ ℕ0)
146145adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (2nd𝑤) ∈ ℕ0)
147 eqid 2735 . . . . . . . . . . . . . . . 16 ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) = ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤)))
148 aks6d1c6.14 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
149148adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑎 ∈ (1...𝐴)𝑁 ((var1𝐾)(+g‘(Poly1𝐾))((algSc‘(Poly1𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
150 aks6d1c6.15 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
151150adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
152130, 73, 131, 132, 133, 134, 135, 137, 141, 78, 142, 144, 146, 147, 149, 151aks6d1c1rh 42107 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) (𝐺𝑈))
153129, 152eqbrtrd 5170 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) (𝐺𝑈))
154130, 88, 60aks6d1c1p1 42089 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤) (𝐺𝑈) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦))))
155153, 154mpbid 232 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)))
15662adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
157109, 155, 156rspcdva 3623 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
158157eqcomd 2741 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)))
159 aks6d1c6.17 . . . . . . . . . . . . . . . . . . 19 𝐻 = ( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀))
160159a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝐻 = ( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀)))
161160reseq1d 5999 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐻𝑆) = (( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀)) ↾ 𝑆))
16281a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑆 = {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)})
163 ssrab2 4090 . . . . . . . . . . . . . . . . . . . 20 {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0m (0...𝐴))
164163a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {𝑠 ∈ (ℕ0m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0m (0...𝐴)))
165162, 164eqsstrd 4034 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ⊆ (ℕ0m (0...𝐴)))
166165resmptd 6060 . . . . . . . . . . . . . . . . 17 (𝜑 → (( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀)) ↾ 𝑆) = (𝑆 ↦ (((eval1𝐾)‘(𝐺))‘𝑀)))
167161, 166eqtrd 2775 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐻𝑆) = (𝑆 ↦ (((eval1𝐾)‘(𝐺))‘𝑀)))
168 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑 = 𝑈) → = 𝑈)
169168fveq2d 6911 . . . . . . . . . . . . . . . . . 18 ((𝜑 = 𝑈) → (𝐺) = (𝐺𝑈))
170169fveq2d 6911 . . . . . . . . . . . . . . . . 17 ((𝜑 = 𝑈) → ((eval1𝐾)‘(𝐺)) = ((eval1𝐾)‘(𝐺𝑈)))
171170fveq1d 6909 . . . . . . . . . . . . . . . 16 ((𝜑 = 𝑈) → (((eval1𝐾)‘(𝐺))‘𝑀) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
172 fvexd 6922 . . . . . . . . . . . . . . . 16 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) ∈ V)
173167, 171, 80, 172fvmptd 7023 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐻𝑆)‘𝑈) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
174173eqcomd 2741 . . . . . . . . . . . . . 14 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = ((𝐻𝑆)‘𝑈))
175 aks6d1c6lem2.3 . . . . . . . . . . . . . 14 (𝜑 → ((𝐻𝑆)‘𝑈) = ((𝐻𝑆)‘𝑉))
176 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 = 𝑉) → = 𝑉)
177176fveq2d 6911 . . . . . . . . . . . . . . . . 17 ((𝜑 = 𝑉) → (𝐺) = (𝐺𝑉))
178177fveq2d 6911 . . . . . . . . . . . . . . . 16 ((𝜑 = 𝑉) → ((eval1𝐾)‘(𝐺)) = ((eval1𝐾)‘(𝐺𝑉)))
179178fveq1d 6909 . . . . . . . . . . . . . . 15 ((𝜑 = 𝑉) → (((eval1𝐾)‘(𝐺))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
180 fvexd 6922 . . . . . . . . . . . . . . 15 (𝜑 → (((eval1𝐾)‘(𝐺𝑉))‘𝑀) ∈ V)
181167, 179, 91, 180fvmptd 7023 . . . . . . . . . . . . . 14 (𝜑 → ((𝐻𝑆)‘𝑉) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
182174, 175, 1813eqtrd 2779 . . . . . . . . . . . . 13 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
183182adantr 480 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
184183oveq2d 7447 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
185158, 184eqtrd 2775 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
186 fveq2 6907 . . . . . . . . . . . . 13 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑉))‘𝑦) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
187186oveq2d 7447 . . . . . . . . . . . 12 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑦)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
188107fveq2d 6911 . . . . . . . . . . . 12 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
189187, 188eqeq12d 2751 . . . . . . . . . . 11 (𝑦 = 𝑀 → (((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
19012, 138elmapd 8879 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑉 ∈ (ℕ0m (0...𝐴)) ↔ 𝑉:(0...𝐴)⟶ℕ0))
19196, 190mpbid 232 . . . . . . . . . . . . . . 15 (𝜑𝑉:(0...𝐴)⟶ℕ0)
192191adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑉:(0...𝐴)⟶ℕ0)
193130, 73, 131, 132, 133, 134, 135, 137, 192, 78, 142, 144, 146, 147, 149, 151aks6d1c1rh 42107 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) (𝐺𝑉))
194129, 193eqbrtrd 5170 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) (𝐺𝑉))
195130, 98, 60aks6d1c1p1 42089 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤) (𝐺𝑉) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦))))
196194, 195mpbid 232 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑦)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)))
197189, 196, 156rspcdva 3623 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
198185, 197eqtrd 2775 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
19946crnggrpd 20265 . . . . . . . . . . 11 (𝜑𝐾 ∈ Grp)
200199adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Grp)
20141, 42, 43, 44, 47, 72, 88fveval1fvcl 22353 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
20241, 42, 43, 44, 47, 72, 98fveval1fvcl 22353 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
203 eqid 2735 . . . . . . . . . . 11 (0g𝐾) = (0g𝐾)
20443, 203, 102grpsubeq0 19057 . . . . . . . . . 10 ((𝐾 ∈ Grp ∧ (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾) ∧ (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾)) → (((((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g𝐾)(((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g𝐾) ↔ (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
205200, 201, 202, 204syl3anc 1370 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g𝐾)(((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g𝐾) ↔ (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
206198, 205mpbird 257 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g𝐾)(((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g𝐾))
207104, 206eqtrd 2775 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g𝐾))
208 fvexd 6922 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
209 elsng 4645 . . . . . . . 8 ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V → ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g𝐾)} ↔ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g𝐾)))
210208, 209syl 17 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g𝐾)} ↔ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g𝐾)))
211207, 210mpbird 257 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g𝐾)})
212 eqid 2735 . . . . . . . . . . . . . . 15 (𝐾s (Base‘𝐾)) = (𝐾s (Base‘𝐾))
21341, 42, 212, 43evl1rhm 22352 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → (eval1𝐾) ∈ ((Poly1𝐾) RingHom (𝐾s (Base‘𝐾))))
21446, 213syl 17 . . . . . . . . . . . . 13 (𝜑 → (eval1𝐾) ∈ ((Poly1𝐾) RingHom (𝐾s (Base‘𝐾))))
215 eqid 2735 . . . . . . . . . . . . . 14 (Base‘(𝐾s (Base‘𝐾))) = (Base‘(𝐾s (Base‘𝐾)))
21644, 215rhmf 20502 . . . . . . . . . . . . 13 ((eval1𝐾) ∈ ((Poly1𝐾) RingHom (𝐾s (Base‘𝐾))) → (eval1𝐾):(Base‘(Poly1𝐾))⟶(Base‘(𝐾s (Base‘𝐾))))
217214, 216syl 17 . . . . . . . . . . . 12 (𝜑 → (eval1𝐾):(Base‘(Poly1𝐾))⟶(Base‘(𝐾s (Base‘𝐾))))
218 fvexd 6922 . . . . . . . . . . . . . 14 (𝜑 → (Base‘𝐾) ∈ V)
219212, 43pwsbas 17534 . . . . . . . . . . . . . 14 ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾s (Base‘𝐾))))
22045, 218, 219syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾s (Base‘𝐾))))
221220feq3d 6724 . . . . . . . . . . . 12 (𝜑 → ((eval1𝐾):(Base‘(Poly1𝐾))⟶((Base‘𝐾) ↑m (Base‘𝐾)) ↔ (eval1𝐾):(Base‘(Poly1𝐾))⟶(Base‘(𝐾s (Base‘𝐾)))))
222217, 221mpbird 257 . . . . . . . . . . 11 (𝜑 → (eval1𝐾):(Base‘(Poly1𝐾))⟶((Base‘𝐾) ↑m (Base‘𝐾)))
22342ply1ring 22265 . . . . . . . . . . . . . 14 (𝐾 ∈ Ring → (Poly1𝐾) ∈ Ring)
22451, 223syl 17 . . . . . . . . . . . . 13 (𝜑 → (Poly1𝐾) ∈ Ring)
225 ringgrp 20256 . . . . . . . . . . . . 13 ((Poly1𝐾) ∈ Ring → (Poly1𝐾) ∈ Grp)
226224, 225syl 17 . . . . . . . . . . . 12 (𝜑 → (Poly1𝐾) ∈ Grp)
22744, 101grpsubcl 19051 . . . . . . . . . . . 12 (((Poly1𝐾) ∈ Grp ∧ (𝐺𝑈) ∈ (Base‘(Poly1𝐾)) ∧ (𝐺𝑉) ∈ (Base‘(Poly1𝐾))) → ((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)) ∈ (Base‘(Poly1𝐾)))
228226, 87, 97, 227syl3anc 1370 . . . . . . . . . . 11 (𝜑 → ((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)) ∈ (Base‘(Poly1𝐾)))
229222, 228ffvelcdmd 7105 . . . . . . . . . 10 (𝜑 → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)))
230218, 218elmapd 8879 . . . . . . . . . 10 (𝜑 → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)) ↔ ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))):(Base‘𝐾)⟶(Base‘𝐾)))
231229, 230mpbid 232 . . . . . . . . 9 (𝜑 → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))):(Base‘𝐾)⟶(Base‘𝐾))
232231ffund 6741 . . . . . . . 8 (𝜑 → Fun ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
233232adantr 480 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → Fun ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
234231ffnd 6738 . . . . . . . . . . 11 (𝜑 → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) Fn (Base‘𝐾))
235234adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) Fn (Base‘𝐾))
236235fndmd 6674 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) = (Base‘𝐾))
237236eqcomd 2741 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (Base‘𝐾) = dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
23872, 237eleqtrd 2841 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
239 fvimacnv 7073 . . . . . . 7 ((Fun ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∧ ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))) → ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g𝐾)} ↔ ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
240233, 238, 239syl2anc 584 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g𝐾)} ↔ ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
241211, 240mpbid 232 . . . . 5 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}))
24240, 241eqeltrd 2839 . . . 4 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽𝑤) ∈ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}))
24329, 33, 242funimassd 6975 . . 3 (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ⊆ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}))
244 hashss 14445 . . 3 (((((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}) ∈ V ∧ (𝐽 “ (ℕ0 × ℕ0)) ⊆ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})) → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ≤ (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
24528, 243, 244syl2anc 584 . 2 (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ≤ (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
2468, 18, 24, 27, 245xrletrd 13201 1 (𝜑𝐷 ≤ (♯‘(((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  wss 3963  {csn 4631  cop 4637   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  cres 5691  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  m cmap 8865  0cc0 11153  1c1 11154   · cmul 11158  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  0cn0 12524  ...cfz 13544  cexp 14099  chash 14366  Σcsu 15719  cdvds 16287   gcd cgcd 16528  cprime 16705  Basecbs 17245  +gcplusg 17298  0gc0g 17486   Σg cgsu 17487  s cpws 17493  Mndcmnd 18760  Grpcgrp 18964  -gcsg 18966  .gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486   RingIso crs 20487  Fieldcfield 20747  ℤRHomczrh 21528  chrcchr 21530  ℤ/nczn 21531  algSccascl 21890  var1cv1 22193  Poly1cpl1 22194  eval1ce1 22334   PrimRoots cprimroots 42073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706  df-phi 16800  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-od 19561  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-rim 20490  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-chr 21534  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evl1 22336  df-primroots 42074
This theorem is referenced by:  aks6d1c6lem3  42154
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