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Theorem aks6d1c6lem2 42173
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 6920 . . . . . 6 (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
42, 3eqeltrid 2844 . . . . 5 (𝜑𝐿 ∈ V)
54imaexd 7939 . . . 4 (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
6 hashxrcl 14397 . . . 4 ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
75, 6syl 17 . . 3 (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
81, 7eqeltrid 2844 . 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 12534 . . . . . . . 8 0 ∈ V
1211a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
1312, 12xpexd 7772 . . . . . 6 (𝜑 → (ℕ0 × ℕ0) ∈ V)
1413mptexd 7245 . . . . 5 (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
1510, 14eqeltrd 2840 . . . 4 (𝜑𝐽 ∈ V)
1615imaexd 7939 . . 3 (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ∈ V)
17 hashxrcl 14397 . . 3 ((𝐽 “ (ℕ0 × ℕ0)) ∈ V → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
1816, 17syl 17 . 2 (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
19 fvexd 6920 . . . . 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 14397 . . 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 5164 . 2 (𝜑𝐷 ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
2822elexd 3503 . . 3 (𝜑 → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}) ∈ V)
29 nfv 1913 . . . 4 𝑤𝜑
30 ovexd 7467 . . . . . 6 ((𝜑𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
3130, 9fmptd 7133 . . . . 5 (𝜑𝐽:(ℕ0 × ℕ0)⟶V)
32 ffun 6738 . . . . 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 6909 . . . . . . 7 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → (𝐸𝑗) = (𝐸𝑤))
3736oveq1d 7447 . . . . . 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 7467 . . . . . 6 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
4034, 37, 38, 39fvmptd 7022 . . . . 5 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽𝑤) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))
41 eqid 2736 . . . . . . . . . 10 (eval1𝐾) = (eval1𝐾)
42 eqid 2736 . . . . . . . . . 10 (Poly1𝐾) = (Poly1𝐾)
43 eqid 2736 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
44 eqid 2736 . . . . . . . . . 10 (Base‘(Poly1𝐾)) = (Base‘(Poly1𝐾))
45 aks6d1c6.3 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Field)
4645fldcrngd 20743 . . . . . . . . . . 11 (𝜑𝐾 ∈ CRing)
4746adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ CRing)
48 eqid 2736 . . . . . . . . . . . 12 (mulGrp‘𝐾) = (mulGrp‘𝐾)
4948, 43mgpbas 20143 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
50 eqid 2736 . . . . . . . . . . 11 (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾))
5146crngringd 20244 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Ring)
5248ringmgp 20237 . . . . . . . . . . . . 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 42120 . . . . . . . . . . . . 13 (𝜑𝐸:(ℕ0 × ℕ0)⟶ℕ)
6059ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) ∈ ℕ)
6160nnnn0d 12589 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) ∈ ℕ0)
62 aks6d1c6.16 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
6348crngmgp 20239 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
6446, 63syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
65 aks6d1c6.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑅 ∈ ℕ)
6665nnnn0d 12589 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ ℕ0)
6764, 66, 50isprimroot 42095 . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (Base‘(mulGrp‘𝐾)))
7069, 49eleqtrrdi 2851 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (Base‘𝐾))
7170adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ (Base‘𝐾))
7249, 50, 54, 61, 71mulgnn0cld 19114 . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (var1𝐾) = (var1𝐾)
77 eqid 2736 . . . . . . . . . . . . . 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 42137 . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . 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 3686 . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . 12 (𝜑 → (𝐺𝑈) ∈ (Base‘(Poly1𝐾)))
8887adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺𝑈) ∈ (Base‘(Poly1𝐾)))
89 eqidd 2737 . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . 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 3686 . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . 12 (𝜑 → (𝐺𝑉) ∈ (Base‘(Poly1𝐾)))
9897adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺𝑉) ∈ (Base‘(Poly1𝐾)))
99 eqidd 2737 . . . . . . . . . . 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 2736 . . . . . . . . . 10 (-g‘(Poly1𝐾)) = (-g‘(Poly1𝐾))
102 eqid 2736 . . . . . . . . . 10 (-g𝐾) = (-g𝐾)
10341, 42, 43, 44, 47, 72, 90, 100, 101, 102evl1subd 22347 . . . . . . . . 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 6905 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑈))‘𝑦) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
106105oveq2d 7448 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑦)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)))
107 oveq2 7440 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀))
108107fveq2d 6909 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
109106, 108eqeq12d 2752 . . . . . . . . . . . . 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 3483 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 ∈ V
111 vex 3483 . . . . . . . . . . . . . . . . . . . . . . 23 𝑙 ∈ V
112110, 111op1std 8025 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = ⟨𝑘, 𝑙⟩ → (1st𝑠) = 𝑘)
113112oveq2d 7448 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st𝑠)) = (𝑃𝑘))
114110, 111op2ndd 8026 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = ⟨𝑘, 𝑙⟩ → (2nd𝑠) = 𝑙)
115114oveq2d 7448 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd𝑠)) = ((𝑁 / 𝑃)↑𝑙))
116113, 115oveq12d 7450 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠))) = ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
117116mpompt 7548 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))
11858eqcomi 2745 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙))) = 𝐸
119117, 118eqtri 2764 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠)))) = 𝐸
120119eqcomi 2745 . . . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (1st𝑠) = (1st𝑤))
124123oveq2d 7448 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (𝑃↑(1st𝑠)) = (𝑃↑(1st𝑤)))
125122fveq2d 6909 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (2nd𝑠) = (2nd𝑤))
126125oveq2d 7448 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑁 / 𝑃)↑(2nd𝑠)) = ((𝑁 / 𝑃)↑(2nd𝑤)))
127124, 126oveq12d 7450 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑃↑(1st𝑠)) · ((𝑁 / 𝑃)↑(2nd𝑠))) = ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))))
128 ovexd 7467 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) ∈ V)
129121, 127, 38, 128fvmptd 7022 . . . . . . . . . . . . . . 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 7467 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...𝐴) ∈ V)
13912, 138elmapd 8881 . . . . . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (ℕ0 × ℕ0) → (1st𝑤) ∈ ℕ0)
144143adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (1st𝑤) ∈ ℕ0)
145 xp2nd 8048 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (ℕ0 × ℕ0) → (2nd𝑤) ∈ ℕ0)
146145adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (2nd𝑤) ∈ ℕ0)
147 eqid 2736 . . . . . . . . . . . . . . . 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 42127 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) (𝐺𝑈))
153129, 152eqbrtrd 5164 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) (𝐺𝑈))
154130, 88, 60aks6d1c1p1 42109 . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
158157eqcomd 2742 . . . . . . . . . . 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 5995 . . . . . . . . . . . . . . . . 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 4079 . . . . . . . . . . . . . . . . . . . 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 4017 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ⊆ (ℕ0m (0...𝐴)))
166165resmptd 6057 . . . . . . . . . . . . . . . . 17 (𝜑 → (( ∈ (ℕ0m (0...𝐴)) ↦ (((eval1𝐾)‘(𝐺))‘𝑀)) ↾ 𝑆) = (𝑆 ↦ (((eval1𝐾)‘(𝐺))‘𝑀)))
167161, 166eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐻𝑆) = (𝑆 ↦ (((eval1𝐾)‘(𝐺))‘𝑀)))
168 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑 = 𝑈) → = 𝑈)
169168fveq2d 6909 . . . . . . . . . . . . . . . . . 18 ((𝜑 = 𝑈) → (𝐺) = (𝐺𝑈))
170169fveq2d 6909 . . . . . . . . . . . . . . . . 17 ((𝜑 = 𝑈) → ((eval1𝐾)‘(𝐺)) = ((eval1𝐾)‘(𝐺𝑈)))
171170fveq1d 6907 . . . . . . . . . . . . . . . 16 ((𝜑 = 𝑈) → (((eval1𝐾)‘(𝐺))‘𝑀) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
172 fvexd 6920 . . . . . . . . . . . . . . . 16 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) ∈ V)
173167, 171, 80, 172fvmptd 7022 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐻𝑆)‘𝑈) = (((eval1𝐾)‘(𝐺𝑈))‘𝑀))
174173eqcomd 2742 . . . . . . . . . . . . . 14 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = ((𝐻𝑆)‘𝑈))
175 aks6d1c6lem2.3 . . . . . . . . . . . . . 14 (𝜑 → ((𝐻𝑆)‘𝑈) = ((𝐻𝑆)‘𝑉))
176 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 = 𝑉) → = 𝑉)
177176fveq2d 6909 . . . . . . . . . . . . . . . . 17 ((𝜑 = 𝑉) → (𝐺) = (𝐺𝑉))
178177fveq2d 6909 . . . . . . . . . . . . . . . 16 ((𝜑 = 𝑉) → ((eval1𝐾)‘(𝐺)) = ((eval1𝐾)‘(𝐺𝑉)))
179178fveq1d 6907 . . . . . . . . . . . . . . 15 ((𝜑 = 𝑉) → (((eval1𝐾)‘(𝐺))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
180 fvexd 6920 . . . . . . . . . . . . . . 15 (𝜑 → (((eval1𝐾)‘(𝐺𝑉))‘𝑀) ∈ V)
181167, 179, 91, 180fvmptd 7022 . . . . . . . . . . . . . 14 (𝜑 → ((𝐻𝑆)‘𝑉) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
182174, 175, 1813eqtrd 2780 . . . . . . . . . . . . 13 (𝜑 → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
183182adantr 480 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘𝑀) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
184183oveq2d 7448 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑈))‘𝑀)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
185158, 184eqtrd 2776 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
186 fveq2 6905 . . . . . . . . . . . . 13 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑉))‘𝑦) = (((eval1𝐾)‘(𝐺𝑉))‘𝑀))
187186oveq2d 7448 . . . . . . . . . . . 12 (𝑦 = 𝑀 → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑦)) = ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)))
188107fveq2d 6909 . . . . . . . . . . . 12 (𝑦 = 𝑀 → (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
189187, 188eqeq12d 2752 . . . . . . . . . . 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 8881 . . . . . . . . . . . . . . . 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 42127 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st𝑤)) · ((𝑁 / 𝑃)↑(2nd𝑤))) (𝐺𝑉))
194129, 193eqbrtrd 5164 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸𝑤) (𝐺𝑉))
195130, 98, 60aks6d1c1p1 42109 . . . . . . . . . . . 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 3622 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))(((eval1𝐾)‘(𝐺𝑉))‘𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
198185, 197eqtrd 2776 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
19946crnggrpd 20245 . . . . . . . . . . 11 (𝜑𝐾 ∈ Grp)
200199adantr 480 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Grp)
20141, 42, 43, 44, 47, 72, 88fveval1fvcl 22338 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑈))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
20241, 42, 43, 44, 47, 72, 98fveval1fvcl 22338 . . . . . . . . . 10 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘(𝐺𝑉))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
203 eqid 2736 . . . . . . . . . . 11 (0g𝐾) = (0g𝐾)
20443, 203, 102grpsubeq0 19045 . . . . . . . . . 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 1372 . . . . . . . . 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 2776 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g𝐾))
208 fvexd 6920 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)))‘((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
209 elsng 4639 . . . . . . . 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 2736 . . . . . . . . . . . . . . 15 (𝐾s (Base‘𝐾)) = (𝐾s (Base‘𝐾))
21341, 42, 212, 43evl1rhm 22337 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → (eval1𝐾) ∈ ((Poly1𝐾) RingHom (𝐾s (Base‘𝐾))))
21446, 213syl 17 . . . . . . . . . . . . 13 (𝜑 → (eval1𝐾) ∈ ((Poly1𝐾) RingHom (𝐾s (Base‘𝐾))))
215 eqid 2736 . . . . . . . . . . . . . 14 (Base‘(𝐾s (Base‘𝐾))) = (Base‘(𝐾s (Base‘𝐾)))
21644, 215rhmf 20486 . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . 14 (𝜑 → (Base‘𝐾) ∈ V)
219212, 43pwsbas 17533 . . . . . . . . . . . . . 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 6722 . . . . . . . . . . . 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 22250 . . . . . . . . . . . . . 14 (𝐾 ∈ Ring → (Poly1𝐾) ∈ Ring)
22451, 223syl 17 . . . . . . . . . . . . 13 (𝜑 → (Poly1𝐾) ∈ Ring)
225 ringgrp 20236 . . . . . . . . . . . . 13 ((Poly1𝐾) ∈ Ring → (Poly1𝐾) ∈ Grp)
226224, 225syl 17 . . . . . . . . . . . 12 (𝜑 → (Poly1𝐾) ∈ Grp)
22744, 101grpsubcl 19039 . . . . . . . . . . . 12 (((Poly1𝐾) ∈ Grp ∧ (𝐺𝑈) ∈ (Base‘(Poly1𝐾)) ∧ (𝐺𝑉) ∈ (Base‘(Poly1𝐾))) → ((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)) ∈ (Base‘(Poly1𝐾)))
228226, 87, 97, 227syl3anc 1372 . . . . . . . . . . 11 (𝜑 → ((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉)) ∈ (Base‘(Poly1𝐾)))
229222, 228ffvelcdmd 7104 . . . . . . . . . 10 (𝜑 → ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)))
230218, 218elmapd 8881 . . . . . . . . . 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 6739 . . . . . . . 8 (𝜑 → Fun ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
233232adantr 480 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → Fun ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
234231ffnd 6736 . . . . . . . . . . 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 6672 . . . . . . . . 9 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) = (Base‘𝐾))
237236eqcomd 2742 . . . . . . . 8 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (Base‘𝐾) = dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
23872, 237eleqtrd 2842 . . . . . . 7 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))))
239 fvimacnv 7072 . . . . . . 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 2840 . . . 4 ((𝜑𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽𝑤) ∈ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}))
24329, 33, 242funimassd 6974 . . 3 (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ⊆ (((eval1𝐾)‘((𝐺𝑈)(-g‘(Poly1𝐾))(𝐺𝑉))) “ {(0g𝐾)}))
244 hashss 14449 . . 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 13205 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 1539  wcel 2107  wne 2939  wral 3060  {crab 3435  Vcvv 3479  wss 3950  {csn 4625  cop 4631   class class class wbr 5142  {copab 5204  cmpt 5224   × cxp 5682  ccnv 5683  dom cdm 5684  cres 5686  cima 5687  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  m cmap 8867  0cc0 11156  1c1 11157   · cmul 11161  *cxr 11295   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  0cn0 12528  ...cfz 13548  cexp 14103  chash 14370  Σcsu 15723  cdvds 16291   gcd cgcd 16532  cprime 16709  Basecbs 17248  +gcplusg 17298  0gc0g 17485   Σg cgsu 17486  s cpws 17492  Mndcmnd 18748  Grpcgrp 18952  -gcsg 18954  .gcmg 19086  CMndccmn 19799  mulGrpcmgp 20138  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470   RingIso crs 20471  Fieldcfield 20731  ℤRHomczrh 21511  chrcchr 21513  ℤ/nczn 21514  algSccascl 21873  var1cv1 22178  Poly1cpl1 22179  eval1ce1 22319   PrimRoots cprimroots 42093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235  ax-mulf 11236
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-fac 14314  df-bc 14343  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-dvds 16292  df-gcd 16533  df-prm 16710  df-phi 16804  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  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 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-od 19547  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-srg 20185  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-dvr 20402  df-rhm 20473  df-rim 20474  df-subrng 20547  df-subrg 20571  df-drng 20732  df-field 20733  df-lmod 20861  df-lss 20931  df-lsp 20971  df-cnfld 21366  df-zring 21459  df-zrh 21515  df-chr 21517  df-assa 21874  df-asp 21875  df-ascl 21876  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-evls 22099  df-evl 22100  df-psr1 22182  df-vr1 22183  df-ply1 22184  df-coe1 22185  df-evl1 22321  df-primroots 42094
This theorem is referenced by:  aks6d1c6lem3  42174
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