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Theorem aks6d1c1p3 42091
Description: In a field with a Frobenius isomorphism (read: algebraic closure or finite field), 𝑁 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)
Hypotheses
Ref Expression
aks6d1c1p3.1 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ((𝑂𝑓)‘𝑦)) = ((𝑂𝑓)‘(𝑒 𝑦)))}
aks6d1c1p3.2 𝑆 = (Poly1𝐾)
aks6d1c1p3.3 𝐵 = (Base‘𝑆)
aks6d1c1p3.4 𝑋 = (var1𝐾)
aks6d1c1p3.5 𝑊 = (mulGrp‘𝑆)
aks6d1c1p3.6 𝑉 = (mulGrp‘𝐾)
aks6d1c1p3.7 = (.g𝑉)
aks6d1c1p3.8 𝐶 = (algSc‘𝑆)
aks6d1c1p3.9 𝐷 = (.g𝑊)
aks6d1c1p3.10 𝑃 = (chr‘𝐾)
aks6d1c1p3.11 𝑂 = (eval1𝐾)
aks6d1c1p3.12 + = (+g𝑆)
aks6d1c1p3.13 (𝜑𝐾 ∈ Field)
aks6d1c1p3.14 (𝜑𝑃 ∈ ℙ)
aks6d1c1p3.15 (𝜑𝑅 ∈ ℕ)
aks6d1c1p3.16 (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1p3.17 (𝜑𝑃𝑁)
aks6d1c1p3.18 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
aks6d1c1p3.19 (𝜑𝐴 ∈ ℤ)
aks6d1c1p3.20 (𝜑𝑁 𝐹)
aks6d1c1p3.21 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾))
Assertion
Ref Expression
aks6d1c1p3 (𝜑 → (𝑁 / 𝑃) 𝐹)
Distinct variable groups:   ,𝑒,𝑓,𝑦   𝑥, ,𝑦   𝑥,𝐴   𝐵,𝑒,𝑓   𝑒,𝐹,𝑓,𝑦   𝑥,𝐾   𝑒,𝑁,𝑓,𝑦   𝑥,𝑁   𝑒,𝑂,𝑓,𝑦   𝑃,𝑒,𝑓,𝑦   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝑒,𝑉,𝑓,𝑦   𝑥,𝑉   𝜑,𝑦,𝑥
Allowed substitution hints:   𝜑(𝑒,𝑓)   𝐴(𝑦,𝑒,𝑓)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑒,𝑓)   𝐷(𝑥,𝑦,𝑒,𝑓)   + (𝑥,𝑦,𝑒,𝑓)   (𝑥,𝑦,𝑒,𝑓)   𝑆(𝑥,𝑦,𝑒,𝑓)   𝐹(𝑥)   𝐾(𝑦,𝑒,𝑓)   𝑂(𝑥)   𝑊(𝑥,𝑦,𝑒,𝑓)   𝑋(𝑥,𝑦,𝑒,𝑓)

Proof of Theorem aks6d1c1p3
Dummy variables 𝑧 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aks6d1c1p3.18 . . . . . . . . 9 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))
21a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
32fveq2d 6910 . . . . . . 7 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑂𝐹) = (𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))))
43fveq1d 6908 . . . . . 6 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) 𝑦)))
5 aks6d1c1p3.11 . . . . . . . 8 𝑂 = (eval1𝐾)
6 aks6d1c1p3.2 . . . . . . . 8 𝑆 = (Poly1𝐾)
7 eqid 2734 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
8 aks6d1c1p3.3 . . . . . . . 8 𝐵 = (Base‘𝑆)
9 aks6d1c1p3.13 . . . . . . . . . 10 (𝜑𝐾 ∈ Field)
109fldcrngd 20758 . . . . . . . . 9 (𝜑𝐾 ∈ CRing)
1110adantr 480 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ CRing)
12 eqid 2734 . . . . . . . . . 10 (Base‘𝑉) = (Base‘𝑉)
13 aks6d1c1p3.7 . . . . . . . . . 10 = (.g𝑉)
14 aks6d1c1p3.6 . . . . . . . . . . . . . 14 𝑉 = (mulGrp‘𝐾)
1514crngmgp 20258 . . . . . . . . . . . . 13 (𝐾 ∈ CRing → 𝑉 ∈ CMnd)
1610, 15syl 17 . . . . . . . . . . . 12 (𝜑𝑉 ∈ CMnd)
1716cmnmndd 19836 . . . . . . . . . . 11 (𝜑𝑉 ∈ Mnd)
1817adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑉 ∈ Mnd)
19 aks6d1c1p3.17 . . . . . . . . . . . . 13 (𝜑𝑃𝑁)
20 aks6d1c1p3.1 . . . . . . . . . . . . . . . 16 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ((𝑂𝑓)‘𝑦)) = ((𝑂𝑓)‘(𝑒 𝑦)))}
21 aks6d1c1p3.20 . . . . . . . . . . . . . . . 16 (𝜑𝑁 𝐹)
2220, 21aks6d1c1p1rcl 42089 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 ∈ ℕ ∧ 𝐹𝐵))
2322simpld 494 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℕ)
24 aks6d1c1p3.14 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ ℙ)
25 prmnn 16707 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
2624, 25syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
27 nndivdvds 16295 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℕ) → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
2823, 26, 27syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℕ))
2919, 28mpbid 232 . . . . . . . . . . . 12 (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
3029nnnn0d 12584 . . . . . . . . . . 11 (𝜑 → (𝑁 / 𝑃) ∈ ℕ0)
3130adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 / 𝑃) ∈ ℕ0)
32 aks6d1c1p3.15 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ ℕ)
3332nnnn0d 12584 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ ℕ0)
3416, 33, 13isprimroot 42074 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) ↔ (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 𝑦) = (0g𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑦) = (0g𝑉) → 𝑅𝑙))))
3534biimpd 229 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝑉 PrimRoots 𝑅) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 𝑦) = (0g𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑦) = (0g𝑉) → 𝑅𝑙))))
3635imp 406 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ∧ (𝑅 𝑦) = (0g𝑉) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 𝑦) = (0g𝑉) → 𝑅𝑙)))
3736simp1d 1141 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝑉))
3812, 13, 18, 31, 37mulgnn0cld 19125 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) 𝑦) ∈ (Base‘𝑉))
3914, 7mgpbas 20157 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝑉)
4039eqcomi 2743 . . . . . . . . . . 11 (Base‘𝑉) = (Base‘𝐾)
4140a1i 11 . . . . . . . . . 10 (𝜑 → (Base‘𝑉) = (Base‘𝐾))
4241adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (Base‘𝑉) = (Base‘𝐾))
4338, 42eleqtrd 2840 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) 𝑦) ∈ (Base‘𝐾))
44 aks6d1c1p3.4 . . . . . . . . 9 𝑋 = (var1𝐾)
455, 44, 7, 6, 8, 11, 43evl1vard 22356 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋𝐵 ∧ ((𝑂𝑋)‘((𝑁 / 𝑃) 𝑦)) = ((𝑁 / 𝑃) 𝑦)))
46 aks6d1c1p3.8 . . . . . . . . 9 𝐶 = (algSc‘𝑆)
4710crngringd 20263 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Ring)
48 eqid 2734 . . . . . . . . . . . . 13 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
4948zrhrhm 21539 . . . . . . . . . . . 12 (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
50 rhmghm 20500 . . . . . . . . . . . 12 ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤring GrpHom 𝐾))
51 zringbas 21481 . . . . . . . . . . . . 13 ℤ = (Base‘ℤring)
5251, 7ghmf 19250 . . . . . . . . . . . 12 ((ℤRHom‘𝐾) ∈ (ℤring GrpHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
5347, 49, 50, 524syl 19 . . . . . . . . . . 11 (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
54 aks6d1c1p3.19 . . . . . . . . . . 11 (𝜑𝐴 ∈ ℤ)
5553, 54ffvelcdmd 7104 . . . . . . . . . 10 (𝜑 → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
5655adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾))
575, 6, 7, 46, 8, 11, 56, 43evl1scad 22354 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘((𝑁 / 𝑃) 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
58 aks6d1c1p3.12 . . . . . . . 8 + = (+g𝑆)
59 eqid 2734 . . . . . . . 8 (+g𝐾) = (+g𝐾)
605, 6, 7, 8, 11, 43, 45, 57, 58, 59evl1addd 22360 . . . . . . 7 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) 𝑦)) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
6160simprd 495 . . . . . 6 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘((𝑁 / 𝑃) 𝑦)) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
624, 61eqtrd 2774 . . . . 5 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦)) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
633fveq1d 6908 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘𝑦) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦))
6463oveq2d 7446 . . . . . . 7 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)) = ((𝑁 / 𝑃) ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
6542eleq2d 2824 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦 ∈ (Base‘𝑉) ↔ 𝑦 ∈ (Base‘𝐾)))
6637, 65mpbid 232 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘𝐾))
675, 44, 40, 6, 8, 11, 37evl1vard 22356 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋𝐵 ∧ ((𝑂𝑋)‘𝑦) = 𝑦))
685, 6, 7, 46, 8, 11, 56, 66evl1scad 22354 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘𝑦) = ((ℤRHom‘𝐾)‘𝐴)))
695, 6, 7, 8, 11, 66, 67, 68, 58, 59evl1addd 22360 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
7069simprd 495 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦) = (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
7170oveq2d 7446 . . . . . . 7 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
7264, 71eqtrd 2774 . . . . . 6 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)) = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
73 aks6d1c1p3.21 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾))
747, 7isrim 20508 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾) ↔ ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)))
7573, 74sylib 218 . . . . . . . . . . . 12 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingHom 𝐾) ∧ (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾)))
7675simprd 495 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7776adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7811crnggrpd 20264 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝐾 ∈ Grp)
797, 59, 78, 43, 56grpcld 18977 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
80 f1ocnvfv1 7295 . . . . . . . . . 10 (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
8177, 79, 80syl2anc 584 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
8281eqcomd 2740 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
83 eqidd 2735 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)))
84 id 22 . . . . . . . . . . . . 13 (𝑥 = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) → 𝑥 = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
8584adantl 481 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
8685oveq2d 7446 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 𝑥) = (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
87 eqid 2734 . . . . . . . . . . . . 13 (mulGrp‘𝐾) = (mulGrp‘𝐾)
8887, 7mgpbas 20157 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
8914fveq2i 6909 . . . . . . . . . . . . 13 (.g𝑉) = (.g‘(mulGrp‘𝐾))
9013, 89eqtri 2762 . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝐾))
9187ringmgp 20256 . . . . . . . . . . . . . 14 (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
9247, 91syl 17 . . . . . . . . . . . . 13 (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
9392adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (mulGrp‘𝐾) ∈ Mnd)
9426nnnn0d 12584 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℕ0)
9594adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℕ0)
9688, 90, 93, 95, 79mulgnn0cld 19125 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
9783, 86, 79, 96fvmptd 7022 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
9897eqcomd 2740 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
9975simpld 494 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingHom 𝐾))
100 rhmghm 20500 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingHom 𝐾) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 GrpHom 𝐾))
10199, 100syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 GrpHom 𝐾))
102101adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 GrpHom 𝐾))
1037, 59, 59ghmlin 19251 . . . . . . . . . . . . 13 (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 GrpHom 𝐾) ∧ ((𝑁 / 𝑃) 𝑦) ∈ (Base‘𝐾) ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) 𝑦))(+g𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((ℤRHom‘𝐾)‘𝐴))))
104102, 43, 56, 103syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) 𝑦))(+g𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((ℤRHom‘𝐾)‘𝐴))))
105 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = ((𝑁 / 𝑃) 𝑦) → 𝑥 = ((𝑁 / 𝑃) 𝑦))
106105adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) 𝑦)) → 𝑥 = ((𝑁 / 𝑃) 𝑦))
107106oveq2d 7446 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) 𝑦)) → (𝑃 𝑥) = (𝑃 ((𝑁 / 𝑃) 𝑦)))
10888, 90, 93, 95, 43mulgnn0cld 19125 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ((𝑁 / 𝑃) 𝑦)) ∈ (Base‘𝐾))
10983, 107, 43, 108fvmptd 7022 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) 𝑦)) = (𝑃 ((𝑁 / 𝑃) 𝑦)))
110 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → 𝑥 = ((ℤRHom‘𝐾)‘𝐴))
111110oveq2d 7446 . . . . . . . . . . . . . . 15 (𝑥 = ((ℤRHom‘𝐾)‘𝐴) → (𝑃 𝑥) = (𝑃 ((ℤRHom‘𝐾)‘𝐴)))
112111adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((ℤRHom‘𝐾)‘𝐴)) → (𝑃 𝑥) = (𝑃 ((ℤRHom‘𝐾)‘𝐴)))
113 aks6d1c1p3.10 . . . . . . . . . . . . . . . . . 18 𝑃 = (chr‘𝐾)
114 eqid 2734 . . . . . . . . . . . . . . . . . 18 ((ℤRHom‘𝐾)‘𝐴) = ((ℤRHom‘𝐾)‘𝐴)
115113, 7, 90, 114, 24, 54, 10fermltlchr 21561 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 ((ℤRHom‘𝐾)‘𝐴)) = ((ℤRHom‘𝐾)‘𝐴))
116115eqcomd 2740 . . . . . . . . . . . . . . . 16 (𝜑 → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ((ℤRHom‘𝐾)‘𝐴)))
117116adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((ℤRHom‘𝐾)‘𝐴) = (𝑃 ((ℤRHom‘𝐾)‘𝐴)))
118117, 56eqeltrrd 2839 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
11983, 112, 56, 118fvmptd 7022 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((ℤRHom‘𝐾)‘𝐴)) = (𝑃 ((ℤRHom‘𝐾)‘𝐴)))
120109, 119oveq12d 7448 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) 𝑦))(+g𝐾)((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ((𝑁 / 𝑃) 𝑦))(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))))
12198, 104, 1203eqtrd 2778 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ((𝑁 / 𝑃) 𝑦))(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))))
12223nncnd 12279 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℂ)
123122adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℂ)
12426nncnd 12279 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ∈ ℂ)
125124adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ∈ ℂ)
12626nnne0d 12313 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ≠ 0)
127126adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑃 ≠ 0)
128123, 125, 127divcan2d 12042 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 · (𝑁 / 𝑃)) = 𝑁)
129128oveq1d 7445 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
13063oveq2d 7446 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ((𝑂𝐹)‘𝑦)) = (𝑁 ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)))
13170oveq2d 7446 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘𝑦)) = (𝑁 (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
132130, 131eqtrd 2774 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ((𝑂𝐹)‘𝑦)) = (𝑁 (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
133132eqcomd 2740 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑁 ((𝑂𝐹)‘𝑦)))
134 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑦 → ((𝑂𝐹)‘𝑧) = ((𝑂𝐹)‘𝑦))
135134oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑦 → (𝑁 ((𝑂𝐹)‘𝑧)) = (𝑁 ((𝑂𝐹)‘𝑦)))
136 oveq2 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑦 → (𝑁 𝑧) = (𝑁 𝑦))
137136fveq2d 6910 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑦 → ((𝑂𝐹)‘(𝑁 𝑧)) = ((𝑂𝐹)‘(𝑁 𝑦)))
138135, 137eqeq12d 2750 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ((𝑁 ((𝑂𝐹)‘𝑧)) = ((𝑂𝐹)‘(𝑁 𝑧)) ↔ (𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦))))
1396ply1crng 22215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐾 ∈ CRing → 𝑆 ∈ CRing)
14010, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑆 ∈ CRing)
141140crnggrpd 20264 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑆 ∈ Grp)
14244, 6, 8vr1cl 22234 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 ∈ Ring → 𝑋𝐵)
14347, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑋𝐵)
1446, 46, 7, 8ply1sclcl 22304 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝐴) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
14547, 55, 144syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
146141, 143, 1453jca 1127 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑆 ∈ Grp ∧ 𝑋𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵))
1478, 58grpcl 18971 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ Grp ∧ 𝑋𝐵 ∧ (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
148146, 147syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵)
1491a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))
150149eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹𝐵 ↔ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵))
151148, 150mpbird 257 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹𝐵)
15220, 151, 23aks6d1c1p1 42088 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑁 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦))))
15321, 152mpbid 232 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦)))
154 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → ((𝑂𝐹)‘𝑦) = ((𝑂𝐹)‘𝑧))
155154oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → (𝑁 ((𝑂𝐹)‘𝑦)) = (𝑁 ((𝑂𝐹)‘𝑧)))
156 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → (𝑁 𝑦) = (𝑁 𝑧))
157156fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → ((𝑂𝐹)‘(𝑁 𝑦)) = ((𝑂𝐹)‘(𝑁 𝑧)))
158155, 157eqeq12d 2750 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → ((𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦)) ↔ (𝑁 ((𝑂𝐹)‘𝑧)) = ((𝑂𝐹)‘(𝑁 𝑧))))
159158cbvralvw 3234 . . . . . . . . . . . . . . . . . . . 20 (∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦)) ↔ ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑧)) = ((𝑂𝐹)‘(𝑁 𝑧)))
160153, 159sylib 218 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑧)) = ((𝑂𝐹)‘(𝑁 𝑧)))
161160adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ∀𝑧 ∈ (𝑉 PrimRoots 𝑅)(𝑁 ((𝑂𝐹)‘𝑧)) = ((𝑂𝐹)‘(𝑁 𝑧)))
162 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (𝑉 PrimRoots 𝑅))
163138, 161, 162rspcdva 3622 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘(𝑁 𝑦)))
1643fveq1d 6908 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘(𝑁 𝑦)) = ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 𝑦)))
16523nnnn0d 12584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℕ0)
166165adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 ∈ ℕ0)
16712, 13, 18, 166, 37mulgnn0cld 19125 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 𝑦) ∈ (Base‘𝑉))
168167, 42eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 𝑦) ∈ (Base‘𝐾))
169143adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑋𝐵)
1705, 44, 7, 6, 8, 11, 168evl1vard 22356 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋𝐵 ∧ ((𝑂𝑋)‘(𝑁 𝑦)) = (𝑁 𝑦)))
171170simprd 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝑋)‘(𝑁 𝑦)) = (𝑁 𝑦))
172169, 171jca 511 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑋𝐵 ∧ ((𝑂𝑋)‘(𝑁 𝑦)) = (𝑁 𝑦)))
173145adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵)
1745, 6, 7, 46, 8, 11, 56, 168evl1scad 22354 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
175174simprd 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 𝑦)) = ((ℤRHom‘𝐾)‘𝐴))
176173, 175jca 511 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝐶‘((ℤRHom‘𝐾)‘𝐴)) ∈ 𝐵 ∧ ((𝑂‘(𝐶‘((ℤRHom‘𝐾)‘𝐴)))‘(𝑁 𝑦)) = ((ℤRHom‘𝐾)‘𝐴)))
1775, 6, 7, 8, 11, 168, 172, 176, 58, 59evl1addd 22360 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) ∈ 𝐵 ∧ ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 𝑦)) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
178177simprd 495 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂‘(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁 𝑦)) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
179164, 178eqtrd 2774 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘(𝑁 𝑦)) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
180163, 179eqtrd 2774 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 ((𝑂𝐹)‘𝑦)) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
181133, 180eqtrd 2774 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
182129, 181eqtrd 2774 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
183128eqcomd 2740 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑁 = (𝑃 · (𝑁 / 𝑃)))
184183oveq1d 7445 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑁 𝑦) = ((𝑃 · (𝑁 / 𝑃)) 𝑦))
185184, 117oveq12d 7448 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) = (((𝑃 · (𝑁 / 𝑃)) 𝑦)(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))))
186182, 185eqtr2d 2775 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) 𝑦)(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
18766, 88eleqtrdi 2848 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → 𝑦 ∈ (Base‘(mulGrp‘𝐾)))
18895, 31, 1873jca 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0𝑦 ∈ (Base‘(mulGrp‘𝐾))))
189 eqid 2734 . . . . . . . . . . . . . . . 16 (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
190189, 90mulgnn0ass 19140 . . . . . . . . . . . . . . 15 (((mulGrp‘𝐾) ∈ Mnd ∧ (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0𝑦 ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) 𝑦) = (𝑃 ((𝑁 / 𝑃) 𝑦)))
19193, 188, 190syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) 𝑦) = (𝑃 ((𝑁 / 𝑃) 𝑦)))
192191oveq1d 7445 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑃 · (𝑁 / 𝑃)) 𝑦)(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ((𝑁 / 𝑃) 𝑦))(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))))
193186, 192eqtr3d 2776 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑃 ((𝑁 / 𝑃) 𝑦))(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))))
1947, 59, 78, 66, 56grpcld 18977 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘𝐾))
195194, 88eleqtrdi 2848 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))
19695, 31, 1953jca 1127 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾))))
197189, 90mulgnn0ass 19140 . . . . . . . . . . . . 13 (((mulGrp‘𝐾) ∈ Mnd ∧ (𝑃 ∈ ℕ0 ∧ (𝑁 / 𝑃) ∈ ℕ0 ∧ (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
19893, 196, 197syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 · (𝑁 / 𝑃)) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
199193, 198eqtr3d 2776 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑃 ((𝑁 / 𝑃) 𝑦))(+g𝐾)(𝑃 ((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
200121, 199eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
201 id 22 . . . . . . . . . . . . . 14 (𝑥 = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) → 𝑥 = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
202201oveq2d 7446 . . . . . . . . . . . . 13 (𝑥 = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) → (𝑃 𝑥) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
203202adantl 481 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) ∧ 𝑥 = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) → (𝑃 𝑥) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
20488, 90, 93, 31, 194mulgnn0cld 19125 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾))
205200, 96eqeltrrd 2839 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) ∈ (Base‘𝐾))
20683, 203, 204, 205fvmptd 7022 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) = (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
207206eqcomd 2740 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (𝑃 ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
20897, 200, 2073eqtrd 2778 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))))
209208fveq2d 6910 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘(((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))) = ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))))
210 f1ocnvfv1 7295 . . . . . . . . 9 (((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)):(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) ∈ (Base‘𝐾)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
21177, 204, 210syl2anc 584 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥))‘((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))) = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
21282, 209, 2113eqtrd 2778 . . . . . . 7 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))))
213212eqcomd 2740 . . . . . 6 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) (𝑦(+g𝐾)((ℤRHom‘𝐾)‘𝐴))) = (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)))
21472, 213eqtr2d 2775 . . . . 5 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → (((𝑁 / 𝑃) 𝑦)(+g𝐾)((ℤRHom‘𝐾)‘𝐴)) = ((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)))
21562, 214eqtrd 2774 . . . 4 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦)) = ((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)))
216215eqcomd 2740 . . 3 ((𝜑𝑦 ∈ (𝑉 PrimRoots 𝑅)) → ((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦)))
217216ralrimiva 3143 . 2 (𝜑 → ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦)))
21820, 151, 29aks6d1c1p1 42088 . 2 (𝜑 → ((𝑁 / 𝑃) 𝐹 ↔ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)((𝑁 / 𝑃) ((𝑂𝐹)‘𝑦)) = ((𝑂𝐹)‘((𝑁 / 𝑃) 𝑦))))
219217, 218mpbird 257 1 (𝜑 → (𝑁 / 𝑃) 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058   class class class wbr 5147  {copab 5209  cmpt 5230  ccnv 5687  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cc 11150  0cc0 11152  1c1 11153   · cmul 11157   / cdiv 11917  cn 12263  0cn0 12523  cz 12610  cdvds 16286   gcd cgcd 16527  cprime 16704  Basecbs 17244  +gcplusg 17297  0gc0g 17485  Mndcmnd 18759  Grpcgrp 18963  .gcmg 19097   GrpHom cghm 19242  CMndccmn 19812  mulGrpcmgp 20151  Ringcrg 20250  CRingccrg 20251   RingHom crh 20485   RingIso crs 20486  Fieldcfield 20746  ringczring 21474  ℤRHomczrh 21527  chrcchr 21529  algSccascl 21889  var1cv1 22192  Poly1cpl1 22193  eval1ce1 22333   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-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-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  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-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  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-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  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-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-subrng 20562  df-subrg 20586  df-drng 20747  df-field 20748  df-lmod 20876  df-lss 20947  df-lsp 20987  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-chr 21533  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-evl1 22335  df-primroots 42073
This theorem is referenced by:  aks6d1c1  42097
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