| Step | Hyp | Ref | 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) | 
| 4 | 2, 3 | eqeltrid 2844 | . . . . 5
⊢ (𝜑 → 𝐿 ∈ V) | 
| 5 | 4 | imaexd 7939 | . . . 4
⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0))) ∈ V) | 
| 6 |  | hashxrcl 14397 | . . . 4
⊢ ((𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0))) ∈ V → (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) ∈ ℝ*) | 
| 7 | 5, 6 | syl 17 | . . 3
⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) ∈ ℝ*) | 
| 8 | 1, 7 | eqeltrid 2844 | . 2
⊢ (𝜑 → 𝐷 ∈
ℝ*) | 
| 9 |  | aks6d1c6lem2.5 | . . . . . 6
⊢ 𝐽 = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) | 
| 10 | 9 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝐽 = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 11 |  | nn0ex 12534 | . . . . . . . 8
⊢
ℕ0 ∈ V | 
| 12 | 11 | a1i 11 | . . . . . . 7
⊢ (𝜑 → ℕ0 ∈
V) | 
| 13 | 12, 12 | xpexd 7772 | . . . . . 6
⊢ (𝜑 → (ℕ0
× ℕ0) ∈ V) | 
| 14 | 13 | mptexd 7245 | . . . . 5
⊢ (𝜑 → (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V) | 
| 15 | 10, 14 | eqeltrd 2840 | . . . 4
⊢ (𝜑 → 𝐽 ∈ V) | 
| 16 | 15 | imaexd 7939 | . . 3
⊢ (𝜑 → (𝐽 “ (ℕ0 ×
ℕ0)) ∈ V) | 
| 17 |  | hashxrcl 14397 | . . 3
⊢ ((𝐽 “ (ℕ0
× ℕ0)) ∈ V → (♯‘(𝐽 “ (ℕ0 ×
ℕ0))) ∈ ℝ*) | 
| 18 | 16, 17 | syl 17 | . 2
⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0
× ℕ0))) ∈ ℝ*) | 
| 19 |  | fvexd 6920 | . . . . 5
⊢ (𝜑 →
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V) | 
| 20 |  | cnvexg 7947 | . . . . 5
⊢
(((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V → ◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V) | 
| 21 | 19, 20 | syl 17 | . . . 4
⊢ (𝜑 → ◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V) | 
| 22 | 21 | imaexd 7939 | . . 3
⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V) | 
| 23 |  | hashxrcl 14397 | . . 3
⊢ ((◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V → (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) ∈ ℝ*) | 
| 24 | 22, 23 | syl 17 | . 2
⊢ (𝜑 → (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) ∈ ℝ*) | 
| 25 | 1 | a1i 11 | . . 3
⊢ (𝜑 → 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0))))) | 
| 26 |  | aks6d1c6lem2.6 | . . 3
⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 ×
ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 ×
ℕ0)))) | 
| 27 | 25, 26 | eqbrtrd 5164 | . 2
⊢ (𝜑 → 𝐷 ≤ (♯‘(𝐽 “ (ℕ0 ×
ℕ0)))) | 
| 28 | 22 | elexd 3503 | . . 3
⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V) | 
| 29 |  | nfv 1913 | . . . 4
⊢
Ⅎ𝑤𝜑 | 
| 30 |  | ovexd 7467 | . . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V) | 
| 31 | 30, 9 | fmptd 7133 | . . . . 5
⊢ (𝜑 → 𝐽:(ℕ0 ×
ℕ0)⟶V) | 
| 32 |  | ffun 6738 | . . . . 5
⊢ (𝐽:(ℕ0 ×
ℕ0)⟶V → Fun 𝐽) | 
| 33 | 31, 32 | syl 17 | . . . 4
⊢ (𝜑 → Fun 𝐽) | 
| 34 | 9 | a1i 11 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐽 = (𝑗 ∈ (ℕ0 ×
ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 35 |  | simpr 484 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑤) → 𝑗 = 𝑤) | 
| 36 | 35 | fveq2d 6909 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑤) → (𝐸‘𝑗) = (𝐸‘𝑤)) | 
| 37 | 36 | oveq1d 7447 | . . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑗 = 𝑤) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) | 
| 38 |  | simpr 484 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑤 ∈ (ℕ0 ×
ℕ0)) | 
| 39 |  | ovexd 7467 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ V) | 
| 40 | 34, 37, 38, 39 | fvmptd 7022 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐽‘𝑤) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) | 
| 41 |  | eqid 2736 | . . . . . . . . . 10
⊢
(eval1‘𝐾) = (eval1‘𝐾) | 
| 42 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Poly1‘𝐾) = (Poly1‘𝐾) | 
| 43 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Base‘𝐾) =
(Base‘𝐾) | 
| 44 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Base‘(Poly1‘𝐾)) =
(Base‘(Poly1‘𝐾)) | 
| 45 |  | aks6d1c6.3 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ Field) | 
| 46 | 45 | fldcrngd 20743 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ CRing) | 
| 47 | 46 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐾 ∈ CRing) | 
| 48 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(mulGrp‘𝐾) =
(mulGrp‘𝐾) | 
| 49 | 48, 43 | mgpbas 20143 | . . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘(mulGrp‘𝐾)) | 
| 50 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(.g‘(mulGrp‘𝐾)) =
(.g‘(mulGrp‘𝐾)) | 
| 51 | 46 | crngringd 20244 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 ∈ Ring) | 
| 52 | 48 | ringmgp 20237 | . . . . . . . . . . . . 13
⊢ (𝐾 ∈ Ring →
(mulGrp‘𝐾) ∈
Mnd) | 
| 53 | 51, 52 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd) | 
| 54 | 53 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (mulGrp‘𝐾) ∈ Mnd) | 
| 55 |  | aks6d1c6.6 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 56 |  | aks6d1c6.4 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑃 ∈ ℙ) | 
| 57 |  | aks6d1c6.7 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑃 ∥ 𝑁) | 
| 58 |  | aks6d1c6.12 | . . . . . . . . . . . . . 14
⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) | 
| 59 | 55, 56, 57, 58 | aks6d1c2p1 42120 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸:(ℕ0 ×
ℕ0)⟶ℕ) | 
| 60 | 59 | ffvelcdmda 7103 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑤) ∈ ℕ) | 
| 61 | 60 | nnnn0d 12589 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑤) ∈
ℕ0) | 
| 62 |  | aks6d1c6.16 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) | 
| 63 | 48 | crngmgp 20239 | . . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ CRing →
(mulGrp‘𝐾) ∈
CMnd) | 
| 64 | 46, 63 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd) | 
| 65 |  | aks6d1c6.5 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑅 ∈ ℕ) | 
| 66 | 65 | nnnn0d 12589 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑅 ∈
ℕ0) | 
| 67 | 64, 66, 50 | isprimroot 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)) → 𝑅 ∥ 𝑜)))) | 
| 68 | 62, 67 | mpbid 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)) → 𝑅 ∥ 𝑜))) | 
| 69 | 68 | simp1d 1142 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾))) | 
| 70 | 69, 49 | eleqtrrdi 2851 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾)) | 
| 71 | 70 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑀 ∈ (Base‘𝐾)) | 
| 72 | 49, 50, 54, 61, 71 | mulgnn0cld 19114 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾)) | 
| 73 |  | aks6d1c6.2 | . . . . . . . . . . . . . 14
⊢ 𝑃 = (chr‘𝐾) | 
| 74 |  | aks6d1c6.11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ∈
ℕ0) | 
| 75 |  | aks6d1c6.9 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 < 𝑃) | 
| 76 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢
(var1‘𝐾) = (var1‘𝐾) | 
| 77 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢
(.g‘(mulGrp‘(Poly1‘𝐾))) =
(.g‘(mulGrp‘(Poly1‘𝐾))) | 
| 78 |  | aks6d1c6.10 | . . . . . . . . . . . . . 14
⊢ 𝐺 = (𝑔 ∈ (ℕ0
↑m (0...𝐴))
↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) | 
| 79 | 45, 56, 73, 74, 75, 76, 77, 78 | aks6d1c5lem0 42137 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:(ℕ0 ↑m
(0...𝐴))⟶(Base‘(Poly1‘𝐾))) | 
| 80 |  | aks6d1c6lem2.1 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ 𝑆) | 
| 81 |  | aks6d1c6.19 | . . . . . . . . . . . . . . . 16
⊢ 𝑆 = {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} | 
| 82 | 81 | eleq2i 2832 | . . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}) | 
| 83 | 80, 82 | sylib 218 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}) | 
| 84 |  | elrabi 3686 | . . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0
↑m (0...𝐴))) | 
| 85 | 84 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0
↑m (0...𝐴)))) | 
| 86 | 83, 85 | mpd 15 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑈 ∈ (ℕ0
↑m (0...𝐴))) | 
| 87 | 79, 86 | ffvelcdmd 7104 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐺‘𝑈) ∈
(Base‘(Poly1‘𝐾))) | 
| 88 | 87 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐺‘𝑈) ∈
(Base‘(Poly1‘𝐾))) | 
| 89 |  | eqidd 2737 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 90 | 88, 89 | jca 511 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐺‘𝑈) ∈
(Base‘(Poly1‘𝐾)) ∧ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 91 |  | aks6d1c6lem2.2 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉 ∈ 𝑆) | 
| 92 | 81 | eleq2i 2832 | . . . . . . . . . . . . . . 15
⊢ (𝑉 ∈ 𝑆 ↔ 𝑉 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}) | 
| 93 | 91, 92 | sylib 218 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}) | 
| 94 |  | elrabi 3686 | . . . . . . . . . . . . . . 15
⊢ (𝑉 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0
↑m (0...𝐴))) | 
| 95 | 94 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑉 ∈ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0
↑m (0...𝐴)))) | 
| 96 | 93, 95 | mpd 15 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑉 ∈ (ℕ0
↑m (0...𝐴))) | 
| 97 | 79, 96 | ffvelcdmd 7104 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐺‘𝑉) ∈
(Base‘(Poly1‘𝐾))) | 
| 98 | 97 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐺‘𝑉) ∈
(Base‘(Poly1‘𝐾))) | 
| 99 |  | eqidd 2737 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 100 | 98, 99 | jca 511 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐺‘𝑉) ∈
(Base‘(Poly1‘𝐾)) ∧ (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 101 |  | eqid 2736 | . . . . . . . . . 10
⊢
(-g‘(Poly1‘𝐾)) =
(-g‘(Poly1‘𝐾)) | 
| 102 |  | eqid 2736 | . . . . . . . . . 10
⊢
(-g‘𝐾) = (-g‘𝐾) | 
| 103 | 41, 42, 43, 44, 47, 72, 90, 100, 101, 102 | evl1subd 22347 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾)) ∧
(((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))) | 
| 104 | 103 | simprd 495 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 105 |  | fveq2 6905 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) | 
| 106 | 105 | oveq2d 7448 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))) | 
| 107 |  | oveq2 7440 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) | 
| 108 | 107 | fveq2d 6909 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 109 | 106, 108 | eqeq12d 2752 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → (((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 110 |  | vex 3483 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑘 ∈ V | 
| 111 |  | vex 3483 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑙 ∈ V | 
| 112 | 110, 111 | op1std 8025 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = 〈𝑘, 𝑙〉 → (1st ‘𝑠) = 𝑘) | 
| 113 | 112 | oveq2d 7448 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 〈𝑘, 𝑙〉 → (𝑃↑(1st ‘𝑠)) = (𝑃↑𝑘)) | 
| 114 | 110, 111 | op2ndd 8026 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = 〈𝑘, 𝑙〉 → (2nd ‘𝑠) = 𝑙) | 
| 115 | 114 | oveq2d 7448 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 〈𝑘, 𝑙〉 → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑𝑙)) | 
| 116 | 113, 115 | oveq12d 7450 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 〈𝑘, 𝑙〉 → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) | 
| 117 | 116 | mpompt 7548 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ (ℕ0
× ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0
↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) | 
| 118 | 58 | eqcomi 2745 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0,
𝑙 ∈
ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = 𝐸 | 
| 119 | 117, 118 | eqtri 2764 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ (ℕ0
× ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = 𝐸 | 
| 120 | 119 | eqcomi 2745 | . . . . . . . . . . . . . . . . 17
⊢ 𝐸 = (𝑠 ∈ (ℕ0 ×
ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) | 
| 121 | 120 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐸 = (𝑠 ∈ (ℕ0 ×
ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))))) | 
| 122 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → 𝑠 = 𝑤) | 
| 123 | 122 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → (1st ‘𝑠) = (1st ‘𝑤)) | 
| 124 | 123 | oveq2d 7448 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → (𝑃↑(1st ‘𝑠)) = (𝑃↑(1st ‘𝑤))) | 
| 125 | 122 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → (2nd ‘𝑠) = (2nd ‘𝑤)) | 
| 126 | 125 | oveq2d 7448 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑(2nd ‘𝑤))) | 
| 127 | 124, 126 | oveq12d 7450 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤)))) | 
| 128 |  | ovexd 7467 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∈ V) | 
| 129 | 121, 127,
38, 128 | fvmptd 7022 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑤) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤)))) | 
| 130 |  | aks6d1c6.1 | . . . . . . . . . . . . . . . 16
⊢  ∼ =
{〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈
(Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))} | 
| 131 | 45 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐾 ∈ Field) | 
| 132 | 56 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑃 ∈ ℙ) | 
| 133 | 65 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑅 ∈ ℕ) | 
| 134 | 55 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑁 ∈ ℕ) | 
| 135 | 57 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑃 ∥ 𝑁) | 
| 136 |  | aks6d1c6.8 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | 
| 137 | 136 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝑁 gcd 𝑅) = 1) | 
| 138 |  | ovexd 7467 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (0...𝐴) ∈ V) | 
| 139 | 12, 138 | elmapd 8881 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑈 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑈:(0...𝐴)⟶ℕ0)) | 
| 140 | 86, 139 | mpbid 232 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑈:(0...𝐴)⟶ℕ0) | 
| 141 | 140 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑈:(0...𝐴)⟶ℕ0) | 
| 142 | 74 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐴 ∈
ℕ0) | 
| 143 |  | xp1st 8047 | . . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ (ℕ0
× ℕ0) → (1st ‘𝑤) ∈
ℕ0) | 
| 144 | 143 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (1st ‘𝑤) ∈
ℕ0) | 
| 145 |  | xp2nd 8048 | . . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ (ℕ0
× ℕ0) → (2nd ‘𝑤) ∈
ℕ0) | 
| 146 | 145 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (2nd ‘𝑤) ∈
ℕ0) | 
| 147 |  | eqid 2736 | . . . . . . . . . . . . . . . 16
⊢ ((𝑃↑(1st
‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) | 
| 148 |  | aks6d1c6.14 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) | 
| 149 | 148 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼
((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎)))) | 
| 150 |  | aks6d1c6.15 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) | 
| 151 | 150 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾)) | 
| 152 | 130, 73, 131, 132, 133, 134, 135, 137, 141, 78, 142, 144, 146, 147, 149, 151 | aks6d1c1rh 42127 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑈)) | 
| 153 | 129, 152 | eqbrtrd 5164 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑈)) | 
| 154 | 130, 88, 60 | aks6d1c1p1 42109 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤) ∼ (𝐺‘𝑈) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)))) | 
| 155 | 153, 154 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦))) | 
| 156 | 62 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)) | 
| 157 | 109, 155,
156 | rspcdva 3622 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 158 | 157 | eqcomd 2742 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))) | 
| 159 |  | aks6d1c6.17 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝐻 = (ℎ ∈ (ℕ0
↑m (0...𝐴))
↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) | 
| 160 | 159 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐻 = (ℎ ∈ (ℕ0
↑m (0...𝐴))
↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))) | 
| 161 | 160 | reseq1d 5995 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐻 ↾ 𝑆) = ((ℎ ∈ (ℕ0
↑m (0...𝐴))
↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ↾ 𝑆)) | 
| 162 | 81 | a1i 11 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 = {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}) | 
| 163 |  | ssrab2 4079 | . . . . . . . . . . . . . . . . . . . 20
⊢ {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0
↑m (0...𝐴)) | 
| 164 | 163 | a1i 11 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {𝑠 ∈ (ℕ0
↑m (0...𝐴))
∣ Σ𝑡 ∈
(0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0
↑m (0...𝐴))) | 
| 165 | 162, 164 | eqsstrd 4017 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑆 ⊆ (ℕ0
↑m (0...𝐴))) | 
| 166 | 165 | resmptd 6057 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((ℎ ∈ (ℕ0
↑m (0...𝐴))
↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))) | 
| 167 | 161, 166 | eqtrd 2776 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐻 ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))) | 
| 168 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ℎ = 𝑈) → ℎ = 𝑈) | 
| 169 | 168 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ℎ = 𝑈) → (𝐺‘ℎ) = (𝐺‘𝑈)) | 
| 170 | 169 | fveq2d 6909 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ℎ = 𝑈) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑈))) | 
| 171 | 170 | fveq1d 6907 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ℎ = 𝑈) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) | 
| 172 |  | fvexd 6920 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) ∈ V) | 
| 173 | 167, 171,
80, 172 | fvmptd 7022 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) | 
| 174 | 173 | eqcomd 2742 | . . . . . . . . . . . . . 14
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = ((𝐻 ↾ 𝑆)‘𝑈)) | 
| 175 |  | aks6d1c6lem2.3 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = ((𝐻 ↾ 𝑆)‘𝑉)) | 
| 176 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ℎ = 𝑉) → ℎ = 𝑉) | 
| 177 | 176 | fveq2d 6909 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ℎ = 𝑉) → (𝐺‘ℎ) = (𝐺‘𝑉)) | 
| 178 | 177 | fveq2d 6909 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ℎ = 𝑉) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑉))) | 
| 179 | 178 | fveq1d 6907 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ℎ = 𝑉) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) | 
| 180 |  | fvexd 6920 | . . . . . . . . . . . . . . 15
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀) ∈ V) | 
| 181 | 167, 179,
91, 180 | fvmptd 7022 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑉) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) | 
| 182 | 174, 175,
181 | 3eqtrd 2780 | . . . . . . . . . . . . 13
⊢ (𝜑 →
(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) | 
| 183 | 182 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) | 
| 184 | 183 | oveq2d 7448 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))) | 
| 185 | 158, 184 | eqtrd 2776 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))) | 
| 186 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) | 
| 187 | 186 | oveq2d 7448 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))) | 
| 188 | 107 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 189 | 187, 188 | eqeq12d 2752 | . . . . . . . . . . 11
⊢ (𝑦 = 𝑀 → (((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 190 | 12, 138 | elmapd 8881 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑉 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑉:(0...𝐴)⟶ℕ0)) | 
| 191 | 96, 190 | mpbid 232 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉:(0...𝐴)⟶ℕ0) | 
| 192 | 191 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝑉:(0...𝐴)⟶ℕ0) | 
| 193 | 130, 73, 131, 132, 133, 134, 135, 137, 192, 78, 142, 144, 146, 147, 149, 151 | aks6d1c1rh 42127 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑉)) | 
| 194 | 129, 193 | eqbrtrd 5164 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑉)) | 
| 195 | 130, 98, 60 | aks6d1c1p1 42109 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤) ∼ (𝐺‘𝑉) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)))) | 
| 196 | 194, 195 | mpbid 232 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦))) | 
| 197 | 189, 196,
156 | rspcdva 3622 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 198 | 185, 197 | eqtrd 2776 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) | 
| 199 | 46 | crnggrpd 20245 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ Grp) | 
| 200 | 199 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → 𝐾 ∈ Grp) | 
| 201 | 41, 42, 43, 44, 47, 72, 88 | fveval1fvcl 22338 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾)) | 
| 202 | 41, 42, 43, 44, 47, 72, 98 | fveval1fvcl 22338 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾)) | 
| 203 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(0g‘𝐾) = (0g‘𝐾) | 
| 204 | 43, 203, 102 | grpsubeq0 19045 | . . . . . . . . . 10
⊢ ((𝐾 ∈ Grp ∧
(((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾) ∧ (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾)) → (((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾) ↔ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 205 | 200, 201,
202, 204 | syl3anc 1372 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾) ↔ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))) | 
| 206 | 198, 205 | mpbird 257 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾)) | 
| 207 | 104, 206 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾)) | 
| 208 |  | fvexd 6920 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V) | 
| 209 |  | elsng 4639 | . . . . . . . 8
⊢
((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾))) | 
| 210 | 208, 209 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾))) | 
| 211 | 207, 210 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)}) | 
| 212 |  | eqid 2736 | . . . . . . . . . . . . . . 15
⊢ (𝐾 ↑s
(Base‘𝐾)) = (𝐾 ↑s
(Base‘𝐾)) | 
| 213 | 41, 42, 212, 43 | evl1rhm 22337 | . . . . . . . . . . . . . 14
⊢ (𝐾 ∈ CRing →
(eval1‘𝐾)
∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾)))) | 
| 214 | 46, 213 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 →
(eval1‘𝐾)
∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾)))) | 
| 215 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢
(Base‘(𝐾
↑s (Base‘𝐾))) = (Base‘(𝐾 ↑s (Base‘𝐾))) | 
| 216 | 44, 215 | rhmf 20486 | . . . . . . . . . . . . 13
⊢
((eval1‘𝐾) ∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾))) →
(eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s
(Base‘𝐾)))) | 
| 217 | 214, 216 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 →
(eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s
(Base‘𝐾)))) | 
| 218 |  | fvexd 6920 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (Base‘𝐾) ∈ V) | 
| 219 | 212, 43 | pwsbas 17533 | . . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Field ∧
(Base‘𝐾) ∈ V)
→ ((Base‘𝐾)
↑m (Base‘𝐾)) = (Base‘(𝐾 ↑s (Base‘𝐾)))) | 
| 220 | 45, 218, 219 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((Base‘𝐾) ↑m
(Base‘𝐾)) =
(Base‘(𝐾
↑s (Base‘𝐾)))) | 
| 221 | 220 | feq3d 6722 | . . . . . . . . . . . 12
⊢ (𝜑 →
((eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶((Base‘𝐾) ↑m
(Base‘𝐾)) ↔
(eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s
(Base‘𝐾))))) | 
| 222 | 217, 221 | mpbird 257 | . . . . . . . . . . 11
⊢ (𝜑 →
(eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶((Base‘𝐾) ↑m
(Base‘𝐾))) | 
| 223 | 42 | ply1ring 22250 | . . . . . . . . . . . . . 14
⊢ (𝐾 ∈ Ring →
(Poly1‘𝐾)
∈ Ring) | 
| 224 | 51, 223 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Ring) | 
| 225 |  | ringgrp 20236 | . . . . . . . . . . . . 13
⊢
((Poly1‘𝐾) ∈ Ring →
(Poly1‘𝐾)
∈ Grp) | 
| 226 | 224, 225 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Grp) | 
| 227 | 44, 101 | grpsubcl 19039 | . . . . . . . . . . . 12
⊢
(((Poly1‘𝐾) ∈ Grp ∧ (𝐺‘𝑈) ∈
(Base‘(Poly1‘𝐾)) ∧ (𝐺‘𝑉) ∈
(Base‘(Poly1‘𝐾))) → ((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾))) | 
| 228 | 226, 87, 97, 227 | syl3anc 1372 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾))) | 
| 229 | 222, 228 | ffvelcdmd 7104 | . . . . . . . . . 10
⊢ (𝜑 →
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾))) | 
| 230 | 218, 218 | elmapd 8881 | . . . . . . . . . 10
⊢ (𝜑 →
(((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)) ↔ ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))):(Base‘𝐾)⟶(Base‘𝐾))) | 
| 231 | 229, 230 | mpbid 232 | . . . . . . . . 9
⊢ (𝜑 →
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))):(Base‘𝐾)⟶(Base‘𝐾)) | 
| 232 | 231 | ffund 6739 | . . . . . . . 8
⊢ (𝜑 → Fun
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) | 
| 233 | 232 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) | 
| 234 | 231 | ffnd 6736 | . . . . . . . . . . 11
⊢ (𝜑 →
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾)) | 
| 235 | 234 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾)) | 
| 236 | 235 | fndmd 6672 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) = (Base‘𝐾)) | 
| 237 | 236 | eqcomd 2742 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (Base‘𝐾) = dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) | 
| 238 | 72, 237 | eleqtrd 2842 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) | 
| 239 |  | fvimacnv 7072 | . . . . . . 7
⊢ ((Fun
((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∧ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))) | 
| 240 | 233, 238,
239 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))) | 
| 241 | 211, 240 | mpbid 232 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) | 
| 242 | 40, 241 | eqeltrd 2840 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 ×
ℕ0)) → (𝐽‘𝑤) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) | 
| 243 | 29, 33, 242 | funimassd 6974 | . . 3
⊢ (𝜑 → (𝐽 “ (ℕ0 ×
ℕ0)) ⊆ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) | 
| 244 |  | hashss 14449 | . . 3
⊢ (((◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V ∧ (𝐽 “ (ℕ0 ×
ℕ0)) ⊆ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) → (♯‘(𝐽 “ (ℕ0 ×
ℕ0))) ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))) | 
| 245 | 28, 243, 244 | syl2anc 584 | . 2
⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0
× ℕ0))) ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))) | 
| 246 | 8, 18, 24, 27, 245 | xrletrd 13205 | 1
⊢ (𝜑 → 𝐷 ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))) |