Proof of Theorem aks6d1c5lem3
| Step | Hyp | Ref
| Expression |
| 1 | | aks6d1c5p3.7 |
. . . . . 6
⊢ 𝑀 =
(mulGrp‘(Poly1‘𝐾)) |
| 2 | | aks6d1p5.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ Field) |
| 3 | 2 | fldcrngd 20742 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ CRing) |
| 4 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Poly1‘𝐾) = (Poly1‘𝐾) |
| 5 | 4 | ply1crng 22200 |
. . . . . . . . 9
⊢ (𝐾 ∈ CRing →
(Poly1‘𝐾)
∈ CRing) |
| 6 | 3, 5 | syl 17 |
. . . . . . . 8
⊢ (𝜑 →
(Poly1‘𝐾)
∈ CRing) |
| 7 | | crngring 20242 |
. . . . . . . 8
⊢
((Poly1‘𝐾) ∈ CRing →
(Poly1‘𝐾)
∈ Ring) |
| 8 | 6, 7 | syl 17 |
. . . . . . 7
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Ring) |
| 9 | | eqid 2737 |
. . . . . . . 8
⊢
(mulGrp‘(Poly1‘𝐾)) =
(mulGrp‘(Poly1‘𝐾)) |
| 10 | 9 | ringmgp 20236 |
. . . . . . 7
⊢
((Poly1‘𝐾) ∈ Ring →
(mulGrp‘(Poly1‘𝐾)) ∈ Mnd) |
| 11 | 8, 10 | syl 17 |
. . . . . 6
⊢ (𝜑 →
(mulGrp‘(Poly1‘𝐾)) ∈ Mnd) |
| 12 | 1, 11 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ Mnd) |
| 13 | 1 | fveq2i 6909 |
. . . . . 6
⊢
(Base‘𝑀) =
(Base‘(mulGrp‘(Poly1‘𝐾))) |
| 14 | | aks6d1c5.7 |
. . . . . 6
⊢ ↑ =
(.g‘(mulGrp‘(Poly1‘𝐾))) |
| 15 | | aks6d1c5p3.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑌 ∈ (ℕ0
↑m (0...𝐴))) |
| 16 | | nn0ex 12532 |
. . . . . . . . . . . . . 14
⊢
ℕ0 ∈ V |
| 17 | 16 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℕ0 ∈
V) |
| 18 | | ovexd 7466 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0...𝐴) ∈ V) |
| 19 | | elmapg 8879 |
. . . . . . . . . . . . 13
⊢
((ℕ0 ∈ V ∧ (0...𝐴) ∈ V) → (𝑌 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑌:(0...𝐴)⟶ℕ0)) |
| 20 | 17, 18, 19 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑌 ∈ (ℕ0
↑m (0...𝐴))
↔ 𝑌:(0...𝐴)⟶ℕ0)) |
| 21 | 15, 20 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌:(0...𝐴)⟶ℕ0) |
| 22 | | aks6d1c5p3.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ (0...𝐴)) |
| 23 | 21, 22 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑌‘𝑊) ∈
ℕ0) |
| 24 | 23 | nn0zd 12639 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌‘𝑊) ∈ ℤ) |
| 25 | | aks6d1c5p3.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈
ℕ0) |
| 26 | 25 | nn0zd 12639 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℤ) |
| 27 | 24, 26 | zsubcld 12727 |
. . . . . . . 8
⊢ (𝜑 → ((𝑌‘𝑊) − 𝐶) ∈ ℤ) |
| 28 | | aks6d1c5p3.4 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ≤ (𝑌‘𝑊)) |
| 29 | 23 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑌‘𝑊) ∈ ℝ) |
| 30 | 25 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 31 | 29, 30 | subge0d 11853 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ ((𝑌‘𝑊) − 𝐶) ↔ 𝐶 ≤ (𝑌‘𝑊))) |
| 32 | 28, 31 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ ((𝑌‘𝑊) − 𝐶)) |
| 33 | 27, 32 | jca 511 |
. . . . . . 7
⊢ (𝜑 → (((𝑌‘𝑊) − 𝐶) ∈ ℤ ∧ 0 ≤ ((𝑌‘𝑊) − 𝐶))) |
| 34 | | elnn0z 12626 |
. . . . . . 7
⊢ (((𝑌‘𝑊) − 𝐶) ∈ ℕ0 ↔ (((𝑌‘𝑊) − 𝐶) ∈ ℤ ∧ 0 ≤ ((𝑌‘𝑊) − 𝐶))) |
| 35 | 33, 34 | sylibr 234 |
. . . . . 6
⊢ (𝜑 → ((𝑌‘𝑊) − 𝐶) ∈
ℕ0) |
| 36 | 8 | ringcmnd 20281 |
. . . . . . . . 9
⊢ (𝜑 →
(Poly1‘𝐾)
∈ CMnd) |
| 37 | | cmnmnd 19815 |
. . . . . . . . 9
⊢
((Poly1‘𝐾) ∈ CMnd →
(Poly1‘𝐾)
∈ Mnd) |
| 38 | 36, 37 | syl 17 |
. . . . . . . 8
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Mnd) |
| 39 | | crngring 20242 |
. . . . . . . . . 10
⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring) |
| 40 | 3, 39 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Ring) |
| 41 | | aks6d1c5.6 |
. . . . . . . . . 10
⊢ 𝑋 = (var1‘𝐾) |
| 42 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘(Poly1‘𝐾)) =
(Base‘(Poly1‘𝐾)) |
| 43 | 41, 4, 42 | vr1cl 22219 |
. . . . . . . . 9
⊢ (𝐾 ∈ Ring → 𝑋 ∈
(Base‘(Poly1‘𝐾))) |
| 44 | 40, 43 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈
(Base‘(Poly1‘𝐾))) |
| 45 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(ℤRHom‘𝐾) = (ℤRHom‘𝐾) |
| 46 | 45 | zrhrhm 21522 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Ring →
(ℤRHom‘𝐾)
∈ (ℤring RingHom 𝐾)) |
| 47 | 40, 46 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤring
RingHom 𝐾)) |
| 48 | | zringbas 21464 |
. . . . . . . . . . . 12
⊢ ℤ =
(Base‘ℤring) |
| 49 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 50 | 48, 49 | rhmf 20485 |
. . . . . . . . . . 11
⊢
((ℤRHom‘𝐾) ∈ (ℤring RingHom
𝐾) →
(ℤRHom‘𝐾):ℤ⟶(Base‘𝐾)) |
| 51 | 47, 50 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾)) |
| 52 | 22 | elfzelzd 13565 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ ℤ) |
| 53 | 51, 52 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝑊) ∈ (Base‘𝐾)) |
| 54 | | aks6d1c5p3.6 |
. . . . . . . . . 10
⊢ 𝑆 =
(algSc‘(Poly1‘𝐾)) |
| 55 | 4, 54, 49, 42 | ply1sclcl 22289 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Ring ∧
((ℤRHom‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (𝑆‘((ℤRHom‘𝐾)‘𝑊)) ∈
(Base‘(Poly1‘𝐾))) |
| 56 | 40, 53, 55 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝑆‘((ℤRHom‘𝐾)‘𝑊)) ∈
(Base‘(Poly1‘𝐾))) |
| 57 | | eqid 2737 |
. . . . . . . . 9
⊢
(+g‘(Poly1‘𝐾)) =
(+g‘(Poly1‘𝐾)) |
| 58 | 42, 57 | mndcl 18755 |
. . . . . . . 8
⊢
(((Poly1‘𝐾) ∈ Mnd ∧ 𝑋 ∈
(Base‘(Poly1‘𝐾)) ∧ (𝑆‘((ℤRHom‘𝐾)‘𝑊)) ∈
(Base‘(Poly1‘𝐾))) → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(Poly1‘𝐾))) |
| 59 | 38, 44, 56, 58 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(Poly1‘𝐾))) |
| 60 | 9, 42 | mgpbas 20142 |
. . . . . . . . 9
⊢
(Base‘(Poly1‘𝐾)) =
(Base‘(mulGrp‘(Poly1‘𝐾))) |
| 61 | 60 | eqcomi 2746 |
. . . . . . . 8
⊢
(Base‘(mulGrp‘(Poly1‘𝐾))) =
(Base‘(Poly1‘𝐾)) |
| 62 | 13, 61 | eqtri 2765 |
. . . . . . 7
⊢
(Base‘𝑀) =
(Base‘(Poly1‘𝐾)) |
| 63 | 59, 62 | eleqtrrdi 2852 |
. . . . . 6
⊢ (𝜑 → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈ (Base‘𝑀)) |
| 64 | 13, 14, 11, 35, 63 | mulgnn0cld 19113 |
. . . . 5
⊢ (𝜑 → (((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘𝑀)) |
| 65 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝑀) =
(Base‘𝑀) |
| 66 | 9 | crngmgp 20238 |
. . . . . . . 8
⊢
((Poly1‘𝐾) ∈ CRing →
(mulGrp‘(Poly1‘𝐾)) ∈ CMnd) |
| 67 | 6, 66 | syl 17 |
. . . . . . 7
⊢ (𝜑 →
(mulGrp‘(Poly1‘𝐾)) ∈ CMnd) |
| 68 | 1, 67 | eqeltrid 2845 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ CMnd) |
| 69 | | fzfid 14014 |
. . . . . . 7
⊢ (𝜑 → (0...𝐴) ∈ Fin) |
| 70 | | diffi 9215 |
. . . . . . 7
⊢
((0...𝐴) ∈ Fin
→ ((0...𝐴) ∖
{𝑊}) ∈
Fin) |
| 71 | 69, 70 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((0...𝐴) ∖ {𝑊}) ∈ Fin) |
| 72 | 11 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) →
(mulGrp‘(Poly1‘𝐾)) ∈ Mnd) |
| 73 | 21 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑌:(0...𝐴)⟶ℕ0) |
| 74 | | eldifi 4131 |
. . . . . . . . . 10
⊢ (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) → 𝑖 ∈ (0...𝐴)) |
| 75 | 74 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑖 ∈ (0...𝐴)) |
| 76 | 73, 75 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑌‘𝑖) ∈
ℕ0) |
| 77 | 38 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (Poly1‘𝐾) ∈ Mnd) |
| 78 | 44 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑋 ∈
(Base‘(Poly1‘𝐾))) |
| 79 | 40 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐾 ∈ Ring) |
| 80 | 79, 46, 50 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾)) |
| 81 | | elfzelz 13564 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (0...𝐴) → 𝑖 ∈ ℤ) |
| 82 | 75, 81 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑖 ∈ ℤ) |
| 83 | 80, 82 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) |
| 84 | 4, 54, 49, 42 | ply1sclcl 22289 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Ring ∧
((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → (𝑆‘((ℤRHom‘𝐾)‘𝑖)) ∈
(Base‘(Poly1‘𝐾))) |
| 85 | 79, 83, 84 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑆‘((ℤRHom‘𝐾)‘𝑖)) ∈
(Base‘(Poly1‘𝐾))) |
| 86 | 42, 57 | mndcl 18755 |
. . . . . . . . . 10
⊢
(((Poly1‘𝐾) ∈ Mnd ∧ 𝑋 ∈
(Base‘(Poly1‘𝐾)) ∧ (𝑆‘((ℤRHom‘𝐾)‘𝑖)) ∈
(Base‘(Poly1‘𝐾))) → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))) ∈
(Base‘(Poly1‘𝐾))) |
| 87 | 77, 78, 85, 86 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))) ∈
(Base‘(Poly1‘𝐾))) |
| 88 | 87, 62 | eleqtrrdi 2852 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘𝑀)) |
| 89 | 13, 14 | mulgnn0cl 19108 |
. . . . . . . 8
⊢
(((mulGrp‘(Poly1‘𝐾)) ∈ Mnd ∧ (𝑌‘𝑖) ∈ ℕ0 ∧ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘𝑀)) → ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘𝑀)) |
| 90 | 72, 76, 88, 89 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘𝑀)) |
| 91 | 90 | ralrimiva 3146 |
. . . . . 6
⊢ (𝜑 → ∀𝑖 ∈ ((0...𝐴) ∖ {𝑊})((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘𝑀)) |
| 92 | 65, 68, 71, 91 | gsummptcl 19985 |
. . . . 5
⊢ (𝜑 → (𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘𝑀)) |
| 93 | | eqid 2737 |
. . . . . 6
⊢
(+g‘𝑀) = (+g‘𝑀) |
| 94 | 65, 93 | mndcl 18755 |
. . . . 5
⊢ ((𝑀 ∈ Mnd ∧ (((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘𝑀) ∧ (𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘𝑀)) → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈ (Base‘𝑀)) |
| 95 | 12, 64, 92, 94 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈ (Base‘𝑀)) |
| 96 | 95, 62 | eleqtrdi 2851 |
. . 3
⊢ (𝜑 → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈
(Base‘(Poly1‘𝐾))) |
| 97 | 65, 93 | cmncom 19816 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ CMnd ∧ (((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘𝑀) ∧ (𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘𝑀)) → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 98 | 68, 64, 92, 97 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 99 | 98 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → (((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = (((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 100 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(.r‘(Poly1‘𝐾)) =
(.r‘(Poly1‘𝐾)) |
| 101 | 1, 100 | mgpplusg 20141 |
. . . . . . . . . . . . 13
⊢
(.r‘(Poly1‘𝐾)) = (+g‘𝑀) |
| 102 | 101 | eqcomi 2746 |
. . . . . . . . . . . 12
⊢
(+g‘𝑀) =
(.r‘(Poly1‘𝐾)) |
| 103 | 102 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (+g‘𝑀) =
(.r‘(Poly1‘𝐾))) |
| 104 | 103 | oveqd 7448 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 105 | 104 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = (((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 106 | 92, 62 | eleqtrdi 2851 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) ∈
(Base‘(Poly1‘𝐾))) |
| 107 | 64, 62 | eleqtrdi 2851 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈
(Base‘(Poly1‘𝐾))) |
| 108 | 60, 14, 11, 25, 59 | mulgnn0cld 19113 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈
(Base‘(Poly1‘𝐾))) |
| 109 | 42, 100, 8, 106, 107, 108 | ringassd 20254 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) |
| 110 | 105, 109 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(+g‘𝑀)(((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) |
| 111 | 99, 110 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) |
| 112 | 111 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) = ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))))) |
| 113 | 29 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑌‘𝑊) ∈ ℂ) |
| 114 | 30 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 115 | 113, 114 | npcand 11624 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑌‘𝑊) − 𝐶) + 𝐶) = (𝑌‘𝑊)) |
| 116 | 115 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑌‘𝑊) = (((𝑌‘𝑊) − 𝐶) + 𝐶)) |
| 117 | 116 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) = ((((𝑌‘𝑊) − 𝐶) + 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) |
| 118 | 60 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(Base‘(Poly1‘𝐾)) =
(Base‘(mulGrp‘(Poly1‘𝐾)))) |
| 119 | 59, 118 | eleqtrd 2843 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(mulGrp‘(Poly1‘𝐾)))) |
| 120 | 35, 25, 119 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑌‘𝑊) − 𝐶) ∈ ℕ0 ∧ 𝐶 ∈ ℕ0
∧ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(mulGrp‘(Poly1‘𝐾))))) |
| 121 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Base‘(mulGrp‘(Poly1‘𝐾))) =
(Base‘(mulGrp‘(Poly1‘𝐾))) |
| 122 | 9, 100 | mgpplusg 20141 |
. . . . . . . . . . . . 13
⊢
(.r‘(Poly1‘𝐾)) =
(+g‘(mulGrp‘(Poly1‘𝐾))) |
| 123 | 121, 14, 122 | mulgnn0dir 19122 |
. . . . . . . . . . . 12
⊢
(((mulGrp‘(Poly1‘𝐾)) ∈ Mnd ∧ (((𝑌‘𝑊) − 𝐶) ∈ ℕ0 ∧ 𝐶 ∈ ℕ0
∧ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(mulGrp‘(Poly1‘𝐾))))) → ((((𝑌‘𝑊) − 𝐶) + 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 124 | 11, 120, 123 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((𝑌‘𝑊) − 𝐶) + 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 125 | 117, 124 | eqtr2d 2778 |
. . . . . . . . . 10
⊢ (𝜑 → ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) |
| 126 | 125 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 127 | | aks6d1c5.8 |
. . . . . . . . . . . . 13
⊢ 𝐺 = (𝑔 ∈ (ℕ0
↑m (0...𝐴))
↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) |
| 128 | 127 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 = (𝑔 ∈ (ℕ0
↑m (0...𝐴))
↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))) |
| 129 | 1 | eqcomi 2746 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘(Poly1‘𝐾)) = 𝑀 |
| 130 | 129 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 = 𝑌) →
(mulGrp‘(Poly1‘𝐾)) = 𝑀) |
| 131 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝑌) |
| 132 | 131 | fveq1d 6908 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) = (𝑌‘𝑖)) |
| 133 | 54 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . 18
⊢
(algSc‘(Poly1‘𝐾)) = 𝑆 |
| 134 | 133 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) →
(algSc‘(Poly1‘𝐾)) = 𝑆) |
| 135 | 134 | fveq1d 6908 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) →
((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) = (𝑆‘((ℤRHom‘𝐾)‘𝑖))) |
| 136 | 135 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) → (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))) |
| 137 | 132, 136 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑔 = 𝑌) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))) |
| 138 | 137 | mpteq2dva 5242 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 = 𝑌) → (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) |
| 139 | 130, 138 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 = 𝑌) →
((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑀 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) |
| 140 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑀 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ V) |
| 141 | 128, 139,
15, 140 | fvmptd 7023 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺‘𝑌) = (𝑀 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) |
| 142 | 22 | snssd 4809 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑊} ⊆ (0...𝐴)) |
| 143 | | undifr 4483 |
. . . . . . . . . . . . . . 15
⊢ ({𝑊} ⊆ (0...𝐴) ↔ (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) = (0...𝐴)) |
| 144 | 142, 143 | sylib 218 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) = (0...𝐴)) |
| 145 | 144 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0...𝐴) = (((0...𝐴) ∖ {𝑊}) ∪ {𝑊})) |
| 146 | 145 | mpteq1d 5237 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) |
| 147 | 146 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑀 Σg (𝑖 ∈ (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) |
| 148 | 141, 147 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝑌) = (𝑀 Σg (𝑖 ∈ (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) |
| 149 | | neldifsnd 4793 |
. . . . . . . . . . 11
⊢ (𝜑 → ¬ 𝑊 ∈ ((0...𝐴) ∖ {𝑊})) |
| 150 | 13, 14, 11, 23, 63 | mulgnn0cld 19113 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘𝑀)) |
| 151 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑊 → (𝑌‘𝑖) = (𝑌‘𝑊)) |
| 152 | | 2fveq3 6911 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑊 → (𝑆‘((ℤRHom‘𝐾)‘𝑖)) = (𝑆‘((ℤRHom‘𝐾)‘𝑊))) |
| 153 | 152 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑊 → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) |
| 154 | 151, 153 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑊 → ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) |
| 155 | 65, 101, 68, 71, 90, 22, 149, 150, 154 | gsumunsn 19978 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 Σg (𝑖 ∈ (((0...𝐴) ∖ {𝑊}) ∪ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))) = ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 156 | 148, 155 | eqtr2d 2778 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((𝑌‘𝑊) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = (𝐺‘𝑌)) |
| 157 | 126, 156 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) = (𝐺‘𝑌)) |
| 158 | 157 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) = ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(𝐺‘𝑌))) |
| 159 | 8 | ringgrpd 20239 |
. . . . . . . 8
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Grp) |
| 160 | | aks6d1p5.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 161 | | aks6d1c5.3 |
. . . . . . . . . 10
⊢ 𝑃 = (chr‘𝐾) |
| 162 | | aks6d1c5.4 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈
ℕ0) |
| 163 | | aks6d1c5.5 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 < 𝑃) |
| 164 | 2, 160, 161, 162, 163, 41, 14, 127 | aks6d1c5lem0 42136 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:(ℕ0 ↑m
(0...𝐴))⟶(Base‘(Poly1‘𝐾))) |
| 165 | 164, 15 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘𝑌) ∈
(Base‘(Poly1‘𝐾))) |
| 166 | | eqid 2737 |
. . . . . . . . 9
⊢
(0g‘(Poly1‘𝐾)) =
(0g‘(Poly1‘𝐾)) |
| 167 | | eqid 2737 |
. . . . . . . . 9
⊢
(-g‘(Poly1‘𝐾)) =
(-g‘(Poly1‘𝐾)) |
| 168 | 42, 166, 167 | grpsubid 19042 |
. . . . . . . 8
⊢
(((Poly1‘𝐾) ∈ Grp ∧ (𝐺‘𝑌) ∈
(Base‘(Poly1‘𝐾))) → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(𝐺‘𝑌)) =
(0g‘(Poly1‘𝐾))) |
| 169 | 159, 165,
168 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(𝐺‘𝑌)) =
(0g‘(Poly1‘𝐾))) |
| 170 | 158, 169 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))((𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))(.r‘(Poly1‘𝐾))((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) =
(0g‘(Poly1‘𝐾))) |
| 171 | 112, 170 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → ((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) =
(0g‘(Poly1‘𝐾))) |
| 172 | 171 | fveq2d 6910 |
. . . 4
⊢ (𝜑 →
((deg1‘𝐾)‘((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) = ((deg1‘𝐾)‘(0g‘(Poly1‘𝐾)))) |
| 173 | | eqid 2737 |
. . . . . . 7
⊢
(deg1‘𝐾) = (deg1‘𝐾) |
| 174 | 173, 4, 166 | deg1z 26126 |
. . . . . 6
⊢ (𝐾 ∈ Ring →
((deg1‘𝐾)‘(0g‘(Poly1‘𝐾))) = -∞) |
| 175 | 40, 174 | syl 17 |
. . . . 5
⊢ (𝜑 →
((deg1‘𝐾)‘(0g‘(Poly1‘𝐾))) = -∞) |
| 176 | 2 | flddrngd 20741 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 ∈ DivRing) |
| 177 | | drngdomn 20749 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ DivRing → 𝐾 ∈ Domn) |
| 178 | 176, 177 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ Domn) |
| 179 | 4 | ply1domn 26163 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Domn →
(Poly1‘𝐾)
∈ Domn) |
| 180 | 178, 179 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 →
(Poly1‘𝐾)
∈ Domn) |
| 181 | 6, 180 | jca 511 |
. . . . . . . . . 10
⊢ (𝜑 →
((Poly1‘𝐾)
∈ CRing ∧ (Poly1‘𝐾) ∈ Domn)) |
| 182 | | isidom 20725 |
. . . . . . . . . 10
⊢
((Poly1‘𝐾) ∈ IDomn ↔
((Poly1‘𝐾)
∈ CRing ∧ (Poly1‘𝐾) ∈ Domn)) |
| 183 | 181, 182 | sylibr 234 |
. . . . . . . . 9
⊢ (𝜑 →
(Poly1‘𝐾)
∈ IDomn) |
| 184 | 173, 4, 42 | deg1xrcl 26121 |
. . . . . . . . . . . . . 14
⊢ ((𝑆‘((ℤRHom‘𝐾)‘𝑊)) ∈
(Base‘(Poly1‘𝐾)) → ((deg1‘𝐾)‘(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
ℝ*) |
| 185 | 56, 184 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
ℝ*) |
| 186 | | 0xr 11308 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ* |
| 187 | 186 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ*) |
| 188 | 173, 4, 42 | deg1xrcl 26121 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈
(Base‘(Poly1‘𝐾)) → ((deg1‘𝐾)‘𝑋) ∈
ℝ*) |
| 189 | 44, 188 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((deg1‘𝐾)‘𝑋) ∈
ℝ*) |
| 190 | 173, 4, 49, 54 | deg1sclle 26151 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Ring ∧
((ℤRHom‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → ((deg1‘𝐾)‘(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ≤ 0) |
| 191 | 40, 53, 190 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ≤ 0) |
| 192 | | 0lt1 11785 |
. . . . . . . . . . . . . . 15
⊢ 0 <
1 |
| 193 | 192 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < 1) |
| 194 | 44, 60 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋 ∈
(Base‘(mulGrp‘(Poly1‘𝐾)))) |
| 195 | 121, 14 | mulg1 19099 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∈
(Base‘(mulGrp‘(Poly1‘𝐾))) → (1 ↑ 𝑋) = 𝑋) |
| 196 | 194, 195 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1 ↑ 𝑋) = 𝑋) |
| 197 | 196 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘(1 ↑ 𝑋)) = ((deg1‘𝐾)‘𝑋)) |
| 198 | | isfld 20740 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing)) |
| 199 | 198 | biimpi 216 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ Field → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing)) |
| 200 | 2, 199 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing)) |
| 201 | 200 | simpld 494 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐾 ∈ DivRing) |
| 202 | | drngnzr 20748 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing) |
| 203 | 201, 202 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ NzRing) |
| 204 | | 1nn0 12542 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℕ0 |
| 205 | 204 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈
ℕ0) |
| 206 | 173, 4, 41, 9, 14 | deg1pw 26160 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ NzRing ∧ 1 ∈
ℕ0) → ((deg1‘𝐾)‘(1 ↑ 𝑋)) = 1) |
| 207 | 203, 205,
206 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘(1 ↑ 𝑋)) = 1) |
| 208 | 197, 207 | eqtr3d 2779 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((deg1‘𝐾)‘𝑋) = 1) |
| 209 | 208 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 =
((deg1‘𝐾)‘𝑋)) |
| 210 | 193, 209 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 <
((deg1‘𝐾)‘𝑋)) |
| 211 | 185, 187,
189, 191, 210 | xrlelttrd 13202 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑆‘((ℤRHom‘𝐾)‘𝑊))) < ((deg1‘𝐾)‘𝑋)) |
| 212 | 4, 173, 40, 42, 57, 44, 56, 211 | deg1add 26142 |
. . . . . . . . . . 11
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) = ((deg1‘𝐾)‘𝑋)) |
| 213 | 208, 205 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝜑 →
((deg1‘𝐾)‘𝑋) ∈
ℕ0) |
| 214 | 212, 213 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ ℕ0) |
| 215 | 173, 4, 166, 42 | deg1nn0clb 26129 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Ring ∧ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ∈
(Base‘(Poly1‘𝐾))) → ((𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ≠
(0g‘(Poly1‘𝐾)) ↔ ((deg1‘𝐾)‘(𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ ℕ0)) |
| 216 | 40, 59, 215 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ≠
(0g‘(Poly1‘𝐾)) ↔ ((deg1‘𝐾)‘(𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ ℕ0)) |
| 217 | 214, 216 | mpbird 257 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))) ≠
(0g‘(Poly1‘𝐾))) |
| 218 | 183, 59, 217, 25, 14 | idomnnzpownz 42133 |
. . . . . . . 8
⊢ (𝜑 → (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ≠
(0g‘(Poly1‘𝐾))) |
| 219 | 173, 4, 166, 42 | deg1nn0clb 26129 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Ring ∧ (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈
(Base‘(Poly1‘𝐾))) → ((𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ≠
(0g‘(Poly1‘𝐾)) ↔ ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) ∈
ℕ0)) |
| 220 | 40, 108, 219 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ≠
(0g‘(Poly1‘𝐾)) ↔ ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) ∈
ℕ0)) |
| 221 | 218, 220 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 →
((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) ∈ ℕ0) |
| 222 | 221 | nn0red 12588 |
. . . . . 6
⊢ (𝜑 →
((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) ∈ ℝ) |
| 223 | 222 | mnfltd 13166 |
. . . . 5
⊢ (𝜑 → -∞ <
((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 224 | 175, 223 | eqbrtrd 5165 |
. . . 4
⊢ (𝜑 →
((deg1‘𝐾)‘(0g‘(Poly1‘𝐾))) < ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 225 | 172, 224 | eqbrtrd 5165 |
. . 3
⊢ (𝜑 →
((deg1‘𝐾)‘((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) < ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) |
| 226 | 96, 225 | jca 511 |
. 2
⊢ (𝜑 → (((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈
(Base‘(Poly1‘𝐾)) ∧ ((deg1‘𝐾)‘((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) < ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) |
| 227 | | eqid 2737 |
. . . . 5
⊢
(Unic1p‘𝐾) = (Unic1p‘𝐾) |
| 228 | 4, 42, 166, 227 | drnguc1p 26213 |
. . . 4
⊢ ((𝐾 ∈ DivRing ∧ (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈
(Base‘(Poly1‘𝐾)) ∧ (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ≠
(0g‘(Poly1‘𝐾))) → (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Unic1p‘𝐾)) |
| 229 | 176, 108,
218, 228 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Unic1p‘𝐾)) |
| 230 | | aks6d1c5p3.5 |
. . . 4
⊢ 𝑄 =
(quot1p‘𝐾) |
| 231 | 230, 4, 42, 173, 167, 100, 227 | q1peqb 26195 |
. . 3
⊢ ((𝐾 ∈ Ring ∧ (𝐺‘𝑌) ∈
(Base‘(Poly1‘𝐾)) ∧ (𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Unic1p‘𝐾)) → ((((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈
(Base‘(Poly1‘𝐾)) ∧ ((deg1‘𝐾)‘((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) < ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) ↔ ((𝐺‘𝑌)𝑄(𝐶
↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀
Σg (𝑖 ∈
((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))))) |
| 232 | 40, 165, 229, 231 | syl3anc 1373 |
. 2
⊢ (𝜑 → ((((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))) ∈
(Base‘(Poly1‘𝐾)) ∧ ((deg1‘𝐾)‘((𝐺‘𝑌)(-g‘(Poly1‘𝐾))(((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))(.r‘(Poly1‘𝐾))(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))))) < ((deg1‘𝐾)‘(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊)))))) ↔ ((𝐺‘𝑌)𝑄(𝐶
↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀
Σg (𝑖 ∈
((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))))) |
| 233 | 226, 232 | mpbid 232 |
1
⊢ (𝜑 → ((𝐺‘𝑌)𝑄(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖)))))))) |