| Step | Hyp | Ref
| Expression |
| 1 | | rtelextdg2.1 |
. . . . 5
⊢ 𝐾 = (𝐸 ↾s 𝐹) |
| 2 | | rtelextdg2.2 |
. . . . 5
⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋}))) |
| 3 | | eqid 2737 |
. . . . 5
⊢
(deg1‘𝐸) = (deg1‘𝐸) |
| 4 | | eqid 2737 |
. . . . 5
⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹) |
| 5 | | rtelextdg2.9 |
. . . . 5
⊢ (𝜑 → 𝐸 ∈ Field) |
| 6 | | rtelextdg2.10 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) |
| 7 | | rtelextdg2.11 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 8 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑝 = 𝐺 → ((𝐸 evalSub1 𝐹)‘𝑝) = ((𝐸 evalSub1 𝐹)‘𝐺)) |
| 9 | 8 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝑝 = 𝐺 → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋)) |
| 10 | 9 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑝 = 𝐺 → ((((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 ↔ (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = 0 )) |
| 11 | | rtelextdg2lem.6 |
. . . . . . . . 9
⊢ 𝐺 = ((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))) |
| 12 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘𝑃) =
(Base‘𝑃) |
| 13 | | rtelextdg2lem.2 |
. . . . . . . . . 10
⊢ ⊕ =
(+g‘𝑃) |
| 14 | | fldsdrgfld 20799 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field) |
| 15 | 5, 6, 14 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field) |
| 16 | 15 | fldcrngd 20742 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ CRing) |
| 17 | 1, 16 | eqeltrid 2845 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 ∈ CRing) |
| 18 | 17 | crngringd 20243 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ Ring) |
| 19 | | rtelextdg2.4 |
. . . . . . . . . . . . 13
⊢ 𝑃 = (Poly1‘𝐾) |
| 20 | 19 | ply1ring 22249 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring) |
| 21 | 18, 20 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ Ring) |
| 22 | 21 | ringgrpd 20239 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ Grp) |
| 23 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(mulGrp‘𝑃) =
(mulGrp‘𝑃) |
| 24 | 23, 12 | mgpbas 20142 |
. . . . . . . . . . 11
⊢
(Base‘𝑃) =
(Base‘(mulGrp‘𝑃)) |
| 25 | | rtelextdg2lem.4 |
. . . . . . . . . . 11
⊢ ∧ =
(.g‘(mulGrp‘𝑃)) |
| 26 | 23 | ringmgp 20236 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ Ring →
(mulGrp‘𝑃) ∈
Mnd) |
| 27 | 21, 26 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd) |
| 28 | | 2nn0 12543 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℕ0 |
| 29 | 28 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 2 ∈
ℕ0) |
| 30 | | rtelextdg2lem.1 |
. . . . . . . . . . . . 13
⊢ 𝑌 = (var1‘𝐾) |
| 31 | 30, 19, 12 | vr1cl 22219 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Ring → 𝑌 ∈ (Base‘𝑃)) |
| 32 | 18, 31 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌 ∈ (Base‘𝑃)) |
| 33 | 24, 25, 27, 29, 32 | mulgnn0cld 19113 |
. . . . . . . . . 10
⊢ (𝜑 → (2 ∧ 𝑌) ∈ (Base‘𝑃)) |
| 34 | | rtelextdg2lem.3 |
. . . . . . . . . . . 12
⊢ ⊗ =
(.r‘𝑃) |
| 35 | | rtelextdg2lem.5 |
. . . . . . . . . . . . 13
⊢ 𝑈 = (algSc‘𝑃) |
| 36 | 5 | fldcrngd 20742 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ CRing) |
| 37 | | sdrgsubrg 20792 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) |
| 38 | 6, 37 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
| 39 | | rtelextdg2.12 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ∈ 𝐹) |
| 40 | 19, 1, 35, 12, 36, 38, 39 | ressasclcl 33596 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑈‘𝐴) ∈ (Base‘𝑃)) |
| 41 | 12, 34, 21, 40, 32 | ringcld 20257 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑈‘𝐴) ⊗ 𝑌) ∈ (Base‘𝑃)) |
| 42 | | rtelextdg2.13 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈ 𝐹) |
| 43 | 19, 1, 35, 12, 36, 38, 42 | ressasclcl 33596 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑈‘𝐵) ∈ (Base‘𝑃)) |
| 44 | 12, 13, 22, 41, 43 | grpcld 18965 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)) ∈ (Base‘𝑃)) |
| 45 | 12, 13, 22, 33, 44 | grpcld 18965 |
. . . . . . . . 9
⊢ (𝜑 → ((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))) ∈ (Base‘𝑃)) |
| 46 | 11, 45 | eqeltrid 2845 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃)) |
| 47 | 11 | fveq2i 6909 |
. . . . . . . . . . . 12
⊢
(coe1‘𝐺) = (coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))) |
| 48 | 47 | fveq1i 6907 |
. . . . . . . . . . 11
⊢
((coe1‘𝐺)‘2) = ((coe1‘((2
∧
𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘2) |
| 49 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐾) = (+g‘𝐾) |
| 50 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ Ring ∧ (2 ∧ 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)) ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0)
→ ((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘2) = (((coe1‘(2
∧
𝑌))‘2)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2))) |
| 51 | 18, 33, 44, 29, 50 | syl31anc 1375 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘2) = (((coe1‘(2
∧
𝑌))‘2)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2))) |
| 52 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(0g‘𝐾) = (0g‘𝐾) |
| 53 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(1r‘𝐾) = (1r‘𝐾) |
| 54 | 19, 30, 25, 18, 29, 52, 53 | coe1mon 33610 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (coe1‘(2
∧
𝑌)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 2, (1r‘𝐾), (0g‘𝐾)))) |
| 55 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 = 2) → 𝑖 = 2) |
| 56 | 55 | iftrued 4533 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 = 2) → if(𝑖 = 2, (1r‘𝐾), (0g‘𝐾)) = (1r‘𝐾)) |
| 57 | | fvexd 6921 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1r‘𝐾) ∈ V) |
| 58 | 54, 56, 29, 57 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((coe1‘(2
∧
𝑌))‘2) =
(1r‘𝐾)) |
| 59 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ Ring ∧ ((𝑈‘𝐴) ⊗ 𝑌) ∈ (Base‘𝑃) ∧ (𝑈‘𝐵) ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0)
→ ((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘2))) |
| 60 | 18, 41, 43, 29, 59 | syl31anc 1375 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘2))) |
| 61 | | rtelextdg2.5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑉 = (Base‘𝐸) |
| 62 | 61 | sdrgss 20794 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝑉) |
| 63 | 1, 61 | ressbas2 17283 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 ⊆ 𝑉 → 𝐹 = (Base‘𝐾)) |
| 64 | 6, 62, 63 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐹 = (Base‘𝐾)) |
| 65 | 39, 64 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ (Base‘𝐾)) |
| 66 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 67 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(.r‘𝐾) = (.r‘𝐾) |
| 68 | 19, 12, 66, 35, 34, 67 | coe1sclmulfv 22286 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0)
→ ((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘2))) |
| 69 | 18, 65, 32, 29, 68 | syl121anc 1377 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘2))) |
| 70 | 19, 30, 18, 52, 53 | coe1vr1 33613 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
(coe1‘𝑌) =
(𝑖 ∈
ℕ0 ↦ if(𝑖 = 1, (1r‘𝐾), (0g‘𝐾)))) |
| 71 | | 1ne2 12474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ≠
2 |
| 72 | 71 | nesymi 2998 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ¬ 2
= 1 |
| 73 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 2 → (𝑖 = 1 ↔ 2 = 1)) |
| 74 | 72, 73 | mtbiri 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 2 → ¬ 𝑖 = 1) |
| 75 | 74 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 = 2) → ¬ 𝑖 = 1) |
| 76 | 75 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 = 2) → if(𝑖 = 1, (1r‘𝐾), (0g‘𝐾)) = (0g‘𝐾)) |
| 77 | | fvexd 6921 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (0g‘𝐾) ∈ V) |
| 78 | 70, 76, 29, 77 | fvmptd 7023 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((coe1‘𝑌)‘2) = (0g‘𝐾)) |
| 79 | 78 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴(.r‘𝐾)((coe1‘𝑌)‘2)) = (𝐴(.r‘𝐾)(0g‘𝐾))) |
| 80 | 66, 67, 52, 18, 65 | ringrzd 20293 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴(.r‘𝐾)(0g‘𝐾)) = (0g‘𝐾)) |
| 81 | 69, 79, 80 | 3eqtrd 2781 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2) = (0g‘𝐾)) |
| 82 | 42, 64 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 ∈ (Base‘𝐾)) |
| 83 | 19, 35, 66, 52 | coe1scl 22290 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) →
(coe1‘(𝑈‘𝐵)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 𝐵, (0g‘𝐾)))) |
| 84 | 18, 82, 83 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(coe1‘(𝑈‘𝐵)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 𝐵, (0g‘𝐾)))) |
| 85 | | 0ne2 12473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ≠
2 |
| 86 | 85 | neii 2942 |
. . . . . . . . . . . . . . . . . . 19
⊢ ¬ 0
= 2 |
| 87 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 0 → (𝑖 = 2 ↔ 0 = 2)) |
| 88 | 86, 87 | mtbiri 327 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 0 → ¬ 𝑖 = 2) |
| 89 | 88, 55 | nsyl3 138 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 = 2) → ¬ 𝑖 = 0) |
| 90 | 89 | iffalsed 4536 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 = 2) → if(𝑖 = 0, 𝐵, (0g‘𝐾)) = (0g‘𝐾)) |
| 91 | 84, 90, 29, 77 | fvmptd 7023 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((coe1‘(𝑈‘𝐵))‘2) = (0g‘𝐾)) |
| 92 | 81, 91 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘2)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘2)) = ((0g‘𝐾)(+g‘𝐾)(0g‘𝐾))) |
| 93 | 18 | ringgrpd 20239 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ Grp) |
| 94 | 66, 52 | grpidcl 18983 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ Grp →
(0g‘𝐾)
∈ (Base‘𝐾)) |
| 95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0g‘𝐾) ∈ (Base‘𝐾)) |
| 96 | 66, 49, 52, 93, 95 | grpridd 18988 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((0g‘𝐾)(+g‘𝐾)(0g‘𝐾)) = (0g‘𝐾)) |
| 97 | 60, 92, 96 | 3eqtrd 2781 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2) = (0g‘𝐾)) |
| 98 | 58, 97 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(((coe1‘(2 ∧ 𝑌))‘2)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘2)) = ((1r‘𝐾)(+g‘𝐾)(0g‘𝐾))) |
| 99 | 66, 53 | ringidcl 20262 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ Ring →
(1r‘𝐾)
∈ (Base‘𝐾)) |
| 100 | 18, 99 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1r‘𝐾) ∈ (Base‘𝐾)) |
| 101 | 66, 49, 52, 93, 100 | grpridd 18988 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((1r‘𝐾)(+g‘𝐾)(0g‘𝐾)) = (1r‘𝐾)) |
| 102 | 36 | crngringd 20243 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸 ∈ Ring) |
| 103 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(1r‘𝐸) = (1r‘𝐸) |
| 104 | 103 | subrg1cl 20580 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 ∈ (SubRing‘𝐸) →
(1r‘𝐸)
∈ 𝐹) |
| 105 | 38, 104 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹) |
| 106 | 6, 62 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 ⊆ 𝑉) |
| 107 | 1, 61, 103 | ress1r 33238 |
. . . . . . . . . . . . . 14
⊢ ((𝐸 ∈ Ring ∧
(1r‘𝐸)
∈ 𝐹 ∧ 𝐹 ⊆ 𝑉) → (1r‘𝐸) = (1r‘𝐾)) |
| 108 | 102, 105,
106, 107 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾)) |
| 109 | 101, 108 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((1r‘𝐾)(+g‘𝐾)(0g‘𝐾)) = (1r‘𝐸)) |
| 110 | 51, 98, 109 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘2) = (1r‘𝐸)) |
| 111 | 48, 110 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (𝜑 →
((coe1‘𝐺)‘2) = (1r‘𝐸)) |
| 112 | 5 | flddrngd 20741 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ∈ DivRing) |
| 113 | | drngnzr 20748 |
. . . . . . . . . . 11
⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing) |
| 114 | | rtelextdg2.3 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘𝐸) |
| 115 | 103, 114 | nzrnz 20515 |
. . . . . . . . . . 11
⊢ (𝐸 ∈ NzRing →
(1r‘𝐸)
≠ 0
) |
| 116 | 112, 113,
115 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1r‘𝐸) ≠ 0 ) |
| 117 | 111, 116 | eqnetrd 3008 |
. . . . . . . . 9
⊢ (𝜑 →
((coe1‘𝐺)‘2) ≠ 0 ) |
| 118 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝐺 = (0g‘𝑃) →
(coe1‘𝐺) =
(coe1‘(0g‘𝑃))) |
| 119 | 118 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝐺 = (0g‘𝑃) →
((coe1‘𝐺)‘2) =
((coe1‘(0g‘𝑃))‘2)) |
| 120 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(0g‘𝑃) = (0g‘𝑃) |
| 121 | 19, 120, 52, 18, 29 | coe1zfv 33612 |
. . . . . . . . . . 11
⊢ (𝜑 →
((coe1‘(0g‘𝑃))‘2) = (0g‘𝐾)) |
| 122 | 102 | ringgrpd 20239 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ Grp) |
| 123 | 122 | grpmndd 18964 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸 ∈ Mnd) |
| 124 | | subrgsubg 20577 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸)) |
| 125 | 38, 124 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸)) |
| 126 | 114 | subg0cl 19152 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ (SubGrp‘𝐸) → 0 ∈ 𝐹) |
| 127 | 125, 126 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ∈ 𝐹) |
| 128 | 1, 61, 114 | ress0g 18775 |
. . . . . . . . . . . 12
⊢ ((𝐸 ∈ Mnd ∧ 0 ∈ 𝐹 ∧ 𝐹 ⊆ 𝑉) → 0 =
(0g‘𝐾)) |
| 129 | 123, 127,
106, 128 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 =
(0g‘𝐾)) |
| 130 | 121, 129 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ (𝜑 →
((coe1‘(0g‘𝑃))‘2) = 0 ) |
| 131 | 119, 130 | sylan9eqr 2799 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺 = (0g‘𝑃)) → ((coe1‘𝐺)‘2) = 0 ) |
| 132 | 117, 131 | mteqand 3033 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ≠ (0g‘𝑃)) |
| 133 | 11 | fveq2i 6909 |
. . . . . . . . . . 11
⊢
((deg1‘𝐾)‘𝐺) = ((deg1‘𝐾)‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))) |
| 134 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(deg1‘𝐾) = (deg1‘𝐾) |
| 135 | | 2re 12340 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ |
| 136 | 135 | rexri 11319 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℝ* |
| 137 | 136 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 2 ∈
ℝ*) |
| 138 | 134, 19, 12 | deg1xrcl 26121 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑈‘𝐴) ⊗ 𝑌) ∈ (Base‘𝑃) → ((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) ∈
ℝ*) |
| 139 | 41, 138 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) ∈
ℝ*) |
| 140 | | 1xr 11320 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℝ* |
| 141 | 140 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 1 ∈
ℝ*) |
| 142 | 134, 19, 66, 12, 34, 35 | deg1mul3le 26156 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ Ring ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) → ((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) ≤ ((deg1‘𝐾)‘𝑌)) |
| 143 | 18, 65, 32, 142 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) ≤ ((deg1‘𝐾)‘𝑌)) |
| 144 | 1, 15 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐾 ∈ Field) |
| 145 | 144 | flddrngd 20741 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐾 ∈ DivRing) |
| 146 | | drngnzr 20748 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing) |
| 147 | 145, 146 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐾 ∈ NzRing) |
| 148 | 134, 19, 30, 147 | deg1vr 33614 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((deg1‘𝐾)‘𝑌) = 1) |
| 149 | 143, 148 | breqtrd 5169 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) ≤ 1) |
| 150 | | 1lt2 12437 |
. . . . . . . . . . . . . . . . 17
⊢ 1 <
2 |
| 151 | 150 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 1 < 2) |
| 152 | 139, 141,
137, 149, 151 | xrlelttrd 13202 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((deg1‘𝐾)‘((𝑈‘𝐴) ⊗ 𝑌)) < 2) |
| 153 | 134, 19, 12 | deg1xrcl 26121 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈‘𝐵) ∈ (Base‘𝑃) → ((deg1‘𝐾)‘(𝑈‘𝐵)) ∈
ℝ*) |
| 154 | 43, 153 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑈‘𝐵)) ∈
ℝ*) |
| 155 | | 0xr 11308 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℝ* |
| 156 | 155 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ∈
ℝ*) |
| 157 | 134, 19, 66, 35 | deg1sclle 26151 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) →
((deg1‘𝐾)‘(𝑈‘𝐵)) ≤ 0) |
| 158 | 18, 82, 157 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑈‘𝐵)) ≤ 0) |
| 159 | | 2pos 12369 |
. . . . . . . . . . . . . . . . 17
⊢ 0 <
2 |
| 160 | 159 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 < 2) |
| 161 | 154, 156,
137, 158, 160 | xrlelttrd 13202 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((deg1‘𝐾)‘(𝑈‘𝐵)) < 2) |
| 162 | 19, 134, 18, 12, 13, 41, 43, 137, 152, 161 | deg1addlt 33620 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((deg1‘𝐾)‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))) < 2) |
| 163 | 134, 19, 30, 23, 25 | deg1pw 26160 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ NzRing ∧ 2 ∈
ℕ0) → ((deg1‘𝐾)‘(2 ∧ 𝑌)) = 2) |
| 164 | 147, 29, 163 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((deg1‘𝐾)‘(2 ∧ 𝑌)) = 2) |
| 165 | 162, 164 | breqtrrd 5171 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((deg1‘𝐾)‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))) < ((deg1‘𝐾)‘(2 ∧ 𝑌))) |
| 166 | 19, 134, 18, 12, 13, 33, 44, 165 | deg1add 26142 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((deg1‘𝐾)‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))) = ((deg1‘𝐾)‘(2 ∧ 𝑌))) |
| 167 | 166, 164 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝜑 →
((deg1‘𝐾)‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))) = 2) |
| 168 | 133, 167 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (𝜑 →
((deg1‘𝐾)‘𝐺) = 2) |
| 169 | 168 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝜑 →
((coe1‘𝐺)‘((deg1‘𝐾)‘𝐺)) = ((coe1‘𝐺)‘2)) |
| 170 | 169, 111,
108 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 →
((coe1‘𝐺)‘((deg1‘𝐾)‘𝐺)) = (1r‘𝐾)) |
| 171 | | eqid 2737 |
. . . . . . . . 9
⊢
(Monic1p‘𝐾) = (Monic1p‘𝐾) |
| 172 | 19, 12, 120, 134, 171, 53 | ismon1p 26182 |
. . . . . . . 8
⊢ (𝐺 ∈
(Monic1p‘𝐾) ↔ (𝐺 ∈ (Base‘𝑃) ∧ 𝐺 ≠ (0g‘𝑃) ∧ ((coe1‘𝐺)‘((deg1‘𝐾)‘𝐺)) = (1r‘𝐾))) |
| 173 | 46, 132, 170, 172 | syl3anbrc 1344 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝐾)) |
| 174 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹) |
| 175 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(eval1‘𝐸) = (eval1‘𝐸) |
| 176 | 174, 61, 19, 1, 12, 175, 36, 38 | ressply1evl 22374 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸 evalSub1 𝐹) = ((eval1‘𝐸) ↾ (Base‘𝑃))) |
| 177 | 176 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = (((eval1‘𝐸) ↾ (Base‘𝑃))‘𝐺)) |
| 178 | 46 | fvresd 6926 |
. . . . . . . . . 10
⊢ (𝜑 →
(((eval1‘𝐸) ↾ (Base‘𝑃))‘𝐺) = ((eval1‘𝐸)‘𝐺)) |
| 179 | 177, 178 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = ((eval1‘𝐸)‘𝐺)) |
| 180 | 179 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = (((eval1‘𝐸)‘𝐺)‘𝑋)) |
| 181 | | eqid 2737 |
. . . . . . . . 9
⊢
(Poly1‘𝐸) = (Poly1‘𝐸) |
| 182 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘(Poly1‘𝐸)) =
(Base‘(Poly1‘𝐸)) |
| 183 | | rtelextdg2.6 |
. . . . . . . . 9
⊢ · =
(.r‘𝐸) |
| 184 | | rtelextdg2.7 |
. . . . . . . . 9
⊢ + =
(+g‘𝐸) |
| 185 | | rtelextdg2.8 |
. . . . . . . . 9
⊢ ↑ =
(.g‘(mulGrp‘𝐸)) |
| 186 | | eqid 2737 |
. . . . . . . . 9
⊢
(coe1‘𝐺) = (coe1‘𝐺) |
| 187 | | eqid 2737 |
. . . . . . . . 9
⊢
((coe1‘𝐺)‘2) = ((coe1‘𝐺)‘2) |
| 188 | | eqid 2737 |
. . . . . . . . 9
⊢
((coe1‘𝐺)‘1) = ((coe1‘𝐺)‘1) |
| 189 | | eqid 2737 |
. . . . . . . . 9
⊢
((coe1‘𝐺)‘0) = ((coe1‘𝐺)‘0) |
| 190 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(PwSer1‘𝐾) = (PwSer1‘𝐾) |
| 191 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘(PwSer1‘𝐾)) =
(Base‘(PwSer1‘𝐾)) |
| 192 | 181, 1, 19, 12, 38, 190, 191, 182 | ressply1bas2 22229 |
. . . . . . . . . . 11
⊢ (𝜑 → (Base‘𝑃) =
((Base‘(PwSer1‘𝐾)) ∩
(Base‘(Poly1‘𝐸)))) |
| 193 | 46, 192 | eleqtrd 2843 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈
((Base‘(PwSer1‘𝐾)) ∩
(Base‘(Poly1‘𝐸)))) |
| 194 | 193 | elin2d 4205 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈
(Base‘(Poly1‘𝐸))) |
| 195 | 1, 3, 19, 12, 46, 38 | ressdeg1 33591 |
. . . . . . . . . 10
⊢ (𝜑 →
((deg1‘𝐸)‘𝐺) = ((deg1‘𝐾)‘𝐺)) |
| 196 | 195, 168 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 →
((deg1‘𝐸)‘𝐺) = 2) |
| 197 | 181, 175,
61, 182, 183, 184, 185, 186, 3, 187, 188, 189, 36, 194, 196, 7 | evl1deg2 33602 |
. . . . . . . 8
⊢ (𝜑 →
(((eval1‘𝐸)‘𝐺)‘𝑋) = ((((coe1‘𝐺)‘2) · (2 ↑ 𝑋)) +
((((coe1‘𝐺)‘1) · 𝑋) +
((coe1‘𝐺)‘0)))) |
| 198 | 111 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝜑 →
(((coe1‘𝐺)‘2) · (2 ↑ 𝑋)) = ((1r‘𝐸) · (2 ↑ 𝑋))) |
| 199 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘𝐸) =
(mulGrp‘𝐸) |
| 200 | 199, 61 | mgpbas 20142 |
. . . . . . . . . . . . 13
⊢ 𝑉 =
(Base‘(mulGrp‘𝐸)) |
| 201 | 199 | ringmgp 20236 |
. . . . . . . . . . . . . 14
⊢ (𝐸 ∈ Ring →
(mulGrp‘𝐸) ∈
Mnd) |
| 202 | 102, 201 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (mulGrp‘𝐸) ∈ Mnd) |
| 203 | 200, 185,
202, 29, 7 | mulgnn0cld 19113 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 ↑ 𝑋) ∈ 𝑉) |
| 204 | 61, 183, 103, 102, 203 | ringlidmd 20269 |
. . . . . . . . . . 11
⊢ (𝜑 →
((1r‘𝐸)
·
(2 ↑
𝑋)) = (2 ↑ 𝑋)) |
| 205 | 198, 204 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 →
(((coe1‘𝐺)‘2) · (2 ↑ 𝑋)) = (2 ↑ 𝑋)) |
| 206 | 47 | fveq1i 6907 |
. . . . . . . . . . . . 13
⊢
((coe1‘𝐺)‘1) = ((coe1‘((2
∧
𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘1) |
| 207 | | 1nn0 12542 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℕ0 |
| 208 | 207 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℕ0) |
| 209 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ Ring ∧ (2 ∧ 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)) ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0)
→ ((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘1) = (((coe1‘(2
∧
𝑌))‘1)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1))) |
| 210 | 18, 33, 44, 208, 209 | syl31anc 1375 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘1) = (((coe1‘(2
∧
𝑌))‘1)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1))) |
| 211 | 71 | neii 2942 |
. . . . . . . . . . . . . . . . . 18
⊢ ¬ 1
= 2 |
| 212 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 1 → (𝑖 = 2 ↔ 1 = 2)) |
| 213 | 212 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 1 → (¬ 𝑖 = 2 ↔ ¬ 1 =
2)) |
| 214 | 213 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 = 1) → (¬ 𝑖 = 2 ↔ ¬ 1 = 2)) |
| 215 | 211, 214 | mpbiri 258 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 = 1) → ¬ 𝑖 = 2) |
| 216 | 215 | iffalsed 4536 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 = 1) → if(𝑖 = 2, (1r‘𝐾), (0g‘𝐾)) = (0g‘𝐾)) |
| 217 | 54, 216, 208, 77 | fvmptd 7023 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((coe1‘(2
∧
𝑌))‘1) =
(0g‘𝐾)) |
| 218 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐾 ∈ Ring ∧ ((𝑈‘𝐴) ⊗ 𝑌) ∈ (Base‘𝑃) ∧ (𝑈‘𝐵) ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0)
→ ((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘1))) |
| 219 | 18, 41, 43, 208, 218 | syl31anc 1375 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘1))) |
| 220 | 19, 12, 66, 35, 34, 67 | coe1sclmulfv 22286 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0)
→ ((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘1))) |
| 221 | 18, 65, 32, 208, 220 | syl121anc 1377 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘1))) |
| 222 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1) |
| 223 | 222 | iftrued 4533 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 = 1) → if(𝑖 = 1, (1r‘𝐾), (0g‘𝐾)) = (1r‘𝐾)) |
| 224 | 70, 223, 208, 57 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
((coe1‘𝑌)‘1) = (1r‘𝐾)) |
| 225 | 224 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴(.r‘𝐾)((coe1‘𝑌)‘1)) = (𝐴(.r‘𝐾)(1r‘𝐾))) |
| 226 | 66, 67, 53, 18, 65 | ringridmd 20270 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴(.r‘𝐾)(1r‘𝐾)) = 𝐴) |
| 227 | 221, 225,
226 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1) = 𝐴) |
| 228 | | 0ne1 12337 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ≠
1 |
| 229 | 228 | nesymi 2998 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ¬ 1
= 0 |
| 230 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0)) |
| 231 | 229, 230 | mtbiri 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 1 → ¬ 𝑖 = 0) |
| 232 | 231 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 = 1) → ¬ 𝑖 = 0) |
| 233 | 232 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 = 1) → if(𝑖 = 0, 𝐵, (0g‘𝐾)) = (0g‘𝐾)) |
| 234 | 84, 233, 208, 77 | fvmptd 7023 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((coe1‘(𝑈‘𝐵))‘1) = (0g‘𝐾)) |
| 235 | 227, 234 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘1)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘1)) = (𝐴(+g‘𝐾)(0g‘𝐾))) |
| 236 | 66, 49, 52, 93, 65 | grpridd 18988 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴(+g‘𝐾)(0g‘𝐾)) = 𝐴) |
| 237 | 219, 235,
236 | 3eqtrd 2781 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1) = 𝐴) |
| 238 | 217, 237 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
(((coe1‘(2 ∧ 𝑌))‘1)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘1)) = ((0g‘𝐾)(+g‘𝐾)𝐴)) |
| 239 | 66, 49, 52, 93, 65 | grplidd 18987 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((0g‘𝐾)(+g‘𝐾)𝐴) = 𝐴) |
| 240 | 210, 238,
239 | 3eqtrd 2781 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘1) = 𝐴) |
| 241 | 206, 240 | eqtrid 2789 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((coe1‘𝐺)‘1) = 𝐴) |
| 242 | 241 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝜑 →
(((coe1‘𝐺)‘1) · 𝑋) = (𝐴 · 𝑋)) |
| 243 | 47 | fveq1i 6907 |
. . . . . . . . . . . 12
⊢
((coe1‘𝐺)‘0) = ((coe1‘((2
∧
𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘0) |
| 244 | | 0nn0 12541 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℕ0 |
| 245 | 244 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 ∈
ℕ0) |
| 246 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ Ring ∧ (2 ∧ 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)) ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0)
→ ((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘0) = (((coe1‘(2
∧
𝑌))‘0)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0))) |
| 247 | 18, 33, 44, 245, 246 | syl31anc 1375 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘0) = (((coe1‘(2
∧
𝑌))‘0)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0))) |
| 248 | 88 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 = 0) → ¬ 𝑖 = 2) |
| 249 | 248 | iffalsed 4536 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 = 0) → if(𝑖 = 2, (1r‘𝐾), (0g‘𝐾)) = (0g‘𝐾)) |
| 250 | 54, 249, 245, 77 | fvmptd 7023 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((coe1‘(2
∧
𝑌))‘0) =
(0g‘𝐾)) |
| 251 | 19, 12, 13, 49 | coe1addfv 22268 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐾 ∈ Ring ∧ ((𝑈‘𝐴) ⊗ 𝑌) ∈ (Base‘𝑃) ∧ (𝑈‘𝐵) ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0)
→ ((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘0))) |
| 252 | 18, 41, 43, 245, 251 | syl31anc 1375 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0) =
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘0))) |
| 253 | 19, 12, 66, 35, 34, 67 | coe1sclmulfv 22286 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0)
→ ((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘0))) |
| 254 | 18, 65, 32, 245, 253 | syl121anc 1377 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0) = (𝐴(.r‘𝐾)((coe1‘𝑌)‘0))) |
| 255 | 228 | neii 2942 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ¬ 0
= 1 |
| 256 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1)) |
| 257 | 255, 256 | mtbiri 327 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 0 → ¬ 𝑖 = 1) |
| 258 | 257 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 = 0) → ¬ 𝑖 = 1) |
| 259 | 258 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 = 0) → if(𝑖 = 1, (1r‘𝐾), (0g‘𝐾)) = (0g‘𝐾)) |
| 260 | 70, 259, 245, 77 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
((coe1‘𝑌)‘0) = (0g‘𝐾)) |
| 261 | 260 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴(.r‘𝐾)((coe1‘𝑌)‘0)) = (𝐴(.r‘𝐾)(0g‘𝐾))) |
| 262 | 254, 261,
80 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0) = (0g‘𝐾)) |
| 263 | 19, 35, 66 | ply1sclid 22291 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → 𝐵 = ((coe1‘(𝑈‘𝐵))‘0)) |
| 264 | 18, 82, 263 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 = ((coe1‘(𝑈‘𝐵))‘0)) |
| 265 | 264 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((coe1‘(𝑈‘𝐵))‘0) = 𝐵) |
| 266 | 262, 265 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
(((coe1‘((𝑈‘𝐴) ⊗ 𝑌))‘0)(+g‘𝐾)((coe1‘(𝑈‘𝐵))‘0)) = ((0g‘𝐾)(+g‘𝐾)𝐵)) |
| 267 | 66, 49, 52, 93, 82 | grplidd 18987 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
((0g‘𝐾)(+g‘𝐾)𝐵) = 𝐵) |
| 268 | 252, 266,
267 | 3eqtrd 2781 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0) = 𝐵) |
| 269 | 250, 268 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
(((coe1‘(2 ∧ 𝑌))‘0)(+g‘𝐾)((coe1‘(((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))‘0)) = ((0g‘𝐾)(+g‘𝐾)𝐵)) |
| 270 | 247, 269,
267 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((coe1‘((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵))))‘0) = 𝐵) |
| 271 | 243, 270 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝜑 →
((coe1‘𝐺)‘0) = 𝐵) |
| 272 | 242, 271 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝜑 →
((((coe1‘𝐺)‘1) · 𝑋) +
((coe1‘𝐺)‘0)) = ((𝐴 · 𝑋) + 𝐵)) |
| 273 | 205, 272 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝜑 →
((((coe1‘𝐺)‘2) · (2 ↑ 𝑋)) +
((((coe1‘𝐺)‘1) · 𝑋) +
((coe1‘𝐺)‘0))) = ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵))) |
| 274 | | rtelextdg2.14 |
. . . . . . . . 9
⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 ) |
| 275 | 273, 274 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 →
((((coe1‘𝐺)‘2) · (2 ↑ 𝑋)) +
((((coe1‘𝐺)‘1) · 𝑋) +
((coe1‘𝐺)‘0))) = 0 ) |
| 276 | 180, 197,
275 | 3eqtrd 2781 |
. . . . . . 7
⊢ (𝜑 → (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = 0 ) |
| 277 | 10, 173, 276 | rspcedvdw 3625 |
. . . . . 6
⊢ (𝜑 → ∃𝑝 ∈ (Monic1p‘𝐾)(((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 ) |
| 278 | 174, 1, 61, 114, 36, 38 | elirng 33736 |
. . . . . 6
⊢ (𝜑 → (𝑋 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑝 ∈ (Monic1p‘𝐾)(((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 ))) |
| 279 | 7, 277, 278 | mpbir2and 713 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹)) |
| 280 | 1, 2, 3, 4, 5, 6, 279 | algextdeg 33766 |
. . . 4
⊢ (𝜑 → (𝐿[:]𝐾) = ((deg1‘𝐸)‘((𝐸 minPoly 𝐹)‘𝑋))) |
| 281 | 1 | fveq2i 6909 |
. . . . . . 7
⊢
(Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹)) |
| 282 | 19, 281 | eqtri 2765 |
. . . . . 6
⊢ 𝑃 =
(Poly1‘(𝐸
↾s 𝐹)) |
| 283 | | eqid 2737 |
. . . . . 6
⊢ {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝑋) = 0 } = {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝑋) = 0 } |
| 284 | | eqid 2737 |
. . . . . 6
⊢
(RSpan‘𝑃) =
(RSpan‘𝑃) |
| 285 | | eqid 2737 |
. . . . . 6
⊢
(idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) |
| 286 | 174, 282,
61, 5, 6, 7, 114, 283, 284, 285, 4 | minplycl 33749 |
. . . . 5
⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝑋) ∈ (Base‘𝑃)) |
| 287 | 1, 3, 19, 12, 286, 38 | ressdeg1 33591 |
. . . 4
⊢ (𝜑 →
((deg1‘𝐸)‘((𝐸 minPoly 𝐹)‘𝑋)) = ((deg1‘𝐾)‘((𝐸 minPoly 𝐹)‘𝑋))) |
| 288 | 280, 287 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (𝐿[:]𝐾) = ((deg1‘𝐾)‘((𝐸 minPoly 𝐹)‘𝑋))) |
| 289 | 1 | fveq2i 6909 |
. . . 4
⊢
(deg1‘𝐾) = (deg1‘(𝐸 ↾s 𝐹)) |
| 290 | 174, 282,
61, 5, 6, 7, 114, 4, 289, 120, 12, 276, 46, 132 | minplymindeg 33751 |
. . 3
⊢ (𝜑 →
((deg1‘𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)) ≤ ((deg1‘𝐾)‘𝐺)) |
| 291 | 288, 290 | eqbrtrd 5165 |
. 2
⊢ (𝜑 → (𝐿[:]𝐾) ≤ ((deg1‘𝐾)‘𝐺)) |
| 292 | 291, 168 | breqtrd 5169 |
1
⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2) |