Proof of Theorem fldgenfldext
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fldgenfldext.l | . . 3
⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) | 
| 2 |  | fldgenfldext.b | . . . 4
⊢ 𝐵 = (Base‘𝐸) | 
| 3 |  | fldgenfldext.e | . . . 4
⊢ (𝜑 → 𝐸 ∈ Field) | 
| 4 |  | fldgenfldext.f | . . . . . 6
⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | 
| 5 | 2 | sdrgss 20795 | . . . . . 6
⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵) | 
| 6 | 4, 5 | syl 17 | . . . . 5
⊢ (𝜑 → 𝐹 ⊆ 𝐵) | 
| 7 |  | fldgenfldext.1 | . . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| 8 | 6, 7 | unssd 4191 | . . . 4
⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ 𝐵) | 
| 9 | 2, 3, 8 | fldgenfld 33323 | . . 3
⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ∈ Field) | 
| 10 | 1, 9 | eqeltrid 2844 | . 2
⊢ (𝜑 → 𝐿 ∈ Field) | 
| 11 |  | fldgenfldext.k | . . 3
⊢ 𝐾 = (𝐸 ↾s 𝐹) | 
| 12 |  | fldsdrgfld 20800 | . . . 4
⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field) | 
| 13 | 3, 4, 12 | syl2anc 584 | . . 3
⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field) | 
| 14 | 11, 13 | eqeltrid 2844 | . 2
⊢ (𝜑 → 𝐾 ∈ Field) | 
| 15 | 1 | oveq1i 7442 | . . . . . 6
⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) | 
| 16 |  | ovexd 7467 | . . . . . . 7
⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V) | 
| 17 |  | ressress 17294 | . . . . . . 7
⊢ (((𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V ∧ 𝐹 ∈ (SubDRing‘𝐸)) → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹))) | 
| 18 | 16, 4, 17 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹))) | 
| 19 | 15, 18 | eqtrid 2788 | . . . . 5
⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹))) | 
| 20 | 3 | flddrngd 20742 | . . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈ DivRing) | 
| 21 | 2, 20, 8 | fldgenssid 33316 | . . . . . . . 8
⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴))) | 
| 22 | 21 | unssad 4192 | . . . . . . 7
⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴))) | 
| 23 |  | sseqin2 4222 | . . . . . . 7
⊢ (𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)) ↔ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹) | 
| 24 | 22, 23 | sylib 218 | . . . . . 6
⊢ (𝜑 → ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹) | 
| 25 | 24 | oveq2d 7448 | . . . . 5
⊢ (𝜑 → (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)) = (𝐸 ↾s 𝐹)) | 
| 26 | 19, 25 | eqtrd 2776 | . . . 4
⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹)) | 
| 27 | 11, 2 | ressbas2 17284 | . . . . . 6
⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘𝐾)) | 
| 28 | 6, 27 | syl 17 | . . . . 5
⊢ (𝜑 → 𝐹 = (Base‘𝐾)) | 
| 29 | 28 | oveq2d 7448 | . . . 4
⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾))) | 
| 30 | 26, 29 | eqtr3d 2778 | . . 3
⊢ (𝜑 → (𝐸 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾))) | 
| 31 | 11, 30 | eqtrid 2788 | . 2
⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾))) | 
| 32 | 10 | fldcrngd 20743 | . . . . 5
⊢ (𝜑 → 𝐿 ∈ CRing) | 
| 33 | 32 | crngringd 20244 | . . . 4
⊢ (𝜑 → 𝐿 ∈ Ring) | 
| 34 | 14 | fldcrngd 20743 | . . . . . . 7
⊢ (𝜑 → 𝐾 ∈ CRing) | 
| 35 | 34 | crngringd 20244 | . . . . . 6
⊢ (𝜑 → 𝐾 ∈ Ring) | 
| 36 | 11, 35 | eqeltrrid 2845 | . . . . 5
⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring) | 
| 37 | 26, 36 | eqeltrd 2840 | . . . 4
⊢ (𝜑 → (𝐿 ↾s 𝐹) ∈ Ring) | 
| 38 | 2, 20, 8 | fldgenssv 33318 | . . . . . . 7
⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) | 
| 39 | 1, 2 | ressbas2 17284 | . . . . . . 7
⊢ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿)) | 
| 40 | 38, 39 | syl 17 | . . . . . 6
⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿)) | 
| 41 | 22, 40 | sseqtrd 4019 | . . . . 5
⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿)) | 
| 42 | 20 | drngringd 20738 | . . . . . . 7
⊢ (𝜑 → 𝐸 ∈ Ring) | 
| 43 |  | sdrgsubrg 20793 | . . . . . . . . 9
⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) | 
| 44 |  | eqid 2736 | . . . . . . . . . 10
⊢
(1r‘𝐸) = (1r‘𝐸) | 
| 45 | 44 | subrg1cl 20581 | . . . . . . . . 9
⊢ (𝐹 ∈ (SubRing‘𝐸) →
(1r‘𝐸)
∈ 𝐹) | 
| 46 | 4, 43, 45 | 3syl 18 | . . . . . . . 8
⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹) | 
| 47 | 22, 46 | sseldd 3983 | . . . . . . 7
⊢ (𝜑 → (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴))) | 
| 48 | 1, 2, 44 | ress1r 33239 | . . . . . . 7
⊢ ((𝐸 ∈ Ring ∧
(1r‘𝐸)
∈ (𝐸 fldGen (𝐹 ∪ 𝐴)) ∧ (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) → (1r‘𝐸) = (1r‘𝐿)) | 
| 49 | 42, 47, 38, 48 | syl3anc 1372 | . . . . . 6
⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐿)) | 
| 50 | 49, 46 | eqeltrrd 2841 | . . . . 5
⊢ (𝜑 → (1r‘𝐿) ∈ 𝐹) | 
| 51 | 41, 50 | jca 511 | . . . 4
⊢ (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹)) | 
| 52 |  | eqid 2736 | . . . . 5
⊢
(Base‘𝐿) =
(Base‘𝐿) | 
| 53 |  | eqid 2736 | . . . . 5
⊢
(1r‘𝐿) = (1r‘𝐿) | 
| 54 | 52, 53 | issubrg 20572 | . . . 4
⊢ (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹))) | 
| 55 | 33, 37, 51, 54 | syl21anbrc 1344 | . . 3
⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿)) | 
| 56 | 28, 55 | eqeltrrd 2841 | . 2
⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿)) | 
| 57 |  | brfldext 33699 | . . 3
⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿)))) | 
| 58 | 57 | biimpar 477 | . 2
⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾) | 
| 59 | 10, 14, 31, 56, 58 | syl22anc 838 | 1
⊢ (𝜑 → 𝐿/FldExt𝐾) |