| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | minplyval.1 | . . 3
⊢ 𝑀 = (𝐸 minPoly 𝐹) | 
| 2 |  | ply1annig1p.e | . . . . 5
⊢ (𝜑 → 𝐸 ∈ Field) | 
| 3 | 2 | elexd 3503 | . . . 4
⊢ (𝜑 → 𝐸 ∈ V) | 
| 4 |  | ply1annig1p.f | . . . . 5
⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | 
| 5 | 4 | elexd 3503 | . . . 4
⊢ (𝜑 → 𝐹 ∈ V) | 
| 6 |  | ply1annig1p.b | . . . . . . 7
⊢ 𝐵 = (Base‘𝐸) | 
| 7 | 6 | fvexi 6919 | . . . . . 6
⊢ 𝐵 ∈ V | 
| 8 | 7 | a1i 11 | . . . . 5
⊢ (𝜑 → 𝐵 ∈ V) | 
| 9 | 8 | mptexd 7245 | . . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈
V) | 
| 10 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸)) | 
| 11 | 10, 6 | eqtr4di 2794 | . . . . . . 7
⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵) | 
| 12 | 11 | adantr 480 | . . . . . 6
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵) | 
| 13 |  | oveq12 7441 | . . . . . . . . 9
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹)) | 
| 14 | 13 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) =
(idlGen1p‘(𝐸 ↾s 𝐹))) | 
| 15 |  | ply1annig1p.g | . . . . . . . 8
⊢ 𝐺 =
(idlGen1p‘(𝐸 ↾s 𝐹)) | 
| 16 | 14, 15 | eqtr4di 2794 | . . . . . . 7
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺) | 
| 17 |  | oveq12 7441 | . . . . . . . . . 10
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹)) | 
| 18 |  | ply1annig1p.o | . . . . . . . . . 10
⊢ 𝑂 = (𝐸 evalSub1 𝐹) | 
| 19 | 17, 18 | eqtr4di 2794 | . . . . . . . . 9
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂) | 
| 20 | 19 | dmeqd 5915 | . . . . . . . 8
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂) | 
| 21 | 19 | fveq1d 6907 | . . . . . . . . . 10
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞)) | 
| 22 | 21 | fveq1d 6907 | . . . . . . . . 9
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥)) | 
| 23 |  | fveq2 6905 | . . . . . . . . . . 11
⊢ (𝑒 = 𝐸 → (0g‘𝑒) = (0g‘𝐸)) | 
| 24 | 23 | adantr 480 | . . . . . . . . . 10
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = (0g‘𝐸)) | 
| 25 |  | ply1annig1p.0 | . . . . . . . . . 10
⊢  0 =
(0g‘𝐸) | 
| 26 | 24, 25 | eqtr4di 2794 | . . . . . . . . 9
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 ) | 
| 27 | 22, 26 | eqeq12d 2752 | . . . . . . . 8
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 )) | 
| 28 | 20, 27 | rabeqbidv 3454 | . . . . . . 7
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → {𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) | 
| 29 | 16, 28 | fveq12d 6912 | . . . . . 6
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)}) = (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) | 
| 30 | 12, 29 | mpteq12dv 5232 | . . . . 5
⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)})) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }))) | 
| 31 |  | df-minply 33744 | . . . . 5
⊢  minPoly
= (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦
((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)}))) | 
| 32 | 30, 31 | ovmpoga 7588 | . . . 4
⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V) →
(𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }))) | 
| 33 | 3, 5, 9, 32 | syl3anc 1372 | . . 3
⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }))) | 
| 34 | 1, 33 | eqtrid 2788 | . 2
⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }))) | 
| 35 |  | fveqeq2 6914 | . . . . . 6
⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 )) | 
| 36 | 35 | rabbidv 3443 | . . . . 5
⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) | 
| 37 |  | ply1annig1p.q | . . . . 5
⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | 
| 38 | 36, 37 | eqtr4di 2794 | . . . 4
⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄) | 
| 39 | 38 | fveq2d 6909 | . . 3
⊢ (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄)) | 
| 40 | 39 | adantl 481 | . 2
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄)) | 
| 41 |  | ply1annig1p.a | . 2
⊢ (𝜑 → 𝐴 ∈ 𝐵) | 
| 42 |  | fvexd 6920 | . 2
⊢ (𝜑 → (𝐺‘𝑄) ∈ V) | 
| 43 | 34, 40, 41, 42 | fvmptd 7022 | 1
⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄)) |