Proof of Theorem dgrlt
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
𝐹 =
0𝑝) | 
| 2 | 1 | fveq2d 6910 | . . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
(deg‘𝐹) =
(deg‘0𝑝)) | 
| 3 |  | dgreq0.1 | . . . . . 6
⊢ 𝑁 = (deg‘𝐹) | 
| 4 |  | dgr0 26302 | . . . . . . 7
⊢
(deg‘0𝑝) = 0 | 
| 5 | 4 | eqcomi 2746 | . . . . . 6
⊢ 0 =
(deg‘0𝑝) | 
| 6 | 2, 3, 5 | 3eqtr4g 2802 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
𝑁 = 0) | 
| 7 |  | nn0ge0 12551 | . . . . . 6
⊢ (𝑀 ∈ ℕ0
→ 0 ≤ 𝑀) | 
| 8 | 7 | ad2antlr 727 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
0 ≤ 𝑀) | 
| 9 | 6, 8 | eqbrtrd 5165 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
𝑁 ≤ 𝑀) | 
| 10 | 1 | fveq2d 6910 | . . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
(coeff‘𝐹) =
(coeff‘0𝑝)) | 
| 11 |  | dgreq0.2 | . . . . . . 7
⊢ 𝐴 = (coeff‘𝐹) | 
| 12 |  | coe0 26295 | . . . . . . . 8
⊢
(coeff‘0𝑝) = (ℕ0 ×
{0}) | 
| 13 | 12 | eqcomi 2746 | . . . . . . 7
⊢
(ℕ0 × {0}) =
(coeff‘0𝑝) | 
| 14 | 10, 11, 13 | 3eqtr4g 2802 | . . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
𝐴 = (ℕ0
× {0})) | 
| 15 | 14 | fveq1d 6908 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
(𝐴‘𝑀) = ((ℕ0 ×
{0})‘𝑀)) | 
| 16 |  | c0ex 11255 | . . . . . . 7
⊢ 0 ∈
V | 
| 17 | 16 | fvconst2 7224 | . . . . . 6
⊢ (𝑀 ∈ ℕ0
→ ((ℕ0 × {0})‘𝑀) = 0) | 
| 18 | 17 | ad2antlr 727 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
((ℕ0 × {0})‘𝑀) = 0) | 
| 19 | 15, 18 | eqtrd 2777 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
(𝐴‘𝑀) = 0) | 
| 20 | 9, 19 | jca 511 | . . 3
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝐹 = 0𝑝) →
(𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) | 
| 21 |  | dgrcl 26272 | . . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈
ℕ0) | 
| 22 | 3, 21 | eqeltrid 2845 | . . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈
ℕ0) | 
| 23 | 22 | nn0red 12588 | . . . . . 6
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℝ) | 
| 24 |  | nn0re 12535 | . . . . . 6
⊢ (𝑀 ∈ ℕ0
→ 𝑀 ∈
ℝ) | 
| 25 |  | ltle 11349 | . . . . . 6
⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀)) | 
| 26 | 23, 24, 25 | syl2an 596 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → (𝑁 < 𝑀 → 𝑁 ≤ 𝑀)) | 
| 27 | 26 | imp 406 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 < 𝑀) → 𝑁 ≤ 𝑀) | 
| 28 | 11, 3 | dgrub 26273 | . . . . . . . 8
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁) | 
| 29 | 28 | 3expia 1122 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝐴‘𝑀) ≠ 0 → 𝑀 ≤ 𝑁)) | 
| 30 |  | lenlt 11339 | . . . . . . . 8
⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) | 
| 31 | 24, 23, 30 | syl2anr 597 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) | 
| 32 | 29, 31 | sylibd 239 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝐴‘𝑀) ≠ 0 → ¬ 𝑁 < 𝑀)) | 
| 33 | 32 | necon4ad 2959 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → (𝑁 < 𝑀 → (𝐴‘𝑀) = 0)) | 
| 34 | 33 | imp 406 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 < 𝑀) → (𝐴‘𝑀) = 0) | 
| 35 | 27, 34 | jca 511 | . . 3
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 < 𝑀) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) | 
| 36 | 20, 35 | jaodan 960 | . 2
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝐹 = 0𝑝 ∨
𝑁 < 𝑀)) → (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) | 
| 37 |  | leloe 11347 | . . . . . . 7
⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))) | 
| 38 | 23, 24, 37 | syl2an 596 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → (𝑁 ≤ 𝑀 ↔ (𝑁 < 𝑀 ∨ 𝑁 = 𝑀))) | 
| 39 | 38 | biimpa 476 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 ≤ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)) | 
| 40 | 39 | adantrr 717 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀)) | 
| 41 |  | fveq2 6906 | . . . . . 6
⊢ (𝑁 = 𝑀 → (𝐴‘𝑁) = (𝐴‘𝑀)) | 
| 42 | 3, 11 | dgreq0 26305 | . . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) | 
| 43 | 42 | ad2antrr 726 | . . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) | 
| 44 |  | simprr 773 | . . . . . . . 8
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐴‘𝑀) = 0) | 
| 45 | 44 | eqeq2d 2748 | . . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝐴‘𝑁) = (𝐴‘𝑀) ↔ (𝐴‘𝑁) = 0)) | 
| 46 | 43, 45 | bitr4d 282 | . . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = (𝐴‘𝑀))) | 
| 47 | 41, 46 | imbitrrid 246 | . . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 = 𝑀 → 𝐹 = 0𝑝)) | 
| 48 | 47 | orim2d 969 | . . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → ((𝑁 < 𝑀 ∨ 𝑁 = 𝑀) → (𝑁 < 𝑀 ∨ 𝐹 = 0𝑝))) | 
| 49 | 40, 48 | mpd 15 | . . 3
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝑁 < 𝑀 ∨ 𝐹 = 0𝑝)) | 
| 50 | 49 | orcomd 872 | . 2
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) ∧ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)) → (𝐹 = 0𝑝 ∨ 𝑁 < 𝑀)) | 
| 51 | 36, 50 | impbida 801 | 1
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝐹 = 0𝑝 ∨
𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))) |