| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | breq2 5146 | . . . 4
⊢ (𝑘 = 𝑛 → (𝑁 ≤s 𝑘 ↔ 𝑁 ≤s 𝑛)) | 
| 2 | 1 | elrab 3691 | . . 3
⊢ (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ↔ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) | 
| 3 |  | zno 28369 | . . . . . . 7
⊢ (𝑛 ∈ ℤs
→ 𝑛 ∈  No ) | 
| 4 | 3 | adantr 480 | . . . . . 6
⊢ ((𝑛 ∈ ℤs
∧ 𝑁 ≤s 𝑛) → 𝑛 ∈  No
) | 
| 5 |  | peano5uzs.1 | . . . . . . 7
⊢ (𝜑 → 𝑁 ∈
ℤs) | 
| 6 | 5 | znod 28370 | . . . . . 6
⊢ (𝜑 → 𝑁 ∈  No
) | 
| 7 |  | npcans 28106 | . . . . . 6
⊢ ((𝑛 ∈ 
No  ∧ 𝑁 ∈
 No ) → ((𝑛 -s 𝑁) +s 𝑁) = 𝑛) | 
| 8 | 4, 6, 7 | syl2anr 597 | . . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) +s 𝑁) = 𝑛) | 
| 9 |  | simprl 770 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ ℤs) | 
| 10 | 5 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑁 ∈
ℤs) | 
| 11 | 9, 10 | zsubscld 28383 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 -s 𝑁) ∈
ℤs) | 
| 12 | 3 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑛 ∈ 
No ) | 
| 13 | 6 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → 𝑁 ∈ 
No ) | 
| 14 | 12, 13 | subsge0d 28130 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℤs) → (
0s ≤s (𝑛
-s 𝑁) ↔
𝑁 ≤s 𝑛)) | 
| 15 | 14 | biimpar 477 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℤs) ∧ 𝑁 ≤s 𝑛) → 0s ≤s (𝑛 -s 𝑁)) | 
| 16 | 15 | anasss 466 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 0s ≤s (𝑛 -s 𝑁)) | 
| 17 | 11, 16 | jca 511 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) ∈ ℤs ∧
0s ≤s (𝑛
-s 𝑁))) | 
| 18 |  | eln0zs 28387 | . . . . . . . . 9
⊢ ((𝑛 -s 𝑁) ∈ ℕ0s ↔ ((𝑛 -s 𝑁) ∈ ℤs ∧
0s ≤s (𝑛
-s 𝑁))) | 
| 19 | 17, 18 | sylibr 234 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → (𝑛 -s 𝑁) ∈
ℕ0s) | 
| 20 | 19 | ex 412 | . . . . . . 7
⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → (𝑛 -s 𝑁) ∈
ℕ0s)) | 
| 21 |  | oveq1 7439 | . . . . . . . . . . 11
⊢ (𝑧 = 0s → (𝑧 +s 𝑁) = ( 0s +s 𝑁)) | 
| 22 | 21 | eleq1d 2825 | . . . . . . . . . 10
⊢ (𝑧 = 0s → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ( 0s +s 𝑁) ∈ 𝐴)) | 
| 23 | 22 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑧 = 0s → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ( 0s +s 𝑁) ∈ 𝐴))) | 
| 24 |  | oveq1 7439 | . . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (𝑧 +s 𝑁) = (𝑦 +s 𝑁)) | 
| 25 | 24 | eleq1d 2825 | . . . . . . . . . 10
⊢ (𝑧 = 𝑦 → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ (𝑦 +s 𝑁) ∈ 𝐴)) | 
| 26 | 25 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑧 = 𝑦 → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → (𝑦 +s 𝑁) ∈ 𝐴))) | 
| 27 |  | oveq1 7439 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑦 +s 1s ) → (𝑧 +s 𝑁) = ((𝑦 +s 1s ) +s
𝑁)) | 
| 28 | 27 | eleq1d 2825 | . . . . . . . . . 10
⊢ (𝑧 = (𝑦 +s 1s ) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑦 +s 1s ) +s
𝑁) ∈ 𝐴)) | 
| 29 | 28 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑧 = (𝑦 +s 1s ) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑦 +s 1s ) +s
𝑁) ∈ 𝐴))) | 
| 30 |  | oveq1 7439 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑛 -s 𝑁) → (𝑧 +s 𝑁) = ((𝑛 -s 𝑁) +s 𝑁)) | 
| 31 | 30 | eleq1d 2825 | . . . . . . . . . 10
⊢ (𝑧 = (𝑛 -s 𝑁) → ((𝑧 +s 𝑁) ∈ 𝐴 ↔ ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)) | 
| 32 | 31 | imbi2d 340 | . . . . . . . . 9
⊢ (𝑧 = (𝑛 -s 𝑁) → ((𝜑 → (𝑧 +s 𝑁) ∈ 𝐴) ↔ (𝜑 → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴))) | 
| 33 |  | addslid 28002 | . . . . . . . . . . 11
⊢ (𝑁 ∈ 
No  → ( 0s +s 𝑁) = 𝑁) | 
| 34 | 6, 33 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → ( 0s
+s 𝑁) = 𝑁) | 
| 35 |  | peano5uzs.2 | . . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ 𝐴) | 
| 36 | 34, 35 | eqeltrd 2840 | . . . . . . . . 9
⊢ (𝜑 → ( 0s
+s 𝑁) ∈
𝐴) | 
| 37 |  | peano5uzs.3 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴) | 
| 38 | 37 | ralrimiva 3145 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) | 
| 39 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 +s 𝑁) → (𝑥 +s 1s ) = ((𝑦 +s 𝑁) +s 1s
)) | 
| 40 | 39 | eleq1d 2825 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑦 +s 𝑁) → ((𝑥 +s 1s ) ∈ 𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴)) | 
| 41 | 40 | rspccv 3618 | . . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
𝐴 (𝑥 +s 1s ) ∈ 𝐴 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴)) | 
| 42 | 38, 41 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴)) | 
| 43 | 42 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴)) | 
| 44 |  | n0sno 28329 | . . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ0s
→ 𝑦 ∈  No ) | 
| 45 | 44 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → 𝑦 ∈ 
No ) | 
| 46 |  | 1sno 27873 | . . . . . . . . . . . . . . 15
⊢ 
1s ∈  No | 
| 47 | 46 | a1i 11 | . . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → 1s
∈  No ) | 
| 48 | 6 | adantl 481 | . . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → 𝑁 ∈ 
No ) | 
| 49 | 45, 47, 48 | adds32d 28041 | . . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → ((𝑦 +s 1s )
+s 𝑁) = ((𝑦 +s 𝑁) +s 1s
)) | 
| 50 | 49 | eleq1d 2825 | . . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → (((𝑦 +s 1s )
+s 𝑁) ∈
𝐴 ↔ ((𝑦 +s 𝑁) +s 1s ) ∈ 𝐴)) | 
| 51 | 43, 50 | sylibrd 259 | . . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ0s
∧ 𝜑) → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s
𝑁) ∈ 𝐴)) | 
| 52 | 51 | ex 412 | . . . . . . . . . 10
⊢ (𝑦 ∈ ℕ0s
→ (𝜑 → ((𝑦 +s 𝑁) ∈ 𝐴 → ((𝑦 +s 1s ) +s
𝑁) ∈ 𝐴))) | 
| 53 | 52 | a2d 29 | . . . . . . . . 9
⊢ (𝑦 ∈ ℕ0s
→ ((𝜑 → (𝑦 +s 𝑁) ∈ 𝐴) → (𝜑 → ((𝑦 +s 1s ) +s
𝑁) ∈ 𝐴))) | 
| 54 | 23, 26, 29, 32, 36, 53 | n0sind 28338 | . . . . . . . 8
⊢ ((𝑛 -s 𝑁) ∈ ℕ0s → (𝜑 → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)) | 
| 55 | 54 | com12 32 | . . . . . . 7
⊢ (𝜑 → ((𝑛 -s 𝑁) ∈ ℕ0s → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)) | 
| 56 | 20, 55 | syld 47 | . . . . . 6
⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴)) | 
| 57 | 56 | imp 406 | . . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → ((𝑛 -s 𝑁) +s 𝑁) ∈ 𝐴) | 
| 58 | 8, 57 | eqeltrrd 2841 | . . . 4
⊢ ((𝜑 ∧ (𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛)) → 𝑛 ∈ 𝐴) | 
| 59 | 58 | ex 412 | . . 3
⊢ (𝜑 → ((𝑛 ∈ ℤs ∧ 𝑁 ≤s 𝑛) → 𝑛 ∈ 𝐴)) | 
| 60 | 2, 59 | biimtrid 242 | . 2
⊢ (𝜑 → (𝑛 ∈ {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} → 𝑛 ∈ 𝐴)) | 
| 61 | 60 | ssrdv 3988 | 1
⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴) |