Proof of Theorem eulerpartlemb
Step | Hyp | Ref
| Expression |
1 | | fzfid 13342 |
. . . 4
⊢ (⊤
→ (1...𝑁) ∈
Fin) |
2 | | fzfi 13341 |
. . . . . 6
⊢
(0...𝑁) ∈
Fin |
3 | | snfi 8594 |
. . . . . 6
⊢ {0}
∈ Fin |
4 | 2, 3 | ifcli 4513 |
. . . . 5
⊢ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin |
5 | 4 | a1i 11 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℕ) → if(𝑥
∈ (1...𝑁), (0...𝑁), {0}) ∈
Fin) |
6 | | eldifn 4104 |
. . . . . 6
⊢ (𝑥 ∈ (ℕ ∖
(1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁)) |
7 | 6 | adantl 484 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁)) |
8 | | iffalse 4476 |
. . . . 5
⊢ (¬
𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0}) |
9 | | eqimss 4023 |
. . . . 5
⊢ (if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0} → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0}) |
10 | 7, 8, 9 | 3syl 18 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (ℕ ∖ (1...𝑁))) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0}) |
11 | 1, 5, 10 | ixpfi2 8822 |
. . 3
⊢ (⊤
→ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin) |
12 | 11 | mptru 1544 |
. 2
⊢ X𝑥 ∈
ℕ if(𝑥 ∈
(1...𝑁), (0...𝑁), {0}) ∈
Fin |
13 | | eulerpart.p |
. . . . 5
⊢ 𝑃 = {𝑓 ∈ (ℕ0
↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑓‘𝑘) · 𝑘) = 𝑁)} |
14 | 13 | eulerpartleme 31621 |
. . . 4
⊢ (𝑔 ∈ 𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁)) |
15 | | ffn 6514 |
. . . . . 6
⊢ (𝑔:ℕ⟶ℕ0 →
𝑔 Fn
ℕ) |
16 | 15 | 3ad2ant1 1129 |
. . . . 5
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ) |
17 | | ffvelrn 6849 |
. . . . . . . . . . . . 13
⊢ ((𝑔:ℕ⟶ℕ0 ∧
𝑥 ∈ ℕ) →
(𝑔‘𝑥) ∈
ℕ0) |
18 | 17 | 3ad2antl1 1181 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈
ℕ0) |
19 | 18 | nn0red 11957 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℝ) |
20 | | nnre 11645 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℝ) |
21 | 20 | adantl 484 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ) |
22 | 19, 21 | remulcld 10671 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ∈ ℝ) |
23 | | cnvimass 5949 |
. . . . . . . . . . . . . . . . . 18
⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔 |
24 | | fdm 6522 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔:ℕ⟶ℕ0 →
dom 𝑔 =
ℕ) |
25 | 24 | adantr 483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → dom
𝑔 =
ℕ) |
26 | 23, 25 | sseqtrid 4019 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ⊆
ℕ) |
27 | 26 | sselda 3967 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ) |
28 | | ffvelrn 6849 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑔:ℕ⟶ℕ0 ∧
𝑘 ∈ ℕ) →
(𝑔‘𝑘) ∈
ℕ0) |
29 | 28 | adantlr 713 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈
ℕ0) |
30 | 27, 29 | syldan 593 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈
ℕ0) |
31 | 27 | nnnn0d 11956 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0) |
32 | 30, 31 | nn0mulcld 11961 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈
ℕ0) |
33 | 32 | nn0cnd 11958 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℂ) |
34 | | simpl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0) |
35 | | nnex 11644 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℕ
∈ V |
36 | | frnnn0supp 11954 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℕ
∈ V ∧ 𝑔:ℕ⟶ℕ0) →
(𝑔 supp 0) = (◡𝑔 “ ℕ)) |
37 | 35, 36 | mpan 688 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔:ℕ⟶ℕ0 →
(𝑔 supp 0) = (◡𝑔 “ ℕ)) |
38 | 37 | adantr 483 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (◡𝑔 “ ℕ)) |
39 | | eqimss 4023 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑔 supp 0) = (◡𝑔 “ ℕ) → (𝑔 supp 0) ⊆ (◡𝑔 “ ℕ)) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) ⊆ (◡𝑔 “ ℕ)) |
41 | 35 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → ℕ
∈ V) |
42 | | 0nn0 11913 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 ∈
ℕ0 |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → 0 ∈
ℕ0) |
44 | 34, 40, 41, 43 | suppssr 7861 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (𝑔‘𝑘) = 0) |
45 | 44 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = (0 · 𝑘)) |
46 | | eldifi 4103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ) |
47 | 46 | adantl 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → 𝑘 ∈ ℕ) |
48 | | nncn 11646 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
49 | | mul02 10818 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℂ → (0
· 𝑘) =
0) |
50 | 47, 48, 49 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (0 · 𝑘) = 0) |
51 | 45, 50 | eqtrd 2856 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = 0) |
52 | | nnuz 12282 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℕ =
(ℤ≥‘1) |
53 | 52 | eqimssi 4025 |
. . . . . . . . . . . . . . . . . 18
⊢ ℕ
⊆ (ℤ≥‘1) |
54 | 53 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → ℕ
⊆ (ℤ≥‘1)) |
55 | 26, 33, 51, 54 | sumss 15081 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) →
Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
56 | | simpr 487 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ∈
Fin) |
57 | 56, 32 | fsumnn0cl 15093 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) →
Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈
ℕ0) |
58 | 55, 57 | eqeltrrd 2914 |
. . . . . . . . . . . . . . 15
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) →
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) ∈
ℕ0) |
59 | | eleq1 2900 |
. . . . . . . . . . . . . . 15
⊢
(Σ𝑘 ∈
ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0 ↔ 𝑁 ∈
ℕ0)) |
60 | 58, 59 | syl5ibcom 247 |
. . . . . . . . . . . . . 14
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) →
(Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁 → 𝑁 ∈
ℕ0)) |
61 | 60 | 3impia 1113 |
. . . . . . . . . . . . 13
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑁 ∈
ℕ0) |
62 | 61 | adantr 483 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈
ℕ0) |
63 | 62 | nn0red 11957 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ) |
64 | 18 | nn0ge0d 11959 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔‘𝑥)) |
65 | | nnge1 11666 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℕ → 1 ≤
𝑥) |
66 | 65 | adantl 484 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥) |
67 | 19, 21, 64, 66 | lemulge11d 11577 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ ((𝑔‘𝑥) · 𝑥)) |
68 | 56 | adantr 483 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → (◡𝑔 “ ℕ) ∈
Fin) |
69 | 32 | nn0red 11957 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ) |
70 | 69 | adantlr 713 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ) |
71 | 32 | nn0ge0d 11959 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘)) |
72 | 71 | adantlr 713 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘)) |
73 | | fveq2 6670 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑥 → (𝑔‘𝑘) = (𝑔‘𝑥)) |
74 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥) |
75 | 73, 74 | oveq12d 7174 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) · 𝑘) = ((𝑔‘𝑥) · 𝑥)) |
76 | | simprr 771 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → 𝑥 ∈ (◡𝑔 “ ℕ)) |
77 | 68, 70, 72, 75, 76 | fsumge1 15152 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
78 | 77 | expr 459 |
. . . . . . . . . . . . . . 15
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))) |
79 | | eldif 3946 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) |
80 | 51 | ralrimiva 3182 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) →
∀𝑘 ∈ (ℕ
∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0) |
81 | 75 | eqeq1d 2823 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑥 → (((𝑔‘𝑘) · 𝑘) = 0 ↔ ((𝑔‘𝑥) · 𝑥) = 0)) |
82 | 81 | rspccva 3622 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑘 ∈
(ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0) |
83 | 80, 82 | sylan 582 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0) |
84 | 79, 83 | sylan2br 596 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬
𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0) |
85 | 56 | adantr 483 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (◡𝑔 “ ℕ) ∈
Fin) |
86 | 32 | adantlr 713 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈
ℕ0) |
87 | 86 | nn0red 11957 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ) |
88 | 86 | nn0ge0d 11959 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘)) |
89 | 85, 87, 88 | fsumge0 15150 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤
Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
90 | 89 | adantrr 715 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬
𝑥 ∈ (◡𝑔 “ ℕ))) → 0 ≤
Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
91 | 84, 90 | eqbrtrd 5088 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬
𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
92 | 91 | expr 459 |
. . . . . . . . . . . . . . 15
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬
𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))) |
93 | 78, 92 | pm2.61d 181 |
. . . . . . . . . . . . . 14
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
94 | 55 | adantr 483 |
. . . . . . . . . . . . . 14
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) →
Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
95 | 93, 94 | breqtrd 5092 |
. . . . . . . . . . . . 13
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
96 | 95 | 3adantl3 1164 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
97 | | simpl3 1189 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) |
98 | 96, 97 | breqtrd 5092 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ 𝑁) |
99 | 19, 22, 63, 67, 98 | letrd 10797 |
. . . . . . . . . 10
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ 𝑁) |
100 | | nn0uz 12281 |
. . . . . . . . . . . 12
⊢
ℕ0 = (ℤ≥‘0) |
101 | 18, 100 | eleqtrdi 2923 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈
(ℤ≥‘0)) |
102 | 62 | nn0zd 12086 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ) |
103 | | elfz5 12901 |
. . . . . . . . . . 11
⊢ (((𝑔‘𝑥) ∈ (ℤ≥‘0)
∧ 𝑁 ∈ ℤ)
→ ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁)) |
104 | 101, 102,
103 | syl2anc 586 |
. . . . . . . . . 10
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁)) |
105 | 99, 104 | mpbird 259 |
. . . . . . . . 9
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (0...𝑁)) |
106 | 105 | adantr 483 |
. . . . . . . 8
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ (0...𝑁)) |
107 | | iftrue 4473 |
. . . . . . . . 9
⊢ (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁)) |
108 | 107 | adantl 484 |
. . . . . . . 8
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁)) |
109 | 106, 108 | eleqtrrd 2916 |
. . . . . . 7
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
110 | | nnge1 11666 |
. . . . . . . . . . . . . 14
⊢ ((𝑔‘𝑥) ∈ ℕ → 1 ≤ (𝑔‘𝑥)) |
111 | | nnnn0 11905 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℕ0) |
112 | 111 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0) |
113 | 112 | nn0ge0d 11959 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥) |
114 | | lemulge12 11503 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔‘𝑥))) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)) |
115 | 114 | expr 459 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))) |
116 | 21, 19, 113, 115 | syl21anc 835 |
. . . . . . . . . . . . . . 15
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))) |
117 | | letr 10734 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ ((𝑔‘𝑥) · 𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 ≤ ((𝑔‘𝑥) · 𝑥) ∧ ((𝑔‘𝑥) · 𝑥) ≤ 𝑁) → 𝑥 ≤ 𝑁)) |
118 | 21, 22, 63, 117 | syl3anc 1367 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑥 ≤ ((𝑔‘𝑥) · 𝑥) ∧ ((𝑔‘𝑥) · 𝑥) ≤ 𝑁) → 𝑥 ≤ 𝑁)) |
119 | 98, 118 | mpan2d 692 |
. . . . . . . . . . . . . . 15
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ ((𝑔‘𝑥) · 𝑥) → 𝑥 ≤ 𝑁)) |
120 | 116, 119 | syld 47 |
. . . . . . . . . . . . . 14
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ 𝑁)) |
121 | 110, 120 | syl5 34 |
. . . . . . . . . . . . 13
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ≤ 𝑁)) |
122 | | simpr 487 |
. . . . . . . . . . . . . . 15
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ) |
123 | 122, 52 | eleqtrdi 2923 |
. . . . . . . . . . . . . 14
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈
(ℤ≥‘1)) |
124 | | elfz5 12901 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈
(ℤ≥‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁)) |
125 | 123, 102,
124 | syl2anc 586 |
. . . . . . . . . . . . 13
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁)) |
126 | 121, 125 | sylibrd 261 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁))) |
127 | 126 | con3d 155 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔‘𝑥) ∈ ℕ)) |
128 | | elnn0 11900 |
. . . . . . . . . . . . 13
⊢ ((𝑔‘𝑥) ∈ ℕ0 ↔ ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0)) |
129 | 18, 128 | sylib 220 |
. . . . . . . . . . . 12
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0)) |
130 | 129 | ord 860 |
. . . . . . . . . . 11
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ (𝑔‘𝑥) ∈ ℕ → (𝑔‘𝑥) = 0)) |
131 | 127, 130 | syld 47 |
. . . . . . . . . 10
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → (𝑔‘𝑥) = 0)) |
132 | 131 | imp 409 |
. . . . . . . . 9
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) = 0) |
133 | | fvex 6683 |
. . . . . . . . . 10
⊢ (𝑔‘𝑥) ∈ V |
134 | 133 | elsn 4582 |
. . . . . . . . 9
⊢ ((𝑔‘𝑥) ∈ {0} ↔ (𝑔‘𝑥) = 0) |
135 | 132, 134 | sylibr 236 |
. . . . . . . 8
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ {0}) |
136 | 8 | adantl 484 |
. . . . . . . 8
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0}) |
137 | 135, 136 | eleqtrrd 2916 |
. . . . . . 7
⊢ ((((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
138 | 109, 137 | pm2.61dan 811 |
. . . . . 6
⊢ (((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
139 | 138 | ralrimiva 3182 |
. . . . 5
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
140 | | vex 3497 |
. . . . . 6
⊢ 𝑔 ∈ V |
141 | 140 | elixp 8468 |
. . . . 5
⊢ (𝑔 ∈ X𝑥 ∈
ℕ if(𝑥 ∈
(1...𝑁), (0...𝑁), {0}) ↔ (𝑔 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))) |
142 | 16, 139, 141 | sylanbrc 585 |
. . . 4
⊢ ((𝑔:ℕ⟶ℕ0 ∧
(◡𝑔 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
143 | 14, 142 | sylbi 219 |
. . 3
⊢ (𝑔 ∈ 𝑃 → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) |
144 | 143 | ssriv 3971 |
. 2
⊢ 𝑃 ⊆ X𝑥 ∈
ℕ if(𝑥 ∈
(1...𝑁), (0...𝑁), {0}) |
145 | | ssfi 8738 |
. 2
⊢ ((X𝑥 ∈
ℕ if(𝑥 ∈
(1...𝑁), (0...𝑁), {0}) ∈ Fin ∧ 𝑃 ⊆ X𝑥 ∈
ℕ if(𝑥 ∈
(1...𝑁), (0...𝑁), {0})) → 𝑃 ∈ Fin) |
146 | 12, 144, 145 | mp2an 690 |
1
⊢ 𝑃 ∈ Fin |