Step | Hyp | Ref
| Expression |
1 | | 3z 11410 |
. . 3
⊢ 3 ∈
ℤ |
2 | 1 | a1i 11 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) → 3
∈ ℤ) |
3 | | fzfid 12772 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) →
(0...𝑁) ∈
Fin) |
4 | | ffvelrn 6357 |
. . . . 5
⊢ ((𝐹:(0...𝑁)⟶ℤ ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℤ) |
5 | 4 | adantll 750 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℤ) |
6 | | 10nnOLD 11193 |
. . . . . 6
⊢ 10 ∈
ℕ |
7 | 6 | nnzi 11401 |
. . . . 5
⊢ 10 ∈
ℤ |
8 | | elfznn0 12433 |
. . . . . 6
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
9 | 8 | adantl 482 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
10 | | zexpcl 12875 |
. . . . 5
⊢ ((10
∈ ℤ ∧ 𝑘
∈ ℕ0) → (10↑𝑘) ∈ ℤ) |
11 | 7, 9, 10 | sylancr 695 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (10↑𝑘) ∈ ℤ) |
12 | 5, 11 | zmulcld 11488 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐹‘𝑘) · (10↑𝑘)) ∈ ℤ) |
13 | 3, 12 | fsumzcl 14466 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) →
Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) ∈ ℤ) |
14 | 3, 5 | fsumzcl 14466 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) →
Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) ∈ ℤ) |
15 | 12, 5 | zsubcld 11487 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘)) ∈ ℤ) |
16 | | ax-1cn 9994 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
17 | 6 | nncni 11030 |
. . . . . . . . . . . 12
⊢ 10 ∈
ℂ |
18 | 16, 17 | negsubdi2i 10367 |
. . . . . . . . . . 11
⊢ -(1
− 10) = (10 − 1) |
19 | | df-10OLD 11087 |
. . . . . . . . . . . 12
⊢ 10 = (9 +
1) |
20 | 19 | oveq1i 6660 |
. . . . . . . . . . 11
⊢ (10
− 1) = ((9 + 1) − 1) |
21 | | 9cn 11108 |
. . . . . . . . . . . 12
⊢ 9 ∈
ℂ |
22 | 21, 16 | pncan3oi 10297 |
. . . . . . . . . . 11
⊢ ((9 + 1)
− 1) = 9 |
23 | 18, 20, 22 | 3eqtri 2648 |
. . . . . . . . . 10
⊢ -(1
− 10) = 9 |
24 | | 3t3e9 11180 |
. . . . . . . . . 10
⊢ (3
· 3) = 9 |
25 | 23, 24 | eqtr4i 2647 |
. . . . . . . . 9
⊢ -(1
− 10) = (3 · 3) |
26 | 17 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ 10 ∈ ℂ) |
27 | | 1re 10039 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℝ |
28 | | 1lt10OLD 11238 |
. . . . . . . . . . . . . . . . 17
⊢ 1 <
10 |
29 | 27, 28 | gtneii 10149 |
. . . . . . . . . . . . . . . 16
⊢ 10 ≠
1 |
30 | 29 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ 10 ≠ 1) |
31 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℕ0) |
32 | 26, 30, 31 | geoser 14599 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ Σ𝑗 ∈
(0...(𝑘 −
1))(10↑𝑗) = ((1
− (10↑𝑘)) / (1
− 10))) |
33 | | fzfid 12772 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ (0...(𝑘 − 1))
∈ Fin) |
34 | | elfznn0 12433 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...(𝑘 − 1)) → 𝑗 ∈ ℕ0) |
35 | 34 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ0
∧ 𝑗 ∈ (0...(𝑘 − 1))) → 𝑗 ∈
ℕ0) |
36 | | zexpcl 12875 |
. . . . . . . . . . . . . . . 16
⊢ ((10
∈ ℤ ∧ 𝑗
∈ ℕ0) → (10↑𝑗) ∈ ℤ) |
37 | 7, 35, 36 | sylancr 695 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ0
∧ 𝑗 ∈ (0...(𝑘 − 1))) →
(10↑𝑗) ∈
ℤ) |
38 | 33, 37 | fsumzcl 14466 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ Σ𝑗 ∈
(0...(𝑘 −
1))(10↑𝑗) ∈
ℤ) |
39 | 32, 38 | eqeltrrd 2702 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ ((1 − (10↑𝑘)) / (1 − 10)) ∈
ℤ) |
40 | | 1z 11407 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℤ |
41 | | zsubcl 11419 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℤ ∧ 10 ∈ ℤ) → (1 − 10) ∈
ℤ) |
42 | 40, 7, 41 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢ (1
− 10) ∈ ℤ |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (1 − 10) ∈ ℤ) |
44 | 27, 28 | ltneii 10150 |
. . . . . . . . . . . . . . . 16
⊢ 1 ≠
10 |
45 | 16, 17 | subeq0i 10361 |
. . . . . . . . . . . . . . . . 17
⊢ ((1
− 10) = 0 ↔ 1 = 10) |
46 | 45 | necon3bii 2846 |
. . . . . . . . . . . . . . . 16
⊢ ((1
− 10) ≠ 0 ↔ 1 ≠ 10) |
47 | 44, 46 | mpbir 221 |
. . . . . . . . . . . . . . 15
⊢ (1
− 10) ≠ 0 |
48 | 47 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (1 − 10) ≠ 0) |
49 | 7, 31, 10 | sylancr 695 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ (10↑𝑘) ∈
ℤ) |
50 | | zsubcl 11419 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℤ ∧ (10↑𝑘) ∈ ℤ) → (1 −
(10↑𝑘)) ∈
ℤ) |
51 | 40, 49, 50 | sylancr 695 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (1 − (10↑𝑘)) ∈ ℤ) |
52 | | dvdsval2 14986 |
. . . . . . . . . . . . . 14
⊢ (((1
− 10) ∈ ℤ ∧ (1 − 10) ≠ 0 ∧ (1 −
(10↑𝑘)) ∈
ℤ) → ((1 − 10) ∥ (1 − (10↑𝑘)) ↔ ((1 − (10↑𝑘)) / (1 − 10)) ∈
ℤ)) |
53 | 43, 48, 51, 52 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ ((1 − 10) ∥ (1 − (10↑𝑘)) ↔ ((1 − (10↑𝑘)) / (1 − 10)) ∈
ℤ)) |
54 | 39, 53 | mpbird 247 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ (1 − 10) ∥ (1 − (10↑𝑘))) |
55 | 49 | zcnd 11483 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ (10↑𝑘) ∈
ℂ) |
56 | | negsubdi2 10340 |
. . . . . . . . . . . . 13
⊢
(((10↑𝑘) ∈
ℂ ∧ 1 ∈ ℂ) → -((10↑𝑘) − 1) = (1 − (10↑𝑘))) |
57 | 55, 16, 56 | sylancl 694 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ -((10↑𝑘)
− 1) = (1 − (10↑𝑘))) |
58 | 54, 57 | breqtrrd 4681 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ (1 − 10) ∥ -((10↑𝑘) − 1)) |
59 | | peano2zm 11420 |
. . . . . . . . . . . . 13
⊢
((10↑𝑘) ∈
ℤ → ((10↑𝑘)
− 1) ∈ ℤ) |
60 | 49, 59 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ ((10↑𝑘) −
1) ∈ ℤ) |
61 | | dvdsnegb 14999 |
. . . . . . . . . . . 12
⊢ (((1
− 10) ∈ ℤ ∧ ((10↑𝑘) − 1) ∈ ℤ) → ((1
− 10) ∥ ((10↑𝑘) − 1) ↔ (1 − 10) ∥
-((10↑𝑘) −
1))) |
62 | 42, 60, 61 | sylancr 695 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((1 − 10) ∥ ((10↑𝑘) − 1) ↔ (1 − 10) ∥
-((10↑𝑘) −
1))) |
63 | 58, 62 | mpbird 247 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ (1 − 10) ∥ ((10↑𝑘) − 1)) |
64 | | negdvdsb 14998 |
. . . . . . . . . . 11
⊢ (((1
− 10) ∈ ℤ ∧ ((10↑𝑘) − 1) ∈ ℤ) → ((1
− 10) ∥ ((10↑𝑘) − 1) ↔ -(1 − 10) ∥
((10↑𝑘) −
1))) |
65 | 42, 60, 64 | sylancr 695 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ ((1 − 10) ∥ ((10↑𝑘) − 1) ↔ -(1 − 10) ∥
((10↑𝑘) −
1))) |
66 | 63, 65 | mpbid 222 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ -(1 − 10) ∥ ((10↑𝑘) − 1)) |
67 | 25, 66 | syl5eqbrr 4689 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (3 · 3) ∥ ((10↑𝑘) − 1)) |
68 | 1 | a1i 11 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ 3 ∈ ℤ) |
69 | | muldvds1 15006 |
. . . . . . . . 9
⊢ ((3
∈ ℤ ∧ 3 ∈ ℤ ∧ ((10↑𝑘) − 1) ∈ ℤ) → ((3
· 3) ∥ ((10↑𝑘) − 1) → 3 ∥ ((10↑𝑘) − 1))) |
70 | 68, 68, 60, 69 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ ((3 · 3) ∥ ((10↑𝑘) − 1) → 3 ∥ ((10↑𝑘) − 1))) |
71 | 67, 70 | mpd 15 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
→ 3 ∥ ((10↑𝑘) − 1)) |
72 | 9, 71 | syl 17 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 3 ∥ ((10↑𝑘) − 1)) |
73 | 1 | a1i 11 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 3 ∈ ℤ) |
74 | 11, 59 | syl 17 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((10↑𝑘) − 1) ∈ ℤ) |
75 | | dvdsmultr2 15021 |
. . . . . . 7
⊢ ((3
∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ ∧ ((10↑𝑘) − 1) ∈ ℤ)
→ (3 ∥ ((10↑𝑘) − 1) → 3 ∥ ((𝐹‘𝑘) · ((10↑𝑘) − 1)))) |
76 | 73, 5, 74, 75 | syl3anc 1326 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (3 ∥ ((10↑𝑘) − 1) → 3 ∥
((𝐹‘𝑘) · ((10↑𝑘) − 1)))) |
77 | 72, 76 | mpd 15 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 3 ∥ ((𝐹‘𝑘) · ((10↑𝑘) − 1))) |
78 | 5 | zcnd 11483 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐹‘𝑘) ∈ ℂ) |
79 | 11 | zcnd 11483 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (10↑𝑘) ∈ ℂ) |
80 | 16 | a1i 11 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ) |
81 | 78, 79, 80 | subdid 10486 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐹‘𝑘) · ((10↑𝑘) − 1)) = (((𝐹‘𝑘) · (10↑𝑘)) − ((𝐹‘𝑘) · 1))) |
82 | 78 | mulid1d 10057 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐹‘𝑘) · 1) = (𝐹‘𝑘)) |
83 | 82 | oveq2d 6666 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → (((𝐹‘𝑘) · (10↑𝑘)) − ((𝐹‘𝑘) · 1)) = (((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘))) |
84 | 81, 83 | eqtrd 2656 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐹‘𝑘) · ((10↑𝑘) − 1)) = (((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘))) |
85 | 77, 84 | breqtrd 4679 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → 3 ∥ (((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘))) |
86 | 3, 2, 15, 85 | fsumdvds 15030 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) → 3
∥ Σ𝑘 ∈
(0...𝑁)(((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘))) |
87 | 12 | zcnd 11483 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐹‘𝑘) · (10↑𝑘)) ∈ ℂ) |
88 | 3, 87, 78 | fsumsub 14520 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) →
Σ𝑘 ∈ (0...𝑁)(((𝐹‘𝑘) · (10↑𝑘)) − (𝐹‘𝑘)) = (Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) − Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) |
89 | 86, 88 | breqtrd 4679 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) → 3
∥ (Σ𝑘 ∈
(0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) − Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) |
90 | | dvdssub2 15023 |
. 2
⊢ (((3
∈ ℤ ∧ Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) ∈ ℤ ∧ Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘) ∈ ℤ) ∧ 3 ∥
(Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) − Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) |
91 | 2, 13, 14, 89, 90 | syl31anc 1329 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(0...𝑁)⟶ℤ) → (3
∥ Σ𝑘 ∈
(0...𝑁)((𝐹‘𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) |