Step | Hyp | Ref
| Expression |
1 | | eqid 2170 |
. 2
⊢ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)) = (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)) |
2 | | elfzelz 9970 |
. . . 4
⊢ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) → 𝑗 ∈ ℤ) |
3 | 2 | adantl 275 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝑗 ∈ ℤ) |
4 | | mptfzshft.1 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | 4 | adantr 274 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → 𝐾 ∈ ℤ) |
6 | 3, 5 | zsubcld 9328 |
. 2
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) → (𝑗 − 𝐾) ∈ ℤ) |
7 | | elfzelz 9970 |
. . . 4
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ) |
8 | 7 | adantl 275 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ) |
9 | 4 | adantr 274 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
10 | 8, 9 | zaddcld 9327 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + 𝐾) ∈ ℤ) |
11 | | simprr 527 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑘 = (𝑗 − 𝐾)) |
12 | 11 | oveq1d 5866 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → (𝑘 + 𝐾) = ((𝑗 − 𝐾) + 𝐾)) |
13 | 2 | ad2antrl 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑗 ∈ ℤ) |
14 | 4 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝐾 ∈ ℤ) |
15 | | zcn 9206 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℤ → 𝑗 ∈
ℂ) |
16 | | zcn 9206 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℂ) |
17 | | npcan 8117 |
. . . . . . . . 9
⊢ ((𝑗 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑗 − 𝐾) + 𝐾) = 𝑗) |
18 | 15, 16, 17 | syl2an 287 |
. . . . . . . 8
⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑗 − 𝐾) + 𝐾) = 𝑗) |
19 | 13, 14, 18 | syl2anc 409 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → ((𝑗 − 𝐾) + 𝐾) = 𝑗) |
20 | 12, 19 | eqtr2d 2204 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑗 = (𝑘 + 𝐾)) |
21 | | simprl 526 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
22 | 20, 21 | eqeltrrd 2248 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → (𝑘 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
23 | | mptfzshft.2 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ ℤ) |
24 | 23 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑀 ∈ ℤ) |
25 | | mptfzshft.3 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
26 | 25 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑁 ∈ ℤ) |
27 | 13, 14 | zsubcld 9328 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → (𝑗 − 𝐾) ∈ ℤ) |
28 | 11, 27 | eqeltrd 2247 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑘 ∈ ℤ) |
29 | | fzaddel 10004 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
30 | 24, 26, 28, 14, 29 | syl22anc 1234 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
31 | 22, 30 | mpbird 166 |
. . . 4
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → 𝑘 ∈ (𝑀...𝑁)) |
32 | 31, 20 | jca 304 |
. . 3
⊢ ((𝜑 ∧ (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) |
33 | | simprr 527 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑗 = (𝑘 + 𝐾)) |
34 | | simprl 526 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑘 ∈ (𝑀...𝑁)) |
35 | 23 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑀 ∈ ℤ) |
36 | 25 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑁 ∈ ℤ) |
37 | 7 | ad2antrl 487 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑘 ∈ ℤ) |
38 | 4 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝐾 ∈ ℤ) |
39 | 35, 36, 37, 38, 29 | syl22anc 1234 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝑘 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) |
40 | 34, 39 | mpbid 146 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → (𝑘 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
41 | 33, 40 | eqeltrd 2247 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) |
42 | 33 | oveq1d 5866 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → (𝑗 − 𝐾) = ((𝑘 + 𝐾) − 𝐾)) |
43 | | zcn 9206 |
. . . . . . 7
⊢ (𝑘 ∈ ℤ → 𝑘 ∈
ℂ) |
44 | | pncan 8114 |
. . . . . . 7
⊢ ((𝑘 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑘 + 𝐾) − 𝐾) = 𝑘) |
45 | 43, 16, 44 | syl2an 287 |
. . . . . 6
⊢ ((𝑘 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑘 + 𝐾) − 𝐾) = 𝑘) |
46 | 37, 38, 45 | syl2anc 409 |
. . . . 5
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → ((𝑘 + 𝐾) − 𝐾) = 𝑘) |
47 | 42, 46 | eqtr2d 2204 |
. . . 4
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → 𝑘 = (𝑗 − 𝐾)) |
48 | 41, 47 | jca 304 |
. . 3
⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾))) → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾))) |
49 | 32, 48 | impbida 591 |
. 2
⊢ (𝜑 → ((𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ∧ 𝑘 = (𝑗 − 𝐾)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝑘 + 𝐾)))) |
50 | 1, 6, 10, 49 | f1od 6050 |
1
⊢ (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁)) |