Step | Hyp | Ref
| Expression |
1 | | eqid 2164 |
. . . 4
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) |
2 | | seq3shft.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | | seq3shft.ex |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ 𝑉) |
4 | 3 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐹 ∈ 𝑉) |
5 | | seq3shft.n |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | 5 | zcnd 9308 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℂ) |
7 | 6 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
8 | | eluzelz 9469 |
. . . . . . . 8
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → 𝑥 ∈ ℤ) |
9 | 8 | adantl 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ ℤ) |
10 | 9 | zcnd 9308 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ ℂ) |
11 | | shftvalg 10772 |
. . . . . 6
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑥) = (𝐹‘(𝑥 − 𝑁))) |
12 | 4, 7, 10, 11 | syl3anc 1227 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝐹 shift 𝑁)‘𝑥) = (𝐹‘(𝑥 − 𝑁))) |
13 | | fveq2 5483 |
. . . . . . 7
⊢ (𝑎 = (𝑥 − 𝑁) → (𝐹‘𝑎) = (𝐹‘(𝑥 − 𝑁))) |
14 | 13 | eleq1d 2233 |
. . . . . 6
⊢ (𝑎 = (𝑥 − 𝑁) → ((𝐹‘𝑎) ∈ 𝑆 ↔ (𝐹‘(𝑥 − 𝑁)) ∈ 𝑆)) |
15 | | seq3shft.fn |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁))) → (𝐹‘𝑥) ∈ 𝑆) |
16 | 15 | ralrimiva 2537 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁))(𝐹‘𝑥) ∈ 𝑆) |
17 | | fveq2 5483 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) |
18 | 17 | eleq1d 2233 |
. . . . . . . . 9
⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑎) ∈ 𝑆)) |
19 | 18 | cbvralv 2690 |
. . . . . . . 8
⊢
(∀𝑥 ∈
(ℤ≥‘(𝑀 − 𝑁))(𝐹‘𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ≥‘(𝑀 − 𝑁))(𝐹‘𝑎) ∈ 𝑆) |
20 | 16, 19 | sylib 121 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘(𝑀 − 𝑁))(𝐹‘𝑎) ∈ 𝑆) |
21 | 20 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈
(ℤ≥‘(𝑀 − 𝑁))(𝐹‘𝑎) ∈ 𝑆) |
22 | 2, 5 | zsubcld 9312 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℤ) |
23 | 22 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝑀 − 𝑁) ∈ ℤ) |
24 | 5 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
25 | 9, 24 | zsubcld 9312 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝑥 − 𝑁) ∈ ℤ) |
26 | 2 | zred 9307 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℝ) |
27 | 26 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℝ) |
28 | 9 | zred 9307 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ ℝ) |
29 | 24 | zred 9307 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℝ) |
30 | | eluzle 9472 |
. . . . . . . . 9
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → 𝑀 ≤ 𝑥) |
31 | 30 | adantl 275 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑀 ≤ 𝑥) |
32 | 27, 28, 29, 31 | lesub1dd 8453 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝑀 − 𝑁) ≤ (𝑥 − 𝑁)) |
33 | | eluz2 9466 |
. . . . . . 7
⊢ ((𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁)) ↔ ((𝑀 − 𝑁) ∈ ℤ ∧ (𝑥 − 𝑁) ∈ ℤ ∧ (𝑀 − 𝑁) ≤ (𝑥 − 𝑁))) |
34 | 23, 25, 32, 33 | syl3anbrc 1170 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))) |
35 | 14, 21, 34 | rspcdva 2833 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑥 − 𝑁)) ∈ 𝑆) |
36 | 12, 35 | eqeltrd 2241 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝐹 shift 𝑁)‘𝑥) ∈ 𝑆) |
37 | | seq3shft.pl |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
38 | 1, 2, 36, 37 | seqf 10390 |
. . 3
⊢ (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)):(ℤ≥‘𝑀)⟶𝑆) |
39 | 38 | ffnd 5335 |
. 2
⊢ (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)) Fn (ℤ≥‘𝑀)) |
40 | | eqid 2164 |
. . . . . 6
⊢
(ℤ≥‘(𝑀 − 𝑁)) = (ℤ≥‘(𝑀 − 𝑁)) |
41 | 40, 22, 15, 37 | seqf 10390 |
. . . . 5
⊢ (𝜑 → seq(𝑀 − 𝑁)( + , 𝐹):(ℤ≥‘(𝑀 − 𝑁))⟶𝑆) |
42 | 41 | ffnd 5335 |
. . . 4
⊢ (𝜑 → seq(𝑀 − 𝑁)( + , 𝐹) Fn (ℤ≥‘(𝑀 − 𝑁))) |
43 | | seqex 10376 |
. . . . 5
⊢ seq(𝑀 − 𝑁)( + , 𝐹) ∈ V |
44 | 43 | shftfn 10760 |
. . . 4
⊢
((seq(𝑀 −
𝑁)( + , 𝐹) Fn (ℤ≥‘(𝑀 − 𝑁)) ∧ 𝑁 ∈ ℂ) → (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))}) |
45 | 42, 6, 44 | syl2anc 409 |
. . 3
⊢ (𝜑 → (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))}) |
46 | | shftuz 10753 |
. . . . . 6
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))} = (ℤ≥‘((𝑀 − 𝑁) + 𝑁))) |
47 | 5, 22, 46 | syl2anc 409 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))} = (ℤ≥‘((𝑀 − 𝑁) + 𝑁))) |
48 | 2 | zcnd 9308 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ ℂ) |
49 | 48, 6 | npcand 8207 |
. . . . . 6
⊢ (𝜑 → ((𝑀 − 𝑁) + 𝑁) = 𝑀) |
50 | 49 | fveq2d 5487 |
. . . . 5
⊢ (𝜑 →
(ℤ≥‘((𝑀 − 𝑁) + 𝑁)) = (ℤ≥‘𝑀)) |
51 | 47, 50 | eqtrd 2197 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))} = (ℤ≥‘𝑀)) |
52 | 51 | fneq2d 5276 |
. . 3
⊢ (𝜑 → ((seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝑁) ∈
(ℤ≥‘(𝑀 − 𝑁))} ↔ (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁) Fn (ℤ≥‘𝑀))) |
53 | 45, 52 | mpbid 146 |
. 2
⊢ (𝜑 → (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁) Fn (ℤ≥‘𝑀)) |
54 | 48, 6 | negsubd 8209 |
. . . . . 6
⊢ (𝜑 → (𝑀 + -𝑁) = (𝑀 − 𝑁)) |
55 | 54 | adantr 274 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (𝑀 + -𝑁) = (𝑀 − 𝑁)) |
56 | 55 | seqeq1d 10380 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → seq(𝑀 + -𝑁)( + , 𝐹) = seq(𝑀 − 𝑁)( + , 𝐹)) |
57 | | eluzelcn 9471 |
. . . . . 6
⊢ (𝑧 ∈
(ℤ≥‘𝑀) → 𝑧 ∈ ℂ) |
58 | 57 | adantl 275 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑧 ∈ ℂ) |
59 | 6 | adantr 274 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
60 | 58, 59 | negsubd 8209 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (𝑧 + -𝑁) = (𝑧 − 𝑁)) |
61 | 56, 60 | fveq12d 5490 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (seq(𝑀 + -𝑁)( + , 𝐹)‘(𝑧 + -𝑁)) = (seq(𝑀 − 𝑁)( + , 𝐹)‘(𝑧 − 𝑁))) |
62 | | simpr 109 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑧 ∈ (ℤ≥‘𝑀)) |
63 | 5 | adantr 274 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
64 | 63 | znegcld 9309 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → -𝑁 ∈ ℤ) |
65 | 3 | ad2antrr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → 𝐹 ∈ 𝑉) |
66 | 59 | adantr 274 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → 𝑁 ∈ ℂ) |
67 | | elfzelz 9954 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝑀...𝑧) → 𝑦 ∈ ℤ) |
68 | 67 | adantl 275 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → 𝑦 ∈ ℤ) |
69 | 68 | zcnd 9308 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → 𝑦 ∈ ℂ) |
70 | | shftvalg 10772 |
. . . . . 6
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
71 | 65, 66, 69, 70 | syl3anc 1227 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 − 𝑁))) |
72 | 69, 66 | negsubd 8209 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → (𝑦 + -𝑁) = (𝑦 − 𝑁)) |
73 | 72 | fveq2d 5487 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → (𝐹‘(𝑦 + -𝑁)) = (𝐹‘(𝑦 − 𝑁))) |
74 | 71, 73 | eqtr4d 2200 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑦 ∈ (𝑀...𝑧)) → ((𝐹 shift 𝑁)‘𝑦) = (𝐹‘(𝑦 + -𝑁))) |
75 | 36 | adantlr 469 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝐹 shift 𝑁)‘𝑥) ∈ 𝑆) |
76 | | simpll 519 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) → 𝜑) |
77 | | simpr 109 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) → 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) |
78 | 54 | fveq2d 5487 |
. . . . . . . 8
⊢ (𝜑 →
(ℤ≥‘(𝑀 + -𝑁)) = (ℤ≥‘(𝑀 − 𝑁))) |
79 | 78 | eleq2d 2234 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁)) ↔ 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁)))) |
80 | 79 | ad2antrr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) → (𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁)) ↔ 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁)))) |
81 | 77, 80 | mpbid 146 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) → 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁))) |
82 | 76, 81, 15 | syl2anc 409 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + -𝑁))) → (𝐹‘𝑥) ∈ 𝑆) |
83 | 37 | adantlr 469 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
84 | 62, 64, 74, 75, 82, 83 | seq3shft2 10402 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , (𝐹 shift 𝑁))‘𝑧) = (seq(𝑀 + -𝑁)( + , 𝐹)‘(𝑧 + -𝑁))) |
85 | | shftvalg 10772 |
. . . 4
⊢
((seq(𝑀 −
𝑁)( + , 𝐹) ∈ V ∧ 𝑁 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)‘𝑧) = (seq(𝑀 − 𝑁)( + , 𝐹)‘(𝑧 − 𝑁))) |
86 | 43, 59, 58, 85 | mp3an2i 1331 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → ((seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)‘𝑧) = (seq(𝑀 − 𝑁)( + , 𝐹)‘(𝑧 − 𝑁))) |
87 | 61, 84, 86 | 3eqtr4d 2207 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , (𝐹 shift 𝑁))‘𝑧) = ((seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)‘𝑧)) |
88 | 39, 53, 87 | eqfnfvd 5583 |
1
⊢ (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)) |