Step | Hyp | Ref
| Expression |
1 | | iseqcaopr3.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | eluzfz2 9051 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
3 | 1, 2 | syl 14 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
4 | | fveq2 5198 |
. . . . 5
⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑀)) |
5 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀)) |
6 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑀)) |
7 | 5, 6 | oveq12d 5550 |
. . . . 5
⊢ (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))) |
8 | 4, 7 | eqeq12d 2095 |
. . . 4
⊢ (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))) |
9 | 8 | imbi2d 228 |
. . 3
⊢ (𝑧 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))))) |
10 | | fveq2 5198 |
. . . . 5
⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑛)) |
11 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛)) |
12 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑛)) |
13 | 11, 12 | oveq12d 5550 |
. . . . 5
⊢ (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛))) |
14 | 10, 13 | eqeq12d 2095 |
. . . 4
⊢ (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))) |
15 | 14 | imbi2d 228 |
. . 3
⊢ (𝑧 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛))))) |
16 | | fveq2 5198 |
. . . . 5
⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1))) |
17 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))) |
18 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) |
19 | 17, 18 | oveq12d 5550 |
. . . . 5
⊢ (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))) |
20 | 16, 19 | eqeq12d 2095 |
. . . 4
⊢ (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))) |
21 | 20 | imbi2d 228 |
. . 3
⊢ (𝑧 = (𝑛 + 1) → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))))) |
22 | | fveq2 5198 |
. . . . 5
⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = (seq𝑀( + , 𝐻, 𝑆)‘𝑁)) |
23 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑧) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁)) |
24 | | fveq2 5198 |
. . . . . 6
⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐺, 𝑆)‘𝑧) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁)) |
25 | 23, 24 | oveq12d 5550 |
. . . . 5
⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |
26 | 22, 25 | eqeq12d 2095 |
. . . 4
⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧)) ↔ (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))) |
27 | 26 | imbi2d 228 |
. . 3
⊢ (𝑧 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑧)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))))) |
28 | | eluzel2 8624 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
29 | 1, 28 | syl 14 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ ℤ) |
30 | | uzid 8633 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
31 | 29, 30 | syl 14 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
32 | | iseqcaopr3.6 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) |
33 | 32 | ralrimiva 2434 |
. . . . . 6
⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) |
34 | | fveq2 5198 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → (𝐻‘𝑘) = (𝐻‘𝑀)) |
35 | | fveq2 5198 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) |
36 | | fveq2 5198 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀)) |
37 | 35, 36 | oveq12d 5550 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀))) |
38 | 34, 37 | eqeq12d 2095 |
. . . . . . 7
⊢ (𝑘 = 𝑀 → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))) |
39 | 38 | rspcv 2697 |
. . . . . 6
⊢ (𝑀 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) → (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))) |
40 | 31, 33, 39 | sylc 61 |
. . . . 5
⊢ (𝜑 → (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀))) |
41 | | iseqcaopr3.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ 𝑉) |
42 | | iseqcaopr3.2 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) |
43 | 42 | ralrimivva 2443 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆) |
44 | 43 | adantr 270 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆) |
45 | | iseqcaopr3.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) |
46 | | iseqcaopr3.5 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) |
47 | | oveq1 5539 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐹‘𝑘) → (𝑥𝑄𝑦) = ((𝐹‘𝑘)𝑄𝑦)) |
48 | 47 | eleq1d 2147 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹‘𝑘)𝑄𝑦) ∈ 𝑆)) |
49 | | oveq2 5540 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝐺‘𝑘) → ((𝐹‘𝑘)𝑄𝑦) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) |
50 | 49 | eleq1d 2147 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘𝑘) → (((𝐹‘𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆)) |
51 | 48, 50 | rspc2v 2713 |
. . . . . . . . . . 11
⊢ (((𝐹‘𝑘) ∈ 𝑆 ∧ (𝐺‘𝑘) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆)) |
52 | 45, 46, 51 | syl2anc 403 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆)) |
53 | 44, 52 | mpd 13 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆) |
54 | 32, 53 | eqeltrd 2155 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) ∈ 𝑆) |
55 | 54 | ralrimiva 2434 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) ∈ 𝑆) |
56 | | fveq2 5198 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥 → (𝐻‘𝑘) = (𝐻‘𝑥)) |
57 | 56 | eleq1d 2147 |
. . . . . . . 8
⊢ (𝑘 = 𝑥 → ((𝐻‘𝑘) ∈ 𝑆 ↔ (𝐻‘𝑥) ∈ 𝑆)) |
58 | 57 | rspcv 2697 |
. . . . . . 7
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) ∈ 𝑆 → (𝐻‘𝑥) ∈ 𝑆)) |
59 | 55, 58 | mpan9 275 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆) |
60 | | iseqcaopr3.1 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
61 | 29, 41, 59, 60 | iseq1 9442 |
. . . . 5
⊢ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = (𝐻‘𝑀)) |
62 | 45 | ralrimiva 2434 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆) |
63 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥)) |
64 | 63 | eleq1d 2147 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘𝑥) ∈ 𝑆)) |
65 | 64 | rspcv 2697 |
. . . . . . . 8
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆 → (𝐹‘𝑥) ∈ 𝑆)) |
66 | 62, 65 | mpan9 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
67 | 29, 41, 66, 60 | iseq1 9442 |
. . . . . 6
⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) |
68 | 46 | ralrimiva 2434 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆) |
69 | | fveq2 5198 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥)) |
70 | 69 | eleq1d 2147 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥 → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆)) |
71 | 70 | rspcv 2697 |
. . . . . . . 8
⊢ (𝑥 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆 → (𝐺‘𝑥) ∈ 𝑆)) |
72 | 68, 71 | mpan9 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
73 | 29, 41, 72, 60 | iseq1 9442 |
. . . . . 6
⊢ (𝜑 → (seq𝑀( + , 𝐺, 𝑆)‘𝑀) = (𝐺‘𝑀)) |
74 | 67, 73 | oveq12d 5550 |
. . . . 5
⊢ (𝜑 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀))) |
75 | 40, 61, 74 | 3eqtr4d 2123 |
. . . 4
⊢ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀))) |
76 | 75 | a1i 9 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑀) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑀)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑀)))) |
77 | | oveq1 5539 |
. . . . . 6
⊢
((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1)))) |
78 | | elfzouz 9161 |
. . . . . . . . 9
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀)) |
79 | 78 | adantl 271 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
80 | 41 | adantr 270 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑆 ∈ 𝑉) |
81 | 59 | adantlr 460 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆) |
82 | 60 | adantlr 460 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
83 | 79, 80, 81, 82 | iseqp1 9445 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1)))) |
84 | | iseqcaopr3.7 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))) |
85 | | fzofzp1 9236 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
86 | | elfzuz 9041 |
. . . . . . . . . . . 12
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈
(ℤ≥‘𝑀)) |
87 | 85, 86 | syl 14 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈
(ℤ≥‘𝑀)) |
88 | 87 | adantl 271 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈
(ℤ≥‘𝑀)) |
89 | 33 | adantr 270 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) |
90 | | fveq2 5198 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 + 1) → (𝐻‘𝑘) = (𝐻‘(𝑛 + 1))) |
91 | | fveq2 5198 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) |
92 | | fveq2 5198 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1))) |
93 | 91, 92 | oveq12d 5550 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) |
94 | 90, 93 | eqeq12d 2095 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑛 + 1) → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))) |
95 | 94 | rspcv 2697 |
. . . . . . . . . 10
⊢ ((𝑛 + 1) ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))) |
96 | 88, 89, 95 | sylc 61 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) |
97 | 96 | oveq2d 5548 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))) |
98 | 66 | adantlr 460 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
99 | 79, 80, 98, 82 | iseqp1 9445 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
100 | 72 | adantlr 460 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
101 | 79, 80, 100, 82 | iseqp1 9445 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))) |
102 | 99, 101 | oveq12d 5550 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))) |
103 | 84, 97, 102 | 3eqtr4rd 2124 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1)))) |
104 | 83, 103 | eqeq12d 2095 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))) ↔ ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + (𝐻‘(𝑛 + 1))))) |
105 | 77, 104 | syl5ibr 154 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1))))) |
106 | 105 | expcom 114 |
. . . 4
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))))) |
107 | 106 | a2d 26 |
. . 3
⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑛) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛))) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺, 𝑆)‘(𝑛 + 1)))))) |
108 | 9, 15, 21, 27, 76, 107 | fzind2 9248 |
. 2
⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))) |
109 | 3, 108 | mpcom 36 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |