Step | Hyp | Ref
| Expression |
1 | | seq3distr.1 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
2 | | seq3distr.4 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
3 | | seq3distr.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | | seq3distr.2 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) |
5 | | seq3distr.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑆) |
6 | 5 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ 𝑆) |
7 | | seq3distr.t |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆) |
8 | 7 | ralrimivva 2539 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆) |
9 | | oveq1 5831 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝑥𝑇𝑦) = (𝑎𝑇𝑦)) |
10 | 9 | eleq1d 2226 |
. . . . . . . . 9
⊢ (𝑥 = 𝑎 → ((𝑥𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑦) ∈ 𝑆)) |
11 | | oveq2 5832 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑏 → (𝑎𝑇𝑦) = (𝑎𝑇𝑏)) |
12 | 11 | eleq1d 2226 |
. . . . . . . . 9
⊢ (𝑦 = 𝑏 → ((𝑎𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑏) ∈ 𝑆)) |
13 | 10, 12 | cbvral2v 2691 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆 ↔ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
14 | 8, 13 | sylib 121 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
15 | 14 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
16 | | oveq1 5831 |
. . . . . . . 8
⊢ (𝑎 = 𝐶 → (𝑎𝑇𝑏) = (𝐶𝑇𝑏)) |
17 | 16 | eleq1d 2226 |
. . . . . . 7
⊢ (𝑎 = 𝐶 → ((𝑎𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑏) ∈ 𝑆)) |
18 | | oveq2 5832 |
. . . . . . . 8
⊢ (𝑏 = (𝑥 + 𝑦) → (𝐶𝑇𝑏) = (𝐶𝑇(𝑥 + 𝑦))) |
19 | 18 | eleq1d 2226 |
. . . . . . 7
⊢ (𝑏 = (𝑥 + 𝑦) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)) |
20 | 17, 19 | rspc2va 2830 |
. . . . . 6
⊢ (((𝐶 ∈ 𝑆 ∧ (𝑥 + 𝑦) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) |
21 | 6, 1, 15, 20 | syl21anc 1219 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) |
22 | | oveq2 5832 |
. . . . . 6
⊢ (𝑧 = (𝑥 + 𝑦) → (𝐶𝑇𝑧) = (𝐶𝑇(𝑥 + 𝑦))) |
23 | | eqid 2157 |
. . . . . 6
⊢ (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) |
24 | 22, 23 | fvmptg 5544 |
. . . . 5
⊢ (((𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦))) |
25 | 1, 21, 24 | syl2anc 409 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦))) |
26 | | simprl 521 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) |
27 | | oveq2 5832 |
. . . . . . . . 9
⊢ (𝑏 = 𝑥 → (𝐶𝑇𝑏) = (𝐶𝑇𝑥)) |
28 | 27 | eleq1d 2226 |
. . . . . . . 8
⊢ (𝑏 = 𝑥 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑥) ∈ 𝑆)) |
29 | 17, 28 | rspc2va 2830 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑥) ∈ 𝑆) |
30 | 6, 26, 15, 29 | syl21anc 1219 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑥) ∈ 𝑆) |
31 | | oveq2 5832 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (𝐶𝑇𝑧) = (𝐶𝑇𝑥)) |
32 | 31, 23 | fvmptg 5544 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑆 ∧ (𝐶𝑇𝑥) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥)) |
33 | 26, 30, 32 | syl2anc 409 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥)) |
34 | | simprr 522 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) |
35 | | oveq2 5832 |
. . . . . . . . 9
⊢ (𝑏 = 𝑦 → (𝐶𝑇𝑏) = (𝐶𝑇𝑦)) |
36 | 35 | eleq1d 2226 |
. . . . . . . 8
⊢ (𝑏 = 𝑦 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑦) ∈ 𝑆)) |
37 | 17, 36 | rspc2va 2830 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑦) ∈ 𝑆) |
38 | 6, 34, 15, 37 | syl21anc 1219 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑦) ∈ 𝑆) |
39 | | oveq2 5832 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → (𝐶𝑇𝑧) = (𝐶𝑇𝑦)) |
40 | 39, 23 | fvmptg 5544 |
. . . . . 6
⊢ ((𝑦 ∈ 𝑆 ∧ (𝐶𝑇𝑦) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦)) |
41 | 34, 38, 40 | syl2anc 409 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦)) |
42 | 33, 41 | oveq12d 5842 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) |
43 | 4, 25, 42 | 3eqtr4d 2200 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦))) |
44 | 5 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐶 ∈ 𝑆) |
45 | 14 | adantr 274 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
46 | | oveq2 5832 |
. . . . . . . 8
⊢ (𝑏 = (𝐺‘𝑥) → (𝐶𝑇𝑏) = (𝐶𝑇(𝐺‘𝑥))) |
47 | 46 | eleq1d 2226 |
. . . . . . 7
⊢ (𝑏 = (𝐺‘𝑥) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆)) |
48 | 17, 47 | rspc2va 2830 |
. . . . . 6
⊢ (((𝐶 ∈ 𝑆 ∧ (𝐺‘𝑥) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) |
49 | 44, 2, 45, 48 | syl21anc 1219 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) |
50 | | oveq2 5832 |
. . . . . 6
⊢ (𝑧 = (𝐺‘𝑥) → (𝐶𝑇𝑧) = (𝐶𝑇(𝐺‘𝑥))) |
51 | 50, 23 | fvmptg 5544 |
. . . . 5
⊢ (((𝐺‘𝑥) ∈ 𝑆 ∧ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥))) |
52 | 2, 49, 51 | syl2anc 409 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥))) |
53 | | seq3distr.5 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥))) |
54 | 52, 53 | eqtr4d 2193 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐹‘𝑥)) |
55 | 53, 49 | eqeltrd 2234 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
56 | 1, 2, 3, 43, 54, 55, 1 | seq3homo 10409 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)) |
57 | | eqid 2157 |
. . . . 5
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) |
58 | | eluzel2 9444 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
59 | 3, 58 | syl 14 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
60 | 57, 59, 2, 1 | seqf 10360 |
. . . 4
⊢ (𝜑 → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶𝑆) |
61 | 60, 3 | ffvelrnd 5603 |
. . 3
⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆) |
62 | 7, 5, 61 | caovcld 5974 |
. . 3
⊢ (𝜑 → (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆) |
63 | | oveq2 5832 |
. . . 4
⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑁) → (𝐶𝑇𝑧) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) |
64 | 63, 23 | fvmptg 5544 |
. . 3
⊢
(((seq𝑀( + , 𝐺)‘𝑁) ∈ 𝑆 ∧ (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) |
65 | 61, 62, 64 | syl2anc 409 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) |
66 | 56, 65 | eqtr3d 2192 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) |