Step | Hyp | Ref
| Expression |
1 | | iseqdistr.1 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
2 | | iseqdistr.4 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) |
3 | | iseqdistr.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ 𝑉) |
4 | | iseqdistr.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | | iseqdistr.2 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) |
6 | | iseqdistr.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑆) |
7 | 6 | adantr 270 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ 𝑆) |
8 | | iseqdistr.t |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆) |
9 | 8 | ralrimivva 2443 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆) |
10 | | oveq1 5539 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝑥𝑇𝑦) = (𝑎𝑇𝑦)) |
11 | 10 | eleq1d 2147 |
. . . . . . . . 9
⊢ (𝑥 = 𝑎 → ((𝑥𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑦) ∈ 𝑆)) |
12 | | oveq2 5540 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑏 → (𝑎𝑇𝑦) = (𝑎𝑇𝑏)) |
13 | 12 | eleq1d 2147 |
. . . . . . . . 9
⊢ (𝑦 = 𝑏 → ((𝑎𝑇𝑦) ∈ 𝑆 ↔ (𝑎𝑇𝑏) ∈ 𝑆)) |
14 | 11, 13 | cbvral2v 2585 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑆 ↔ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
15 | 9, 14 | sylib 120 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
16 | 15 | adantr 270 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) |
17 | | oveq1 5539 |
. . . . . . . 8
⊢ (𝑎 = 𝐶 → (𝑎𝑇𝑏) = (𝐶𝑇𝑏)) |
18 | 17 | eleq1d 2147 |
. . . . . . 7
⊢ (𝑎 = 𝐶 → ((𝑎𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑏) ∈ 𝑆)) |
19 | | oveq2 5540 |
. . . . . . . 8
⊢ (𝑏 = (𝑥 + 𝑦) → (𝐶𝑇𝑏) = (𝐶𝑇(𝑥 + 𝑦))) |
20 | 19 | eleq1d 2147 |
. . . . . . 7
⊢ (𝑏 = (𝑥 + 𝑦) → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆)) |
21 | 18, 20 | rspc2va 2714 |
. . . . . 6
⊢ (((𝐶 ∈ 𝑆 ∧ (𝑥 + 𝑦) ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) |
22 | 7, 1, 16, 21 | syl21anc 1168 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) |
23 | | oveq2 5540 |
. . . . . 6
⊢ (𝑧 = (𝑥 + 𝑦) → (𝐶𝑇𝑧) = (𝐶𝑇(𝑥 + 𝑦))) |
24 | | eqid 2081 |
. . . . . 6
⊢ (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧)) |
25 | 23, 24 | fvmptg 5269 |
. . . . 5
⊢ (((𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐶𝑇(𝑥 + 𝑦)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦))) |
26 | 1, 22, 25 | syl2anc 403 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (𝐶𝑇(𝑥 + 𝑦))) |
27 | | simprl 497 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) |
28 | | oveq2 5540 |
. . . . . . . . 9
⊢ (𝑏 = 𝑥 → (𝐶𝑇𝑏) = (𝐶𝑇𝑥)) |
29 | 28 | eleq1d 2147 |
. . . . . . . 8
⊢ (𝑏 = 𝑥 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑥) ∈ 𝑆)) |
30 | 18, 29 | rspc2va 2714 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑥) ∈ 𝑆) |
31 | 7, 27, 16, 30 | syl21anc 1168 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑥) ∈ 𝑆) |
32 | | oveq2 5540 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (𝐶𝑇𝑧) = (𝐶𝑇𝑥)) |
33 | 32, 24 | fvmptg 5269 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑆 ∧ (𝐶𝑇𝑥) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥)) |
34 | 27, 31, 33 | syl2anc 403 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) = (𝐶𝑇𝑥)) |
35 | | simprr 498 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) |
36 | | oveq2 5540 |
. . . . . . . . 9
⊢ (𝑏 = 𝑦 → (𝐶𝑇𝑏) = (𝐶𝑇𝑦)) |
37 | 36 | eleq1d 2147 |
. . . . . . . 8
⊢ (𝑏 = 𝑦 → ((𝐶𝑇𝑏) ∈ 𝑆 ↔ (𝐶𝑇𝑦) ∈ 𝑆)) |
38 | 18, 37 | rspc2va 2714 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎𝑇𝑏) ∈ 𝑆) → (𝐶𝑇𝑦) ∈ 𝑆) |
39 | 7, 35, 16, 38 | syl21anc 1168 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇𝑦) ∈ 𝑆) |
40 | | oveq2 5540 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → (𝐶𝑇𝑧) = (𝐶𝑇𝑦)) |
41 | 40, 24 | fvmptg 5269 |
. . . . . 6
⊢ ((𝑦 ∈ 𝑆 ∧ (𝐶𝑇𝑦) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦)) |
42 | 35, 39, 41 | syl2anc 403 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦) = (𝐶𝑇𝑦)) |
43 | 34, 42 | oveq12d 5550 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) |
44 | 5, 26, 43 | 3eqtr4d 2123 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝑥 + 𝑦)) = (((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑥) + ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘𝑦))) |
45 | | iseqdistr.5 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥))) |
46 | | iseqdistr.f |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
47 | 45, 46 | eqeltrrd 2156 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) |
48 | | oveq2 5540 |
. . . . . 6
⊢ (𝑧 = (𝐺‘𝑥) → (𝐶𝑇𝑧) = (𝐶𝑇(𝐺‘𝑥))) |
49 | 48, 24 | fvmptg 5269 |
. . . . 5
⊢ (((𝐺‘𝑥) ∈ 𝑆 ∧ (𝐶𝑇(𝐺‘𝑥)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥))) |
50 | 2, 47, 49 | syl2anc 403 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐶𝑇(𝐺‘𝑥))) |
51 | 50, 45 | eqtr4d 2116 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(𝐺‘𝑥)) = (𝐹‘𝑥)) |
52 | 1, 2, 3, 4, 44, 51, 46, 1 | iseqhomo 9468 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺, 𝑆)‘𝑁)) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁)) |
53 | 4, 3, 2, 1 | iseqcl 9443 |
. . 3
⊢ (𝜑 → (seq𝑀( + , 𝐺, 𝑆)‘𝑁) ∈ 𝑆) |
54 | 8, 6, 53 | caovcld 5674 |
. . 3
⊢ (𝜑 → (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)) ∈ 𝑆) |
55 | | oveq2 5540 |
. . . 4
⊢ (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑁) → (𝐶𝑇𝑧) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |
56 | 55, 24 | fvmptg 5269 |
. . 3
⊢
(((seq𝑀( + , 𝐺, 𝑆)‘𝑁) ∈ 𝑆 ∧ (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)) ∈ 𝑆) → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺, 𝑆)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |
57 | 53, 54, 56 | syl2anc 403 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑆 ↦ (𝐶𝑇𝑧))‘(seq𝑀( + , 𝐺, 𝑆)‘𝑁)) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |
58 | 52, 57 | eqtr3d 2115 |
1
⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁))) |