Step | Hyp | Ref
| Expression |
1 | | addcl 7888 |
. . . 4
⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ) |
2 | 1 | adantl 275 |
. . 3
⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ) |
3 | | addcom 8045 |
. . . 4
⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚)) |
4 | 3 | adantl 275 |
. . 3
⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚)) |
5 | | addass 7893 |
. . . 4
⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦))) |
6 | 5 | adantl 275 |
. . 3
⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦))) |
7 | | isummolem3.5 |
. . . . 5
⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
8 | 7 | simpld 111 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℕ) |
9 | | nnuz 9511 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
10 | 8, 9 | eleqtrdi 2263 |
. . 3
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
11 | | isummolem3.6 |
. . . . . 6
⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴) |
12 | | f1ocnv 5453 |
. . . . . 6
⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(1...𝑀)) |
13 | 11, 12 | syl 14 |
. . . . 5
⊢ (𝜑 → ◡𝑓:𝐴–1-1-onto→(1...𝑀)) |
14 | | isummolem3.7 |
. . . . 5
⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴) |
15 | | f1oco 5463 |
. . . . 5
⊢ ((◡𝑓:𝐴–1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto→𝐴) → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)) |
16 | 13, 14, 15 | syl2anc 409 |
. . . 4
⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)) |
17 | 7, 11, 14 | nnf1o 11328 |
. . . . . . 7
⊢ (𝜑 → 𝑁 = 𝑀) |
18 | 17 | eqcomd 2176 |
. . . . . 6
⊢ (𝜑 → 𝑀 = 𝑁) |
19 | 18 | oveq2d 5867 |
. . . . 5
⊢ (𝜑 → (1...𝑀) = (1...𝑁)) |
20 | | f1oeq2 5430 |
. . . . 5
⊢
((1...𝑀) =
(1...𝑁) → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))) |
21 | 19, 20 | syl 14 |
. . . 4
⊢ (𝜑 → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))) |
22 | 16, 21 | mpbird 166 |
. . 3
⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀)) |
23 | | elnnuz 9512 |
. . . 4
⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈
(ℤ≥‘1)) |
24 | | isummolem3.g |
. . . . . . 7
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) |
25 | | breq1 3990 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑀 ↔ 𝑚 ≤ 𝑀)) |
26 | | fveq2 5494 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚)) |
27 | 26 | csbeq1d 3056 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵) |
28 | 25, 27 | ifbieq1d 3547 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0)) |
29 | | simplr 525 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ) |
30 | | elfzle2 9973 |
. . . . . . . . . 10
⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ≤ 𝑀) |
31 | 30 | adantl 275 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ≤ 𝑀) |
32 | 31 | iftrued 3532 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵) |
33 | | f1of 5440 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴) |
34 | 11, 33 | syl 14 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴) |
35 | 34 | ffvelrnda 5629 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴) |
36 | | isummo.2 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
37 | 36 | ralrimiva 2543 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
38 | 37 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
39 | | nfcsb1v 3082 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 |
40 | 39 | nfel1 2323 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ |
41 | | csbeq1a 3058 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵) |
42 | 41 | eleq1d 2239 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
43 | 40, 42 | rspc 2828 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
44 | 35, 38, 43 | sylc 62 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ) |
45 | 44 | adantlr 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ) |
46 | 32, 45 | eqeltrd 2247 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) |
47 | 24, 28, 29, 46 | fvmptd3 5587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0)) |
48 | 47, 46 | eqeltrd 2247 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ) |
49 | | simplr 525 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈
ℕ) |
50 | 8 | ad2antrr 485 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℕ) |
51 | 50 | nnzd 9322 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ) |
52 | | eluzp1l 9500 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚) |
53 | 51, 52 | sylancom 418 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚) |
54 | 49 | nnzd 9322 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈
ℤ) |
55 | | zltnle 9247 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀)) |
56 | 51, 54, 55 | syl2anc 409 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀)) |
57 | 53, 56 | mpbid 146 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑀) |
58 | 57 | iffalsed 3535 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = 0) |
59 | | 0cn 7901 |
. . . . . . . 8
⊢ 0 ∈
ℂ |
60 | 58, 59 | eqeltrdi 2261 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) |
61 | 24, 28, 49, 60 | fvmptd3 5587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0)) |
62 | 61, 60 | eqeltrd 2247 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) ∈ ℂ) |
63 | | nnsplit 10082 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ → ℕ =
((1...𝑀) ∪
(ℤ≥‘(𝑀 + 1)))) |
64 | 8, 63 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → ℕ = ((1...𝑀) ∪
(ℤ≥‘(𝑀 + 1)))) |
65 | 64 | eleq2d 2240 |
. . . . . . 7
⊢ (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))) |
66 | 65 | biimpa 294 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1)))) |
67 | | elun 3268 |
. . . . . 6
⊢ (𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1)))) |
68 | 66, 67 | sylib 121 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1)))) |
69 | 48, 62, 68 | mpjaodan 793 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℂ) |
70 | 23, 69 | sylan2br 286 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ (𝐺‘𝑚) ∈
ℂ) |
71 | 17 | oveq2d 5867 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝑁) = (1...𝑀)) |
72 | 71 | eleq2d 2240 |
. . . . . . . 8
⊢ (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀))) |
73 | 72 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀))) |
74 | 73 | pm5.32i 451 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀))) |
75 | | isummolem3.4 |
. . . . . . . 8
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0)) |
76 | | breq1 3990 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁)) |
77 | | fveq2 5494 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚)) |
78 | 77 | csbeq1d 3056 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵) |
79 | 76, 78 | ifbieq1d 3547 |
. . . . . . . 8
⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0)) |
80 | | simplr 525 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ) |
81 | | elfzle2 9973 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ≤ 𝑁) |
82 | 81 | adantl 275 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ≤ 𝑁) |
83 | 82 | iftrued 3532 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵) |
84 | | f1of 5440 |
. . . . . . . . . . . . 13
⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴) |
85 | 14, 84 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴) |
86 | 85 | ffvelrnda 5629 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝐾‘𝑚) ∈ 𝐴) |
87 | 37 | adantr 274 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
88 | | nfcsb1v 3082 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 |
89 | 88 | nfel1 2323 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ |
90 | | csbeq1a 3058 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵) |
91 | 90 | eleq1d 2239 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
92 | 89, 91 | rspc 2828 |
. . . . . . . . . . 11
⊢ ((𝐾‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
93 | 86, 87, 92 | sylc 62 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ) |
94 | 93 | adantlr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ) |
95 | 83, 94 | eqeltrd 2247 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) |
96 | 75, 79, 80, 95 | fvmptd3 5587 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0)) |
97 | 96, 95 | eqeltrd 2247 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) ∈ ℂ) |
98 | 74, 97 | sylbir 134 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻‘𝑚) ∈ ℂ) |
99 | 17 | breq2d 3999 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑚 ≤ 𝑁 ↔ 𝑚 ≤ 𝑀)) |
100 | 99 | notbid 662 |
. . . . . . . . . . 11
⊢ (𝜑 → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀)) |
101 | 100 | ad2antrr 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀)) |
102 | 57, 101 | mpbird 166 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑁) |
103 | 102 | iffalsed 3535 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = 0) |
104 | 103, 59 | eqeltrdi 2261 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) |
105 | 75, 79, 49, 104 | fvmptd3 5587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0)) |
106 | 105, 104 | eqeltrd 2247 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) ∈ ℂ) |
107 | 98, 106, 68 | mpjaodan 793 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℂ) |
108 | 23, 107 | sylan2br 286 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ (𝐻‘𝑚) ∈
ℂ) |
109 | | f1oeq2 5430 |
. . . . . . . . . . 11
⊢
((1...𝑀) =
(1...𝑁) → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴)) |
110 | 19, 109 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴)) |
111 | 14, 110 | mpbird 166 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴) |
112 | | f1of 5440 |
. . . . . . . . 9
⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴) |
113 | 111, 112 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴) |
114 | | fvco3 5565 |
. . . . . . . 8
⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖))) |
115 | 113, 114 | sylan 281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖))) |
116 | 115 | fveq2d 5498 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(◡𝑓‘(𝐾‘𝑖)))) |
117 | 11 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴) |
118 | 113 | ffvelrnda 5629 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴) |
119 | | f1ocnvfv2 5755 |
. . . . . . 7
⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖)) |
120 | 117, 118,
119 | syl2anc 409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖)) |
121 | 116, 120 | eqtr2d 2204 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖))) |
122 | 121 | csbeq1d 3056 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) |
123 | | elfznn 9999 |
. . . . . 6
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ) |
124 | | elfzle2 9973 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀) |
125 | 124 | adantl 275 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀) |
126 | 18 | breq2d 3999 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁)) |
127 | 126 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁)) |
128 | 125, 127 | mpbid 146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑁) |
129 | 128 | iftrued 3532 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵) |
130 | 37 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
131 | | nfcsb1v 3082 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 |
132 | 131 | nfel1 2323 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ |
133 | | csbeq1a 3058 |
. . . . . . . . . 10
⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵) |
134 | 133 | eleq1d 2239 |
. . . . . . . . 9
⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ)) |
135 | 132, 134 | rspc 2828 |
. . . . . . . 8
⊢ ((𝐾‘𝑖) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ)) |
136 | 118, 130,
135 | sylc 62 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ) |
137 | 129, 136 | eqeltrd 2247 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ) |
138 | | breq1 3990 |
. . . . . . . 8
⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) |
139 | | fveq2 5494 |
. . . . . . . . 9
⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖)) |
140 | 139 | csbeq1d 3056 |
. . . . . . . 8
⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵) |
141 | 138, 140 | ifbieq1d 3547 |
. . . . . . 7
⊢ (𝑛 = 𝑖 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0)) |
142 | 141, 75 | fvmptg 5570 |
. . . . . 6
⊢ ((𝑖 ∈ ℕ ∧ if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0)) |
143 | 123, 137,
142 | syl2an2 589 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0)) |
144 | 143, 129 | eqtrd 2203 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵) |
145 | | breq1 3990 |
. . . . . . 7
⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑛 ≤ 𝑀 ↔ ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)) |
146 | | fveq2 5494 |
. . . . . . . 8
⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖))) |
147 | 146 | csbeq1d 3056 |
. . . . . . 7
⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) |
148 | 145, 147 | ifbieq1d 3547 |
. . . . . 6
⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0)) |
149 | | f1of 5440 |
. . . . . . . . 9
⊢ ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀)) |
150 | 22, 149 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀)) |
151 | 150 | ffvelrnda 5629 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀)) |
152 | | elfznn 9999 |
. . . . . . 7
⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ) |
153 | 151, 152 | syl 14 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ) |
154 | | elfzle2 9973 |
. . . . . . . . . 10
⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀) |
155 | 151, 154 | syl 14 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀) |
156 | 155 | iftrued 3532 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) |
157 | 156, 122 | eqtr4d 2206 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵) |
158 | 157, 136 | eqeltrd 2247 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ) |
159 | 24, 148, 153, 158 | fvmptd3 5587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0)) |
160 | 159, 156 | eqtrd 2203 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) |
161 | 122, 144,
160 | 3eqtr4d 2213 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖))) |
162 | 2, 4, 6, 10, 22, 70, 108, 161 | seq3f1o 10449 |
. 2
⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀)) |
163 | 18 | fveq2d 5498 |
. 2
⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁)) |
164 | 162, 163 | eqtr3d 2205 |
1
⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁)) |