Detailed syntax breakdown of Definition df-sum
Step | Hyp | Ref
| Expression |
1 | | cA |
. . 3
class 𝐴 |
2 | | cB |
. . 3
class 𝐵 |
3 | | vk |
. . 3
setvar 𝑘 |
4 | 1, 2, 3 | csu 15406 |
. 2
class
Σ𝑘 ∈
𝐴 𝐵 |
5 | | vm |
. . . . . . . . 9
setvar 𝑚 |
6 | 5 | cv 1538 |
. . . . . . . 8
class 𝑚 |
7 | | cuz 12591 |
. . . . . . . 8
class
ℤ≥ |
8 | 6, 7 | cfv 6437 |
. . . . . . 7
class
(ℤ≥‘𝑚) |
9 | 1, 8 | wss 3888 |
. . . . . 6
wff 𝐴 ⊆
(ℤ≥‘𝑚) |
10 | | caddc 10883 |
. . . . . . . 8
class
+ |
11 | | vn |
. . . . . . . . 9
setvar 𝑛 |
12 | | cz 12328 |
. . . . . . . . 9
class
ℤ |
13 | 11 | cv 1538 |
. . . . . . . . . . 11
class 𝑛 |
14 | 13, 1 | wcel 2107 |
. . . . . . . . . 10
wff 𝑛 ∈ 𝐴 |
15 | 3, 13, 2 | csb 3833 |
. . . . . . . . . 10
class
⦋𝑛 /
𝑘⦌𝐵 |
16 | | cc0 10880 |
. . . . . . . . . 10
class
0 |
17 | 14, 15, 16 | cif 4460 |
. . . . . . . . 9
class if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) |
18 | 11, 12, 17 | cmpt 5158 |
. . . . . . . 8
class (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) |
19 | 10, 18, 6 | cseq 13730 |
. . . . . . 7
class seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) |
20 | | vx |
. . . . . . . 8
setvar 𝑥 |
21 | 20 | cv 1538 |
. . . . . . 7
class 𝑥 |
22 | | cli 15202 |
. . . . . . 7
class
⇝ |
23 | 19, 21, 22 | wbr 5075 |
. . . . . 6
wff seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 |
24 | 9, 23 | wa 396 |
. . . . 5
wff (𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) |
25 | 24, 5, 12 | wrex 3066 |
. . . 4
wff
∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) |
26 | | c1 10881 |
. . . . . . . . 9
class
1 |
27 | | cfz 13248 |
. . . . . . . . 9
class
... |
28 | 26, 6, 27 | co 7284 |
. . . . . . . 8
class
(1...𝑚) |
29 | | vf |
. . . . . . . . 9
setvar 𝑓 |
30 | 29 | cv 1538 |
. . . . . . . 8
class 𝑓 |
31 | 28, 1, 30 | wf1o 6436 |
. . . . . . 7
wff 𝑓:(1...𝑚)–1-1-onto→𝐴 |
32 | | cn 11982 |
. . . . . . . . . . 11
class
ℕ |
33 | 13, 30 | cfv 6437 |
. . . . . . . . . . . 12
class (𝑓‘𝑛) |
34 | 3, 33, 2 | csb 3833 |
. . . . . . . . . . 11
class
⦋(𝑓‘𝑛) / 𝑘⦌𝐵 |
35 | 11, 32, 34 | cmpt 5158 |
. . . . . . . . . 10
class (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵) |
36 | 10, 35, 26 | cseq 13730 |
. . . . . . . . 9
class seq1( + ,
(𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) |
37 | 6, 36 | cfv 6437 |
. . . . . . . 8
class (seq1( +
, (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) |
38 | 21, 37 | wceq 1539 |
. . . . . . 7
wff 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) |
39 | 31, 38 | wa 396 |
. . . . . 6
wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
40 | 39, 29 | wex 1782 |
. . . . 5
wff
∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
41 | 40, 5, 32 | wrex 3066 |
. . . 4
wff
∃𝑚 ∈
ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
42 | 25, 41 | wo 844 |
. . 3
wff
(∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) |
43 | 42, 20 | cio 6393 |
. 2
class
(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |
44 | 4, 43 | wceq 1539 |
1
wff
Σ𝑘 ∈
𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |