| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-sum 15723 | . 2
⊢
Σ𝑘 ∈
𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | 
| 2 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑥ℤ | 
| 3 |  | nfsum.1 | . . . . . . 7
⊢
Ⅎ𝑥𝐴 | 
| 4 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑥(ℤ≥‘𝑚) | 
| 5 | 3, 4 | nfss 3976 | . . . . . 6
⊢
Ⅎ𝑥 𝐴 ⊆
(ℤ≥‘𝑚) | 
| 6 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥𝑚 | 
| 7 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥
+ | 
| 8 | 3 | nfcri 2897 | . . . . . . . . . 10
⊢
Ⅎ𝑥 𝑛 ∈ 𝐴 | 
| 9 |  | nfcv 2905 | . . . . . . . . . . 11
⊢
Ⅎ𝑥𝑛 | 
| 10 |  | nfsum.2 | . . . . . . . . . . 11
⊢
Ⅎ𝑥𝐵 | 
| 11 | 9, 10 | nfcsbw 3925 | . . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵 | 
| 12 |  | nfcv 2905 | . . . . . . . . . 10
⊢
Ⅎ𝑥0 | 
| 13 | 8, 11, 12 | nfif 4556 | . . . . . . . . 9
⊢
Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) | 
| 14 | 2, 13 | nfmpt 5249 | . . . . . . . 8
⊢
Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) | 
| 15 | 6, 7, 14 | nfseq 14052 | . . . . . . 7
⊢
Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) | 
| 16 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑥
⇝ | 
| 17 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑥𝑧 | 
| 18 | 15, 16, 17 | nfbr 5190 | . . . . . 6
⊢
Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧 | 
| 19 | 5, 18 | nfan 1899 | . . . . 5
⊢
Ⅎ𝑥(𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) | 
| 20 | 2, 19 | nfrexw 3313 | . . . 4
⊢
Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) | 
| 21 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑥ℕ | 
| 22 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥𝑓 | 
| 23 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑥(1...𝑚) | 
| 24 | 22, 23, 3 | nff1o 6846 | . . . . . . 7
⊢
Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴 | 
| 25 |  | nfcv 2905 | . . . . . . . . . 10
⊢
Ⅎ𝑥1 | 
| 26 |  | nfcv 2905 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑓‘𝑛) | 
| 27 | 26, 10 | nfcsbw 3925 | . . . . . . . . . . 11
⊢
Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵 | 
| 28 | 21, 27 | nfmpt 5249 | . . . . . . . . . 10
⊢
Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) | 
| 29 | 25, 7, 28 | nfseq 14052 | . . . . . . . . 9
⊢
Ⅎ𝑥seq1(
+ , (𝑛 ∈ ℕ
↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) | 
| 30 | 29, 6 | nffv 6916 | . . . . . . . 8
⊢
Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) | 
| 31 | 30 | nfeq2 2923 | . . . . . . 7
⊢
Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) | 
| 32 | 24, 31 | nfan 1899 | . . . . . 6
⊢
Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) | 
| 33 | 32 | nfex 2324 | . . . . 5
⊢
Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) | 
| 34 | 21, 33 | nfrexw 3313 | . . . 4
⊢
Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) | 
| 35 | 20, 34 | nfor 1904 | . . 3
⊢
Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) | 
| 36 | 35 | nfiotaw 6518 | . 2
⊢
Ⅎ𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | 
| 37 | 1, 36 | nfcxfr 2903 | 1
⊢
Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵 |