Theorem nfsum1 11297

Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypothesis

Ref	Expression
nfsum1.1	⊢ Ⅎ𝑘𝐴

Assertion

Ref	Expression
nfsum1	⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Proof of Theorem nfsum1

Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sumdc 11295	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
2		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfsum1.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
4		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3135	. . . . . 6 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2302	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
7	6	nfdc 1647	. . . . . . 7 ⊢ Ⅎ𝑘DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2504	. . . . . 6 ⊢ Ⅎ𝑘∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘𝑚
10		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘 +
11	3	nfcri 2302	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
12		nfcsb1v 3078	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
13		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑘0
14	11, 12, 13	nfif 3548	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
15	2, 14	nfmpt 4074	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
16	9, 10, 15	nfseq 10390	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
17		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑘 ⇝
18		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
19	16, 17, 18	nfbr 4028	. . . . . 6 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
20	5, 8, 19	nf3an 1554	. . . . 5 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
21	2, 20	nfrexya 2507	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
22		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑘ℕ
23		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
24		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
25	23, 24, 3	nff1o 5430	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
26		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
27		nfv 1516	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ≤ 𝑚
28		nfcsb1v 3078	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
29	27, 28, 13	nfif 3548	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
30	22, 29	nfmpt 4074	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
31	26, 10, 30	nfseq 10390	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))
32	31, 9	nffv 5496	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
33	32	nfeq2 2320	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
34	25, 33	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑘(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
35	34	nfex 1625	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
36	22, 35	nfrexya 2507	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
37	21, 36	nfor 1562	. . 3 ⊢ Ⅎ𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)))
38	37	nfiotaw 5157	. 2 ⊢ Ⅎ𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
39	1, 38	nfcxfr 2305	1 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 103   ∨ wo 698  DECID wdc 824   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Ⅎwnfc 2295  ∀wral 2444  ∃wrex 2445  ⦋csb 3045   ⊆ wss 3116  ifcif 3520   class class class wbr 3982   ↦ cmpt 4043  ℩cio 5151  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842  0cc0 7753  1c1 7754   + caddc 7756   ≤ cle 7934  ℕcn 8857  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  seqcseq 10380   ⇝ cli 11219  Σcsu 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-seqfrec 10381  df-sumdc 11295

This theorem is referenced by:  mertenslem2  11477