Theorem nfsum1 15382

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

Step	Hyp	Ref	Expression
1		df-sum 15379	. 2 ⊢ 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))))
2		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfsum1.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
4		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3917	. . . . . 6 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑘𝑚
7		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑘 +
8	3	nfcri 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
9		nfcsb1v 3861	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
10		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑘0
11	8, 9, 10	nfif 4494	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
12	2, 11	nfmpt 5185	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
13	6, 7, 12	nfseq 13712	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
14		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑘 ⇝
15		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
16	13, 14, 15	nfbr 5125	. . . . . 6 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
17	5, 16	nfan 1905	. . . . 5 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
18	2, 17	nfrex 3239	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
19		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑘ℕ
20		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
21		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
22	20, 21, 3	nff1o 6710	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
23		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
24		nfcsb1v 3861	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
25	19, 24	nfmpt 5185	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
26	23, 7, 25	nfseq 13712	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
27	26, 6	nffv 6778	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
28	27	nfeq2 2925	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
29	22, 28	nfan 1905	. . . . . 6 ⊢ Ⅎ𝑘(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
30	29	nfex 2321	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
31	19, 30	nfrex 3239	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
32	18, 31	nfor 1910	. . 3 ⊢ Ⅎ𝑘(∃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)))
33	32	nfiotaw 6392	. 2 ⊢ Ⅎ𝑘(℩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))))
34	1, 33	nfcxfr 2906	1 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 843   = wceq 1541  ∃wex 1785   ∈ wcel 2109  Ⅎwnfc 2888  ∃wrex 3066  ⦋csb 3836   ⊆ wss 3891  ifcif 4464   class class class wbr 5078   ↦ cmpt 5161  ℩cio 6386  –1-1-onto→wf1o 6429  ‘cfv 6430  (class class class)co 7268  0cc0 10855  1c1 10856   + caddc 10858  ℕcn 11956  ℤcz 12302  ℤ_≥cuz 12564  ...cfz 13221  seqcseq 13702   ⇝ cli 15174  Σcsu 15378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-seq 13703  df-sum 15379

This theorem is referenced by:  dvmptfprod  43440  dvnprodlem1  43441  fourierdlem112  43713  etransclem32  43761  sge0reuz  43939