Theorem nfsum1 15663

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 15660	. 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 2899	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfsum1.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
4		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3971	. . . . . 6 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑘𝑚
7		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑘 +
8	3	nfcri 2886	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
9		nfcsb1v 3915	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
10		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑘0
11	8, 9, 10	nfif 4555	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
12	2, 11	nfmpt 5250	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
13	6, 7, 12	nfseq 14003	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
14		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘 ⇝
15		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
16	13, 14, 15	nfbr 5190	. . . . . 6 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
17	5, 16	nfan 1895	. . . . 5 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
18	2, 17	nfrexw 3306	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
19		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑘ℕ
20		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
21		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
22	20, 21, 3	nff1o 6832	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
23		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
24		nfcsb1v 3915	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
25	19, 24	nfmpt 5250	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
26	23, 7, 25	nfseq 14003	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
27	26, 6	nffv 6902	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
28	27	nfeq2 2916	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
29	22, 28	nfan 1895	. . . . . 6 ⊢ Ⅎ𝑘(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
30	29	nfex 2313	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
31	19, 30	nfrexw 3306	. . . 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 1900	. . 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 6499	. 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 2897	1 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 846   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Ⅎwnfc 2879  ∃wrex 3066  ⦋csb 3890   ⊆ wss 3945  ifcif 4525   class class class wbr 5143   ↦ cmpt 5226  ℩cio 6493  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7415  0cc0 11133  1c1 11134   + caddc 11136  ℕcn 12237  ℤcz 12583  ℤ_≥cuz 12847  ...cfz 13511  seqcseq 13993   ⇝ cli 15455  Σcsu 15659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-seq 13994  df-sum 15660

This theorem is referenced by:  dvmptfprod  45324  dvnprodlem1  45325  fourierdlem112  45597  etransclem32  45645  sge0reuz  45826