Theorem nfsum1 15038

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 15035	. 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 2955	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfsum1.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
4		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3907	. . . . . 6 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘𝑚
7		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘 +
8	3	nfcri 2943	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
9		nfcsb1v 3852	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
10		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑘0
11	8, 9, 10	nfif 4454	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
12	2, 11	nfmpt 5127	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
13	6, 7, 12	nfseq 13374	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
14		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑘 ⇝
15		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
16	13, 14, 15	nfbr 5077	. . . . . 6 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥
17	5, 16	nfan 1900	. . . . 5 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
18	2, 17	nfrex 3268	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
19		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑘ℕ
20		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
21		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
22	20, 21, 3	nff1o 6588	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
23		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
24		nfcsb1v 3852	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
25	19, 24	nfmpt 5127	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
26	23, 7, 25	nfseq 13374	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
27	26, 6	nffv 6655	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
28	27	nfeq2 2972	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
29	22, 28	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑘(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
30	29	nfex 2332	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
31	19, 30	nfrex 3268	. . . 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 1905	. . 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 6287	. 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 2953	1 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Ⅎwnfc 2936  ∃wrex 3107  ⦋csb 3828   ⊆ wss 3881  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  ℩cio 6281  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529  ℕcn 11625  ℤcz 11969  ℤ_≥cuz 12231  ...cfz 12885  seqcseq 13364   ⇝ cli 14833  Σcsu 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seq 13365  df-sum 15035

This theorem is referenced by:  dvmptfprod  42587  dvnprodlem1  42588  fourierdlem112  42860  etransclem32  42908  sge0reuz  43086