Theorem nfsum1 15592

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 15589	. 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 2894	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfsum1.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
4		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3922	. . . . . 6 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑘𝑚
7		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑘 +
8	3	nfcri 2886	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
9		nfcsb1v 3869	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
10		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑘0
11	8, 9, 10	nfif 4501	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
12	2, 11	nfmpt 5184	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
13	6, 7, 12	nfseq 13913	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
14		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑘 ⇝
15		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
16	13, 14, 15	nfbr 5133	. . . . . 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	nfrexw 3280	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
19		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑘ℕ
20		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
21		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
22	20, 21, 3	nff1o 6756	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
23		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
24		nfcsb1v 3869	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
25	19, 24	nfmpt 5184	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
26	23, 7, 25	nfseq 13913	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
27	26, 6	nffv 6827	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
28	27	nfeq2 2912	. . . . . . 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 2325	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
31	19, 30	nfrexw 3280	. . . 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 6436	. 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 2892	1 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Ⅎwnfc 2879  ∃wrex 3056  ⦋csb 3845   ⊆ wss 3897  ifcif 4470   class class class wbr 5086   ↦ cmpt 5167  ℩cio 6430  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341  0cc0 11001  1c1 11002   + caddc 11004  ℕcn 12120  ℤcz 12463  ℤ_≥cuz 12727  ...cfz 13402  seqcseq 13903   ⇝ cli 15386  Σcsu 15588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-seq 13904  df-sum 15589

This theorem is referenced by:  dvmptfprod  45983  dvnprodlem1  45984  fourierdlem112  46256  etransclem32  46304  sge0reuz  46485