Theorem nfsum 15616

Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. Version of nfsum 15616 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 11-Dec-2005.) (Revised by GG, 24-Feb-2024.)

Hypotheses

Ref	Expression
nfsum.1	⊢ Ⅎ𝑥𝐴
nfsum.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfsum	⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑘

Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)

Proof of Theorem nfsum

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

Step	Hyp	Ref	Expression
1		df-sum 15612	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsum.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3930	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
7		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥 +
8	3	nfcri 2883	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
9		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
10		nfsum.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
11	9, 10	nfcsbw 3879	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
12		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
13	8, 11, 12	nfif 4509	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
14	2, 13	nfmpt 5193	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
15	6, 7, 14	nfseq 13936	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
16		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
17		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
18	15, 16, 17	nfbr 5142	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧
19	5, 18	nfan 1899	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
20	2, 19	nfrexw 3278	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
21		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥ℕ
22		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
23		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
24	22, 23, 3	nff1o 6766	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
25		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
26		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
27	26, 10	nfcsbw 3879	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
28	21, 27	nfmpt 5193	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
29	25, 7, 28	nfseq 13936	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
30	29, 6	nffv 6836	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
31	30	nfeq2 2909	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
32	24, 31	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
33	32	nfex 2323	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
34	21, 33	nfrexw 3278	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
35	20, 34	nfor 1904	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
36	35	nfiotaw 6446	. 2 ⊢ Ⅎ𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
37	1, 36	nfcxfr 2889	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Ⅎwnfc 2876  ∃wrex 3053  ⦋csb 3853   ⊆ wss 3905  ifcif 4478   class class class wbr 5095   ↦ cmpt 5176  ℩cio 6440  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   + caddc 11031  ℕcn 12146  ℤcz 12489  ℤ_≥cuz 12753  ...cfz 13428  seqcseq 13926   ⇝ cli 15409  Σcsu 15611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seq 13927  df-sum 15612

This theorem is referenced by:  fsum2dlem  15695  fsumcom2  15699  fsumrlim  15736  fsumiun  15746  fsumcn  24777  fsum2cn  24778  nfitg1  25691  nfitg  25692  dvmptfsum  25895  fsumdvdscom  27111  binomcxplemdvsum  44328  binomcxplemnotnn0  44329  fsumcnf  44999  fsumiunss  45557  dvmptfprod  45927  sge0iunmptlemre  46397