Theorem nfsum 15739

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 15739 with a disjoint variable condition, which does not require ax-13 2380. (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 15735	. 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 2908	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsum.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 4001	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
7		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥 +
8	3	nfcri 2900	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
9		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
10		nfsum.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
11	9, 10	nfcsbw 3948	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
12		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
13	8, 11, 12	nfif 4578	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
14	2, 13	nfmpt 5273	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
15	6, 7, 14	nfseq 14062	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
16		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
17		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
18	15, 16, 17	nfbr 5213	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧
19	5, 18	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
20	2, 19	nfrexw 3319	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
21		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥ℕ
22		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
23		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
24	22, 23, 3	nff1o 6860	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
25		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
26		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
27	26, 10	nfcsbw 3948	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
28	21, 27	nfmpt 5273	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
29	25, 7, 28	nfseq 14062	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
30	29, 6	nffv 6930	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
31	30	nfeq2 2926	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
32	24, 31	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
33	32	nfex 2328	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
34	21, 33	nfrexw 3319	. . . 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 1903	. . 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 6529	. 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 2906	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Ⅎwnfc 2893  ∃wrex 3076  ⦋csb 3921   ⊆ wss 3976  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ℩cio 6523  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185   + caddc 11187  ℕcn 12293  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  seqcseq 14052   ⇝ cli 15530  Σcsu 15734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seq 14053  df-sum 15735

This theorem is referenced by:  fsum2dlem  15818  fsumcom2  15822  fsumrlim  15859  fsumiun  15869  fsumcn  24913  fsum2cn  24914  nfitg1  25829  nfitg  25830  dvmptfsum  26033  fsumdvdscom  27246  binomcxplemdvsum  44324  binomcxplemnotnn0  44325  fsumcnf  44921  fsumiunss  45496  dvmptfprod  45866  sge0iunmptlemre  46336