Theorem nfsumw 15043

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 15044 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.)

Hypotheses

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

Assertion

Ref	Expression
nfsumw	⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑘

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

Proof of Theorem nfsumw

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

Step	Hyp	Ref	Expression
1		df-sum 15039	. 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 2976	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsumw.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2976	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3957	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2976	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
7		nfcv 2976	. . . . . . . 8 ⊢ Ⅎ𝑥 +
8	3	nfcri 2970	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
9		nfcv 2976	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
10		nfsumw.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
11	9, 10	nfcsbw 3906	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
12		nfcv 2976	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
13	8, 11, 12	nfif 4493	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
14	2, 13	nfmpt 5160	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
15	6, 7, 14	nfseq 13377	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
16		nfcv 2976	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
17		nfcv 2976	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
18	15, 16, 17	nfbr 5110	. . . . . 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	nfrex 3308	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
21		nfcv 2976	. . . . 5 ⊢ Ⅎ𝑥ℕ
22		nfcv 2976	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
23		nfcv 2976	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
24	22, 23, 3	nff1o 6610	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
25		nfcv 2976	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
26		nfcv 2976	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
27	26, 10	nfcsbw 3906	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
28	21, 27	nfmpt 5160	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
29	25, 7, 28	nfseq 13377	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
30	29, 6	nffv 6677	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
31	30	nfeq2 2994	. . . . . . 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 2342	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
34	21, 33	nfrex 3308	. . . 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 6315	. 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 2974	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 398   ∨ wo 843   = wceq 1536  ∃wex 1779   ∈ wcel 2113  Ⅎwnfc 2960  ∃wrex 3138  ⦋csb 3880   ⊆ wss 3933  ifcif 4464   class class class wbr 5063   ↦ cmpt 5143  ℩cio 6309  –1-1-onto→wf1o 6351  ‘cfv 6352  (class class class)co 7153  0cc0 10534  1c1 10535   + caddc 10537  ℕcn 11635  ℤcz 11979  ℤ_≥cuz 12241  ...cfz 12890  seqcseq 13367   ⇝ cli 14837  Σcsu 15038

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7156  df-oprab 7157  df-mpo 7158  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-seq 13368  df-sum 15039

This theorem is referenced by:  fsum2dlem  15121  fsumcom2  15125  fsumrlim  15162  fsumiun  15172  fsumcn  23474  fsum2cn  23475  nfitg1  24370  nfitg  24371  dvmptfsum  24570  fsumdvdscom  25760  binomcxplemdvsum  40761  binomcxplemnotnn0  40762  fsumcnf  41352  fsumiunss  41930  dvmptfprod  42304  sge0iunmptlemre  42771