Theorem nfsum 9876

Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

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

Assertion

Ref	Expression
nfsum	⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

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 9873	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚))))
2		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsum.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2178	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 2938	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
7		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥 +
8	3	nfcri 2172	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
9		nfcv 2178	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
10		nfsum.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
11	9, 10	nfcsb 2884	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
12		nfcv 2178	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
13	8, 11, 12	nfif 3356	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
14	2, 13	nfmpt 3849	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
15		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥ℂ
16	6, 7, 14, 15	nfiseq 9218	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ)
17		nfcv 2178	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
18		nfcv 2178	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
19	16, 17, 18	nfbr 3808	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧
20	5, 19	nfan 1457	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧)
21	2, 20	nfrexya 2363	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧)
22		nfcv 2178	. . . . 5 ⊢ Ⅎ𝑥ℕ
23		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
24		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
25	23, 24, 3	nff1o 5124	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
26		nfcv 2178	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
27		nfcv 2178	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
28	27, 10	nfcsb 2884	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
29	22, 28	nfmpt 3849	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
30	26, 7, 29, 15	nfiseq 9218	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)
31	30, 6	nffv 5185	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚)
32	31	nfeq2 2189	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚)
33	25, 32	nfan 1457	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚))
34	33	nfex 1528	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚))
35	22, 34	nfrexya 2363	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚))
36	21, 35	nfor 1466	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚)))
37	36	nfiotaxy 4871	. 2 ⊢ Ⅎ𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵), ℂ)‘𝑚))))
38	1, 37	nfcxfr 2175	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 97   ∨ wo 629   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  ∃wrex 2307  ⦋csb 2852   ⊆ wss 2917  ifcif 3331   class class class wbr 3764   ↦ cmpt 3818  ℩cio 4865  –1-1-onto→wf1o 4901  ‘cfv 4902  (class class class)co 5512  ℂcc 6887  0cc0 6889  1c1 6890   + caddc 6892  ℕcn 7914  ℤcz 8245  ℤ_≥cuz 8473  ...cfz 8874  seqcseq 9211   ⇝ cli 9799  Σcsu 9872

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212  df-sum 9873

This theorem is referenced by: (None)