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Theorem nfsum 10107
 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 𝑥𝐴
nfsum.2 𝑥𝐵
Assertion
Ref Expression
nfsum 𝑥Σ𝑘𝐴 𝐵

Proof of Theorem nfsum
Dummy variables 𝑓 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 10104 . 2 Σ𝑘𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
2 nfcv 2194 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2194 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 2966 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2194 . . . . . . . 8 𝑥𝑚
7 nfcv 2194 . . . . . . . 8 𝑥 +
83nfcri 2188 . . . . . . . . . 10 𝑥 𝑛𝐴
9 nfcv 2194 . . . . . . . . . . 11 𝑥𝑛
10 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
119, 10nfcsb 2912 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
12 nfcv 2194 . . . . . . . . . 10 𝑥0
138, 11, 12nfif 3384 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
142, 13nfmpt 3877 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
15 nfcv 2194 . . . . . . . 8 𝑥
166, 7, 14, 15nfiseq 9382 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ)
17 nfcv 2194 . . . . . . 7 𝑥
18 nfcv 2194 . . . . . . 7 𝑥𝑧
1916, 17, 18nfbr 3836 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧
205, 19nfan 1473 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧)
212, 20nfrexya 2380 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧)
22 nfcv 2194 . . . . 5 𝑥
23 nfcv 2194 . . . . . . . 8 𝑥𝑓
24 nfcv 2194 . . . . . . . 8 𝑥(1...𝑚)
2523, 24, 3nff1o 5152 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
26 nfcv 2194 . . . . . . . . . 10 𝑥1
27 nfcv 2194 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
2827, 10nfcsb 2912 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
2922, 28nfmpt 3877 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3026, 7, 29, 15nfiseq 9382 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)
3130, 6nffv 5213 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
3231nfeq2 2205 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
3325, 32nfan 1473 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3433nfex 1544 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3522, 34nfrexya 2380 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3621, 35nfor 1482 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)))
3736nfiotaxy 4899 . 2 𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
381, 37nfcxfr 2191 1 𝑥Σ𝑘𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ∨ wo 639   = wceq 1259  ∃wex 1397   ∈ wcel 1409  Ⅎwnfc 2181  ∃wrex 2324  ⦋csb 2880   ⊆ wss 2945  ifcif 3359   class class class wbr 3792   ↦ cmpt 3846  ℩cio 4893  –1-1-onto→wf1o 4929  ‘cfv 4930  (class class class)co 5540  ℂcc 6945  0cc0 6947  1c1 6948   + caddc 6950  ℕcn 7990  ℤcz 8302  ℤ≥cuz 8569  ...cfz 8976  seqcseq 9375   ⇝ cli 10030  Σcsu 10103 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-if 3360  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-recs 5951  df-frec 6009  df-iseq 9376  df-sum 10104 This theorem is referenced by: (None)
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